Introduction
Consider an instructor tasked with grading essays. She typically first inspects a sample of the essays to inform her grading policy: some may show excellent writing while others have an impressive vocabulary, etc. These comparable attributes between essays are called features . From this small sample, she can draw up a rubric, namely the detailed features she will use to grade with limited bias . As she grades, she is likely to realize that the original sample that first calibrated her grading policy was too small to contain every important dimension: as the grading goes on, she realizes that grammar is also distinguishing and important, so she needs to go back to the first essays to revise their grading.
This is clearly inefficient, and the instructor would hope to use AI tools to assist in this process of discriminative feature extraction, namely the extraction of the relevant features of the data on which she can base her decisions. If she had essays from previous years together with the grade they obtained, she could train a machine learning model to predict the essays’ grades from their embeddings , which would serve as features for grading. This would automate the grading, but the features used would be opaque and uninterpretable, which is incompatible with transparent decision-making: a crucial aspect in such applications. Current techniques cannot automate the interpretable rubric extraction step. A crucial challenge for decision-makers is therefore: given an unstructured set of items, discover interpretable and discriminative features for this dataset , along with per-item per-feature scores, that they can reliably base their decisions on. Additionally, when decision-making is delegated to an LLM, the same features should make the LLM’s decisions auditable.
A natural approach is to give an LLM the whole set of items and ask it directly for the features that distinguish them (a grading rubric, in the case of essays). However, processing an entire dataset requires an extremely long context, and LLM performance typically degrades as input length increases [1 , 2 ] . Consistent with this, we find that direct LLM extraction misses features that are present in the data and conflates others (§ 4 ). The difficulty is therefore one of context management : how to use LLMs on the set of items so that they can reliably identify the features that matter. We introduce an end-to-end approach that, given an unstructured dataset, discovers human-interpretable, dataset-relevant features using LLMs.
Contributions. Our idea is that the relevance of a feature can be revealed by a set of pairwise comparisons of items, and that LLM judges can articulate these comparisons in human language. We call such an articulated comparison a rationale : which features the judge cites, and which item it favors on each. The key is that, with enough comparisons, a statistical model can reverse-engineer the judge’s rationales and recover which features distinguish the items and how each item scores along each feature.
We treat the rationales as data. A statistical choice model (§ 3 ) describes each rationale as a two-stage outcome: whether the LLM cites a feature, and, when it does, which item wins on it. The model absorbs the LLM’s potential biases, such as a preference for the first or second position in the prompt. Fitting it yields per-item, per-feature scores with confidence intervals, together with diagnostics that filter out features with unreliably estimated scores. Estimation is non-trivial: the likelihood is non-convex and the number of parameters grows with the number of items. We nonetheless prove, under mild assumptions, that the likelihood admits a local maximizer that is consistent for every item score at once in the regime where the number of items grows (Theorems 1 and 2).
We validate our approach extensively in three settings. First, we benchmark it on a synthetic hiring dataset of cover letters with known ground-truth features, where our method correctly recovers all planted features, and most discovered features track a single planted one, whereas methods that prompt an LLM with the whole dataset/a sample of it miss some features and blur several together (§ 4 ). We then apply it to a real dataset of online debates (the ChangeMyView subreddit) to discover and quantify which argumentation techniques are the most effective. We find that our method reaches an accuracy comparable to the state of the art while providing interpretable features (§ 5 ) along with insights on argumentation techniques. Finally, we show in Appendix § 9 how an LLM’s rationales can also be used to audit its decisions and represent, in interpretable terms, what actually mattered in its decision when comparing two items, yielding interpretability insights for LLM-based scoring.
Our approach provides a concrete procedure that can be supplied as instructions to an LLM when tasked with feature discovery. It offers a principled way to organize multiple LLM agents for this purpose: a lead agent delegates pairwise comparisons along a graph to parallel sub-agents and aggregates their rationales through a statistical model. This aligns with recent efforts to scale AI performance through the orchestration of multiple agents [3 ] , where how the work is split and recombined is critical to performance [4 ] .
Related work
The LLM-as-a-Judge paradigm The LLM-as-a-Judge paradigm is an approach to several problems that uses LLMs to evaluate or rank items where one could have solicited a human judge. Examples include model benchmarking: in their foundational work, Zheng et al. [5 ] asked both LLMs and humans which of two AI-generated answers to a prompt they preferred, and found substantial agreement between the two, suggesting that LLMs could replace humans for comparing items. Nowadays, LLMs are used as judges in a variety of domains, spanning information retrieval [6 ] , recommendation systems [7 ] , alignment training [8 ] , automated essay scoring [9 ] , etc. LLM biases. Nonetheless, several works [5 , 10 , 11 ] document biases in LLMs, such as a positional bias (when asked to compare two items, the LLM prefers the item that appears first/second in the prompt for no particular reason), a verbosity bias (a tendency to prefer more verbose items, such as longer essays if the items being assessed are essays), etc. These phenomena can severely undermine LLM-based decision-making if not accounted for.
Feature evaluation with LLMs. Closer to our work, general LLM-based approaches typically work from a predetermined pool of features: the features are scored, but never discovered from the data . Evaluation features are either fixed in the prompt [5 , 12 ] , drawn from a predefined skill taxonomy [13 ] , or supplied by the user as a scoring rubric [14 ] . These approaches work by asking the judge to output a few tokens per feature: either a pointwise score [12 , 14 ] or the verdict of a pairwise comparison [5 ] (Liusie et al. 2024) (“A” if item A is better, “B” otherwise). The judge’s rationales are often discarded or used as a qualitative confirmation, while we argue that they could be treated as data useful for feature discovery. Some approaches [16 ] leverage them to create scoring guidelines to explain existing scores and preference labels, but to our knowledge no approach uses them for interpretable feature extraction.
Method
Our method proceeds in four steps, illustrated in Figure 1 . First, LLMs compare pairs of items along a comparison graph, describing the features on which they differ and indicating which item shows more of each feature. These local comparisons require only a small context but are myopic, missing dataset-level information. Second, we group these descriptions into a shared set of features. Third, we fit a statistical choice model that aggregates these local observations into dataset-level, per-item feature scores, using both the cited differences and the features left unmentioned. Finally, we derive per-feature identifiability and reliability diagnostics that flag features whose scores are not pinned down by the observations.
Figure 1. Overview of the pipeline described in § 3 . Items are first compared in pairs by the LLM, cited features are then clustered, and a statistical model learns the relevant features from the LLM’s choices. Finally, statistical diagnostics identify the reliable scores. The orange running example follows one pair of cover letters (the setup of § 4 ) through the pipeline.
Rationale collection. Let 1 , … , n 1,\ldots,n 1 , … , n denote the items. We compare them along a comparison graph , a set of N N N pairs p = ( i , j ) p=(i,j) p = ( i , j ) where item i i i is shown first. For each pair, an LLM judge is asked how the two items differ: it answers in free text, naming the features on which they differ and which item is stronger on each. These raw features, phrased differently across pairs, are then passed to a second LLM call that groups them into K K K shared features. The outcome is a sparse verdict tensor T ∈ { − 1 , 0 , + 1 } N × K T \in\{-1,0,+1\}^{N \times K} T ∈ { − 1 , 0 , + 1 } N × K : T p , k = 0 T_{p,k}=0 T p , k = 0 if feature k k k was not invoked for pair p p p , and T p , k = + 1 T_{p,k}=+1 T p , k = + 1 (resp. − 1 -1 − 1 ) if it was invoked and the first (resp. second) item of the pair was cited as stronger.
Statistical choice model. Given the verdicts T T T , our objective is to reverse-engineer the items’ scores along each feature and the judge’s biases. Our statistical model takes inspiration from Bradley–Terry models [17 ] to estimate latent scores from pairwise comparisons (see Appendix § 10 .§ 1 for related work).
We assume that each item has a latent score along each feature, collected in a score matrix S ∈ R n × K S \in\mathbb{R}^{n \times K} S ∈ R n × K . If pair p p p compares items i i i (shown first) and j j j , we write Δ p , k = S i , k − S j , k \Delta_{p,k}=S_{i,k}-S_{j,k} Δ p , k = S i , k − S j , k for the true difference of their scores along feature k k k . Our model posits that these score differences drive the LLM’s behavior in two ways: a) the more the items’ scores differ on k k k , the more likely the LLM is to cite feature k k k ; b) when feature k k k is cited, the item having the larger score along it is more likely to be chosen.
Importantly, we do not assume the LLM does any of this reliably: its citations and choices are noisy and may also carry the judge’s biases. We hence model the citation of feature k k k on pair p p p as a coin flip of probability σ ( f ( Δ p , k ) + γ k ) \sigma(f(\Delta_{p,k})+\gamma_k) σ ( f ( Δ p , k ) + γ k ) , where σ \sigma σ is the sigmoid function and f f f is an even function increasing in ∣ Δ ∣ |\Delta| ∣Δ∣ (we use f = log ( 2 cosh ) f=\log(2\cosh) f = log ( 2 cosh ) , a smooth surrogate of the absolute value), so that the non-citation probability is
P ( T p , k = 0 ) = 1 − σ ( f ( Δ p , k ) + γ k ) . (1) \mathbb{P}(T_{p,k}=0)=1-\sigma(f(\Delta_{p,k})+\gamma_k). \tag*{(1)} P ( T p , k = 0 ) = 1 − σ ( f ( Δ p , k ) + γ k ) . (1)
Here γ k \gamma_k γ k is the baseline citation propensity of feature k k k , which accounts for a feature-specific eagerness or reluctance of the LLM to cite it. Under this model, silence is also informative: a non-citation suggests that the judge did not find the pair’s difference on that feature salient enough to cite it.
Our model also assumes that, conditioned on citing feature k k k , the LLM declares the first item the winner with probability σ ( Δ p , k + β ) \sigma(\Delta_{p,k}+\beta) σ ( Δ p , k + β ) , where β \beta β captures the positional bias widely reported in the literature [5 , 10 , 11 ] , which we also observe in our experiments. Multiplying the two gives:
P ( T p , k = 1 ) = σ ( Δ p , k + β ) σ ( f ( Δ p , k ) + γ k ) \mathbb{P}(T_{p,k}=1)=\sigma(\Delta_{p,k}+\beta)\,\sigma(f(\Delta_{p,k})+\gamma_k) P ( T p , k = 1 ) = σ ( Δ p , k + β ) σ ( f ( Δ p , k ) + γ k )
P ( T p , k = − 1 ) = ( 1 − σ ( Δ p , k + β ) ) σ ( f ( Δ p , k ) + γ k ) . (2) \mathbb{P}(T_{p,k}=-1)=(1-\sigma(\Delta_{p,k}+\beta))\,\sigma(f(\Delta_{p,k})+\gamma_k). \tag*{(2)} P ( T p , k = − 1 ) = ( 1 − σ ( Δ p , k + β )) σ ( f ( Δ p , k ) + γ k ) . (2)
Hence, β \beta β and γ k \gamma_k γ k capture the LLM’s biases (a preference for a position in the prompt, and feature-specific citation habits) so that they do not contaminate the estimated scores. The position coefficient β \beta β is shared across features; as we show in Appendix § 10 .§ 2 , performing comparisons with presentation order randomized makes the scores essentially insensitive to this choice, and a per-feature refit confirms it empirically.
Parameter estimation. We estimate θ : = ( vec ( S ) , ( γ k ) k = 1 K , β ) ∈ R n K + K + 1 \theta:= (\operatorname{vec}(S),(\gamma_k)_{k=1}^{K},\beta) \in\mathbb{R}^{nK+K+1} θ := ( vec ( S ) , ( γ k ) k = 1 K , β ) ∈ R n K + K + 1 from the verdict tensor T T T by penalized maximum likelihood,
θ ^ ∈ argmin θ { L ( θ ) + λ 2 ∥ S ∥ F 2 } , (3) \widehat{\theta}\in\underset{\theta}{\operatorname{argmin}}\left\{\mathcal{L}(\theta)+\frac{\lambda}{2}\lVert S\rVert_F^2\right\}, \tag*{(3)} θ ∈ θ argmin { L ( θ ) + 2 λ ∥ S ∥ F 2 } , (3)
where L \mathcal{L} L is the negative log-likelihood of T T T and λ ≥ 0 \lambda\ge0 λ ≥ 0 is a ridge parameter. Under conditional independence of the verdicts across pairs and features given θ \theta θ , L \mathcal{L} L factorizes as
L ( θ ) = − ∑ pairs p ∑ features k [ 1 { T p , k ≠ 0 } log σ ( f ( Δ p , k ) + γ k ) + 1 { T p , k = 0 } log ( 1 − σ ( f ( Δ p , k ) + γ k ) ) + 1 { T p , k = 1 } log σ ( Δ p , k + β ) + 1 { T p , k = − 1 } log ( 1 − σ ( Δ p , k + β ) ) ] . (4) \begin{aligned}
\mathcal{L}(\theta)={}&-\sum_{\text{pairs }p}\sum_{\text{features }k}\Bigl[1_{\{T_{p,k}\ne0\}}\log\sigma\bigl(f(\Delta_{p,k})+\gamma_k\bigr)+1_{\{T_{p,k}=0\}}\log\bigl(1-\sigma(f(\Delta_{p,k})+\gamma_k)\bigr)\\
&\qquad\qquad+1_{\{T_{p,k}=1\}}\log\sigma(\Delta_{p,k}+\beta)+1_{\{T_{p,k}=-1\}}\log\bigl(1-\sigma(\Delta_{p,k}+\beta)\bigr)\Bigr]. \tag*{(4)}
\end{aligned} L ( θ ) = − pairs p ∑ features k ∑ [ 1 { T p , k = 0 } log σ ( f ( Δ p , k ) + γ k ) + 1 { T p , k = 0 } log ( 1 − σ ( f ( Δ p , k ) + γ k ) ) + 1 { T p , k = 1 } log σ ( Δ p , k + β ) + 1 { T p , k = − 1 } log ( 1 − σ ( Δ p , k + β ) ) ] . (4)
The likelihood is invariant to shifting all scores of a feature by a common scalar; the ridge penalty resolves this by selecting, within each invariance class, the centered representative 1 ⊤ s k = 0 \mathbf{1}^{\top}s_k=0 1 ⊤ s k = 0 , where s k s_k s k is the k k k -th column of S S S .
Theoretical guarantees. Our estimator raises a nonstandard consistency question: the number of parameters grows with the number of items n n n , and the negative log-likelihood (4) is mildly non-convex. We prove, under reasonable assumptions, that the likelihood admits a local maximizer that is consistent for every item score at once as n n n grows with a bounded number of comparisons per pair, in the case K = 1 K = 1 K = 1 (single feature).
Theorem 1 (Local MLE and uniform consistency, dense regime). Assume A1–A3 (every pair is compared at least once and at most a fixed number of times, independent of n n n ; the true scores are bounded; f f f is even and C 3 C^3 C 3 ) and A6 (the population information for ( γ , β ) (\gamma, \beta) ( γ , β ) , after profiling out s s s , has smallest eigenvalue of order n 2 n^2 n 2 ). For every c > 0 c > 0 c > 0 there exist constants C , n 0 C, n_0 C , n 0 depending only on c c c and the constants in the assumptions such that for all n ≥ n 0 n \ge n_0 n ≥ n 0 , with probability at least 1 − O ( n − c ) 1 - O(n^{-c}) 1 − O ( n − c ) , the negative log-likelihood (4) admits a stationary point ( s ^ , γ ^ , β ^ ) (\hat{s}, \hat{\gamma}, \hat{\beta}) ( s ^ , γ ^ , β ^ ) on the hyperplane { 1 ⊤ s = 0 } \{1^\top s = 0\} { 1 ⊤ s = 0 } satisfying
∥ s ^ − s ∗ ∥ ∞ ≤ C log n n , max { ∣ γ ^ − γ ∗ ∣ , ∣ β ^ − β ∗ ∣ } ≤ C log n n . (5) \lVert\hat{s} - s^* \rVert_\infty\le C\sqrt{\frac{\log n}{n}}, \qquad\max\{|\hat{\gamma} - \gamma^*|, |\hat{\beta} - \beta^*|\} \le C\sqrt{\frac{\log n}{n}}. \tag*{(5)} ∥ s ^ − s ∗ ∥ ∞ ≤ C n log n , max { ∣ γ ^ − γ ∗ ∣ , ∣ β ^ − β ∗ ∣ } ≤ C n log n . (5) Moreover, this point is the unique stationary point in a neighborhood B ∞ ( s ∗ , ϵ n ) × B ( γ ∗ , r n ) × B ( β ∗ , r n ) \mathcal{B}_\infty(s^*, \epsilon_n) \times\mathcal{B}(\gamma^*, r_n) \times\mathcal{B}(\beta^*, r_n) B ∞ ( s ∗ , ϵ n ) × B ( γ ∗ , r n ) × B ( β ∗ , r n ) with ϵ n , r n = Θ ( log n n ) \epsilon_n, r_n = \Theta\left(\sqrt{\frac{\log n}{n}}\right) ϵ n , r n = Θ ( n l o g n ) , and it maximizes the likelihood there: it is the local MLE.
In Appendix B, we prove a more general version of Theorem 1 (Theorem 2) in which both channels carry arbitrary comparison-level covariates and γ \gamma γ and β \beta β are coefficient vectors.
In practice, we optimize the regularized likelihood (3) numerically on sparse comparison graphs, and the fitted model is well calibrated out of fold (Figure 13 and Figure 14 ). Theorem 1 assumes a dense design and does not cover this regime; what it guarantees is that, near the truth, the non-convex likelihood has a unique stationary point, which is a consistent local maximizer, so a local optimizer targets a well-defined estimator rather than an arbitrary critical point.
Figure 13. Out-of-fold reliability of the citation gate (left, all cells) and direction gate (right, cited cells) of the feature-discovery model. Points: observed rate per bin of predicted probability, with 95% bootstrap intervals over comparisons; faint line: in-sample; dashed: perfect calibration; bottom: cells per bin (log scale).
Figure 14. Out-of-fold Brier skill of the two gates against chance and against the training-fold empirical rates; intervals as in Figure 13 .
Extracting score reliability. Because θ ^ \hat{\theta} θ ^ is an M-estimator, it satisfies a central limit theorem as the number of comparisons grows, which yields confidence intervals on every parameter (see Appendix § 10 .§ 3 for more details). The same machinery gives two per-feature diagnostics: a condition number κ k \kappa_k κ k of the score block of the Fisher information, which flags features whose scores are poorly identified by the data ([18 ] ), and a reliability coefficient ρ k \rho_k ρ k , the fraction of the observed score spread attributable to true between-item differences rather than estimation noise. Features with κ k > κ max \kappa_k > \kappa_{\max} κ k > κ m a x or ρ k < ρ min \rho_k < \rho_{\min} ρ k < ρ m i n , for thresholds fixed a priori, are filtered out before a final refit.
Scoring new items. Running the full pipeline on the entire dataset allows the statistical model to aggregate comparisons globally, correct the judge’s biases, and provide reliability diagnostics. With d d d comparisons per item, this requires d n / 2 dn/2 d n /2 pairwise LLM calls. A cheaper alternative is to discover features on a representative subset and then ask an LLM to score additional items independently along those features, requiring one call per item. This pointwise scoring requires specifying precisely how each feature should be scored, which may require human contextual knowledge of the dataset. It is also subject to well-documented biases of pointwise LLM scoring ((Liusie et al., 2024)). Alternatively, one can compare each new item pairwise against a fixed number of already-scored items along the fixed features and fit a Bradley–Terry model for each feature. This retains dataset-level aggregation and position-bias correction without repeating feature discovery. The choice trades cost against robustness to pointwise biases; both are evaluated in § 5 .
Empirical Results on Interpretable Feature Extraction
We first evaluate our approach in a synthetic hiring application where we control the ground-truth features, allowing us to test which features each method recovers and how accurately it recovers them. We construct a dataset of n = 200 n = 200 n = 200 cover letters responding to a real Google security software engineering job advertisement, with known features (which we call dials ) embedded in each letter. See Appendix § 11 for details about the experiment.
Dataset. Each letter realizes 18 engineered ground-truth dials : 8 primary ones (years of experience, education level, school prestige, number of languages, production launches, skill coverage, employer prestige and prose fluency ), 6 binary role-skill presences (e.g. Python, security assessments), and 4 three-level niche dials (e.g. open-source involvement, security certifications ) exactly two of which are present in each letter.
Dials are drawn near-orthogonally and with capped correlation to letter length, so that each dial is a separately recoverable signal and length cannot proxy for quality. Of the 18 generated features, D = 16 D = 16 D = 16 are relevant recovery targets, while two serve as controls: skill coverage is a redundant composite of the individual skill features, and prose fluency is irrelevant to the job.
Metrics. A discovered feature is said to track a dial if the cover letters’ scores along the feature correlate sufficiently with the dial. A discovered feature is said to be about the dial if its name and the dial’s name are sufficiently aligned in TEXT-EMBEDDING-3-LARGE embedding space. We will compare methods along the following metrics (defined mathematically in Appendix C.2.1).
Cost. Total method cost in USD.
Statistical recovery (Tracked & Stat.R.). The fraction of tracked dials (Tracked), and how strongly on average each dial is tracked (Stat.R.).
Semantic recovery (Sem.R.). The fraction of dials that have a discovered feature that is both about it and tracks it. By definition, it is at most Tracked: the gap between the two counts dials tracked under a misaligned name.
Attributable features (Attrib.). The fraction of the discovered features that track a single dial.
Leakage (LK). The average correlation between a dial and the attributable features other than the one that best tracks it. Since dials are near-orthogonal by construction, a feature that measures one dial should be uncorrelated with the others; leakage quantifies how much of the other dials nonetheless bleeds into its scores (lower is better).
Benchmarks. We compare our approach to direct LLM feature extraction and its variants. Specifically, we provide a powerful LLM (GPT-5.4 (HIGH REASONING)) with the whole dataset or a subset and ask it to identify features and formulate dataset-specific guidelines for scoring them. We then use a cheaper model (GPT-5.4 MINI) to score each cover letter on each feature. More precisely, we consider the following variants:
Direct LLM: An LLM is queried on the full dataset to discover features, after which a cheaper LLM scores each item along each feature. We also consider Direct LLM–Multi-score , where multiple agents score each item with slightly different prompts, and their scores are averaged. This variant approximately matches our approach’s cost, allowing us to evaluate whether the benefits arise from using multiple agents or from our specific way of orchestrating them.
Local LLMs: Multiple agents each examine a randomly selected subset of the dataset and discover features. We pool their features and use an LLM to merge equivalent descriptions into a shared set (see Remark 7 in Appendix C for the choice of merging procedure). A single agent then scores each item along these features. Analogously, we also consider Local LLMs–Multi-score . These variants help assess whether the performance benefits arise from distributing feature discovery across multiple agents or from our pairwise comparisons and statistical aggregation.
Ablations. To examine the contributions of statistical scoring and reliability filtering, we retain the features discovered through our sub-agents’ pairwise comparisons and consider two ablations, each with single-agent and multi-agent scoring.
Pairwise comparisons + LLM scoring: LLMs directly score all our discovered features, replacing statistical scoring and its reliability filtering.
Reliable features + LLM scoring: An LLM scores only the features retained by our statistical model’s reliability filter. Comparing these two ablations isolates the contribution of filtering under LLM scoring.
Results. Table 1 reports the metrics of each benchmark and ablation. Our approach outperforms direct LLM feature extraction, even though that benchmark uses one of OPENAI’s most capable models available at the time of the experiments. It tracks all 16 ground-truth features, compared with 13 for Direct LLM, and achieves complete semantic recovery: the discovered features both track the underlying signals and describe them meaningfully. The three missing dials are a discovery failure rather than a scoring failure: at matched cost, Direct LLM–Multi-score still tracks the same
Table 1. Per-method estimates on the single full-data run. Costs are Batch-API prices in USD; K K K is the number of features the method reports. Tracked, Sem.R., Attrib. in %; Stat.R. and LK are correlations.
More of our features are attributable to a single ground-truth feature (16 of 21 versus 13 of 21), and leakage is lower (0.090 versus 0.109). Direct LLM is however cheaper ($0.26 versus $3.45).
Our approach also outperforms the multi-agent alternatives, whether multiple agents are used for feature discovery (Local LLMs), scoring (Multi-score), or both, with statistically significant gains in tracking, attribution, and leakage over Local LLMs (Figure 15 ). Multi-agent scoring improves statistical recovery but increases leakage, while multi-agent feature discovery reduces recovery compared with a single agent with the full dataset. The benefits therefore do not follow simply from using multiple agents: how we divide their work into pairwise comparisons and aggregate their outputs through a statistical model matters.
Figure 15. Paired comparisons over B = 50 B = 50 B = 50 bootstrap draws (with replacement) of the 200-letter corpus; every arm sees the same draws, and differences are taken within replicate. Each row shows the mean paired difference with its 95% percentile CI (⋆ \star ⋆ : excludes zero); the right column gives Ours better , the share of replicates our method wins (Δ K \Delta K Δ K , the difference of discovered features, is annotated below each panel and carries no share). The panels bracket discovery variance from opposite sides and are not comparable to one another: (a) re-runs discovery per replicate on both sides, (b) freezes discovery on both sides, pooled over fusion and corpus draws. Both panels score our method at a reduced budget (average degree 25.3 against the designed 40); it leads on several metrics regardless. Exact values: Table 7 .
The ablations also achieve high tracking and semantic recovery, consistent with their use of the same features as the full pipeline, but without statistical scoring and its reliability filter, attribution is substantially lower (13–14 attributable features versus 16) and leakage higher (0.103–0.104 for Multi-score methods versus 0.090), despite a slightly higher Stat.R. (0.860–0.868 versus 0.850) and twice the cost. The resulting scores therefore separate the ground-truth signals less clearly. Filtering alone raises the attributable fraction under multi-agent scoring from 14 of 32 to 14 of 21 while preserving complete tracking and semantic recovery, showing that it removes features that track no dial. Statistical scoring on the same 21 features then raises it to 16.
Application to a Real Dataset
We next evaluate our approach on a real-life dataset whose underlying features are unknown: online debates from Reddit’s ChangeMyView forum. The central question is whether automatically discovered features provide interpretable insights into persuasion while retaining sufficient signal to predict which arguments change a participant’s view.
The ChangeMyView dataset. In the /r/ChangeMyView (CMV) subreddit, a user (the original poster, OP) posts an opinion with their reasoning and invites counterarguments. The OP then awards an explicit delta (Δ \Delta Δ ) to any reply that changed their view, providing labels of persuasion success. Tan et al. [19 ] turned this signal into a controlled paired prediction task: predicting which of two responses to a post actually convinced the OP. Hence, the items being assessed by our feature discovery pipeline will be exchanges , namely pairs of the form (OP post, reply). See Appendix § 12 for details about the experiment. Prior work on this task either hand-engineers features (word overlap, style and bag-of-words statistics in Tan et al. [19 ] ; argumentation-theoretic features in Wachsmuth et al. [20 ] ) or learns non-interpretable embeddings (Durmus and Cardie [21 ] ).
Extracting interpretable features. We apply our pairwise feature-discovery procedure to a subset of 300 training exchanges, grouping the resulting descriptions into 29 reply-side features, of which our reliability diagnostics retain 16. These features capture human-recognizable aspects of argumentation, including concrete illustrations, structured rebuttals, conceptual clarification, and confrontational tone , without requiring hand-engineering. Table 10 in Appendix § 12 .3.2 lists all 29 features and their descriptions. Figure 3 illustrates their associations with persuasion: it reports how often a successful reply scores higher on each feature than its matched unsuccessful reply. For instance, features related to evidence and cases show stronger positive associations with persuasion than those related to tone and appeals . Confrontational tone shows a negative association with persuasion, although this association is not statistically significant.
Figure 3. Effectiveness of the discovered argumentation techniques. Discovered features are gathered in five categories. For each discovered feature (D + PW), we report the effectiveness , i.e. the fraction of the test pairs in which the reply that earned the Δ \Delta Δ scores higher on that feature; 0.5 is chance. BH q q q is the Benjamini–Hochberg false-discovery rate across the 29 tests. See Figure 21 for the figure showcasing every feature’s effectiveness with confidence intervals.
Table 10. Which argumentation techniques separate winners from losers (D + PW). This table details the results shown in Figure 3 and Figure 21 . It presents each discovered feature (its name and description) together with its effectiveness , i.e. the fraction of the test pairs in which the reply that earned the Δ \Delta Δ scores higher on that feature. 0.5 is chance. Brackets give 95% Wald intervals and q q q is the Benjamini–Hochberg false-discovery rate across the 29 tests; features in bold satisfy q < 0.05 q < 0.05 q < 0.05 . 19 of the 29 features separate winners, all of them positively. Features marked ∗ ^{*} ∗ form the reliable subset Dr _r r (16 features, κ ≤ 50 \kappa\le50 κ ≤ 50 and ρ ≥ 0.75 \rho\ge0.75 ρ ≥ 0.75 ).
Benchmarks and ablations. We write D for the 29 discovered features and Dr _r r for the 16 that pass the reliability filter, and compare both to isolate the contribution of the statistical model. In addition, we compare our approach to features obtained from two types of direct LLM call. In the first, similarly to the benchmarks of § 4 , GPT-5.4 (HIGH REASONING) is given a sample of exchanges and asked to propose distinct features, each with a name and a short definition (S, for samples; 42 features). In the second, the model is given no data, only a brief description of the persuasion task (P, for prior; 16 features).
For each feature set, the features are held fixed and the training and held-out exchanges are scored along them with GPT-5.4 MINI, following the two scoring protocols of § 3 (Scoring new items ). In pointwise scoring (PW), one call per exchange assigns it a 0–10 score on every feature (Appendix § 12 .§ 2 ). In pairwise scoring (BT), the LLM compares two exchanges on every feature at once and a Bradley–Terry model with a positional-bias term is fitted per feature: training exchanges are compared among themselves, and each held-out exchange is then compared against 16 training exchanges only, whose scores are kept frozen, so that all scores lie on a common scale and new items are scored at fixed cost. Since S has 42 features, its pairwise calls would be prohibitively costly, so we report S under PW only.
Expressiveness of the discovered features. The left panel of Figure 2 reports the noise-corrected participation ratio ([22 ] ) of random subsets of each feature set, which summarizes the effective number of uncorrelated dimensions in the feature scores. Features derived from the data, whether from a sample of exchanges (S) or from our pairwise comparisons (D, Dr _r r ), are far less redundant than features elicited from the model’s prior (P): at 16 features, D and Dr _r r span about eight effective dimensions under BT scoring and six to seven under pointwise scoring, against about four for P, whose curve has flattened before 16. The higher dimensionality under BT scoring is consistent with our intuition that aggregating comparisons at the dataset level preserves distinctions that myopic pointwise LLM scoring blurs.
Figure 2. Discovered features span more effective dimensions than prior ones while retaining predictive accuracy. In both panels, each point is an average over random k k k -subsets of the feature set, so that sets of different sizes are compared at matched size. Left: noise-corrected participation ratio of the feature-score correlation matrix (Spearman) over the 1,600 held-out exchanges; dotted: y = k y=k y = k . Right: held-out accuracy of the ℓ 1 \ell_1 ℓ 1 logistic regression on the 800 held-out pairs; chance = 50%. D: discovered (29 features); Dr _r r : reliable subset of D (16); P: prior (16); S: sample-based (42); BT/PW: Bradley–Terry/pointwise scoring; #words: reply length. Bands: 95% bootstrap percentile intervals over held-out exchanges (left) and exchanges + pairs (right). Shaded: k > 16 k>16 k > 16 , where P and Dr _r r are not defined, and k > 29 k>29 k > 29 , where only S is defined. Details in Appendix § 12 .3.1.
Predictive signal. We fit an ℓ 1 \ell_1 ℓ 1 -regularized logistic regression on the scores of each feature set to predict which of two replies to a post persuaded the OP. Table 2 reports the held-out accuracy of each approach, alongside hand-engineered features, embeddings, and simpler baselines. Our approach (D and Dᵣ) achieves accuracy comparable to the best methods (see Table 9 for full data estimates) while providing interpretable features.
Table 2. Held-out accuracy of each method , with 95% bootstrap percentile intervals over the 800 held-out pairs; chance = 50%. Emb.: TEXT-EMBEDDING-3-LARGE embeddings. Tan: hand-engineered features of Tan et al. [19 ] . Zero-shot: the judge predicts the Δ \Delta Δ -winning reply. #words: reply length.
Table 9. Held-out accuracy of each method. Dᵣ: the reliable subset of the discovered features D. Emb. denotes TEXT-EMBEDDING-3-LARGE embeddings. Arms marked (full) are trained on all 3,411 training pairs; all others on 400. Accuracies for PW arms are obtained from the average of two scoring runs. Zero-shot asks the judge directly to predict the Δ \Delta Δ -winning reply, with the query issued in both presentation orders and the two predictions averaged. Chance = 50%; 95% bootstrap percentile intervals (4,000 resamples of the 800 held-out pairs, shared across arms).
Beyond raw accuracy, how much predictive signal a feature set carries relative to its size matters for interpretability. The right panel of Figure 2 therefore reports held-out accuracy against the number k k k of features, averaged over random k k k -subsets of each set. Among the expressive, non-redundant feature sets identified in Figure 2 (S, D, Dᵣ), ours are the most accurate: the direct-LLM sample-elicited set S has more features and a high effective dimensionality, yet it is the least accurate at almost every k k k , 4.2 points below Dᵣ + PW at k = 16 k = 16 k = 16 and 3.6 points below Dᵣ + PW on average over k k k (paired 95% CI [1.8, 5.4]; see Figure 19 in Appendix § 12 .3.1). This indicates that part of the variation S captures is irrelevant to persuasion. Conversely, the features elicited from the LLM without data access (P) are among the most predictive at low k k k but plateau quickly, illustrating the limits of data-blind feature elicitation.
Figure 19. The reliable discovered features are more predictive, feature for feature, than the full discovered set and than the sample-elicited set. Each panel shows, as a function of the number of features k k k , the difference in held-out accuracy between Dr _{\mathrm{r}} r (the 16 reliable discovered features) and another feature set. At each k k k , an ℓ 1 \ell_1 ℓ 1 -regularized logistic regression is fitted on up to 50 random k k k -subsets of each set (all subsets when there are fewer), and accuracy is averaged over subsets; PW scores are averaged over two scoring runs. Shaded: 95% bootstrap interval at each k k k (4,000 resamples of the 800 held-out pairs and of the drawn subsets). (a, b) Against the full discovered set D, with the same scoring: Dr _{\mathrm{r}} r is ahead at every k k k , by 1.0 point on average over k k k under both scorings (95% CIs [−0.1, 2.2] for PW, [0.0, 2.1] for BT). (c, d) Against the sample-elicited set S, scored pointwise only: Dr _{\mathrm{r}} r + PW is ahead at every k k k , by 3.6 points on average ([1.8, 5.4]); Dr _{\mathrm{r}} r + BT’s advantage grows with k k k , reaching 3.5 points at k = 16 k = 16 k = 16 ([0.5, 6.6]).
The results also show that the statistical model plays an important role. The filtered set Dᵣ has about half as many features as D, yet loses at most half a point of accuracy. Hence, the statistical model retains the informative features, yielding a compact, highly interpretable set with a strong predictive signal. The right panel of Figure 2 confirms this: at every k k k , random subsets of Dᵣ are more accurate than equally sized subsets of D, under both scorings, by one point on average over k ≤ 16 k \le16 k ≤ 16 (paired 95% CIs [−0.1, 2.2] for PW and [0.0, 2.1] for BT), showing that features retained by the filter carry more signal than the unfiltered ones.
Conclusion
We introduced an approach that discovers human-interpretable features from unstructured data using LLMs: judges compare items pairwise, and a statistical choice model aggregates their rationales. On both a controlled experiment and real-life data, it recovers features significantly better than a direct LLM call, even with GPT-5.4 (HIGH REASONING). The recovered features capture most of the predictive signal with fewer, less redundant features, while matching the accuracy of established baselines. Appendix § 9 further extends the framework to auditing an LLM’s own decisions.
Reproducibility Statement
Code to reproduce every experiment in the paper is available at the repository https://github.com/ArnaudoB/Interpretable-Feature-Discovery-LLM. The mathematical guarantees of the introduced statistical choice model are proved in Appendix § 10 , and full details of every experiment are given in Appendices § 11 and § 12 .
Statement on LLM Usage
Because LLMs are the object of study, the judge outputs, pairwise comparisons and pointwise scores analyzed throughout, as well as the synthetic cover-letter corpus of § 4 , were produced by LLMs through API calls.
Beyond this experimental role, our use of generative AI was as follows. The text of the paper was written by the authors, with LLMs used for occasional rewording. Portions of the experimental code were written with LLM assistance; all AI-assisted code was reviewed and tested by the authors, who take full responsibility for its correctness. The mathematical claims and proofs were developed by the authors without AI assistance; as an additional verification step, we used Claude Fable 5 (Anthropic) to audit the proofs, which surfaced an error; the authors diagnosed and corrected it by hand.
Extension: Interpretable LLM-based scoring
The goal of this small extension is to provide a method inspired by the feature discovery pipeline but well-suited to the goal of interpretable scoring. LLMs are increasingly used to score items (essays, résumés, or other LLMs) through pairwise comparisons, and the resulting verdicts are typically aggregated by fitting a Bradley–Terry model. This yields a single scalar score per item, whereas a decision-maker would want an overall score that decomposes into per-feature scores along interpretable dimensions. For example, for an LLM grading scientific research papers, have an overall score per-paper that aggregates scores on atomic features discovered from the data : scientific rigor, novelty of the results, etc.
Method. Established methods (e.g. Li et al. [23 ] ) have the LLM make pairwise comparisons and fit a Bradley–Terry model on their outputs to yield scores. Performing this on many features gives several scores that can be aggregated to one overall score that decomposes cleanly on atomic features, but it requires knowing the assessable features.
Our method works by asking the LLM judge for a holistic verdict, and asking it to justify it by specifying along which feature the winning item is better than the losing one. The canonicalization of cited qualities into K K K shared features is identical to § 3 . The data for a compared pair p p p are therefore the overall verdict w p = 1 first-presented item wins w_p = 1_{\text{first-presented item wins}} w p = 1 first-presented item wins and the citation indicators c p ∈ { 0 , 1 } K \mathbf{c}_p \in\{0,1\}^{K} c p ∈ { 0 , 1 } K , where c p , k = 1 c_{p,k}=1 c p , k = 1 if feature k k k was cited as a quality the winner exhibits more; the direction of each citation is thus implied by w p w_p w p .
Statistical model. To infer per-item per-feature scores from the judge’s justifications, our model links the holistic verdict to the per-feature scores by setting:
P ( w p = 1 ) = σ ( ∑ k = 1 K Δ p , k + β ) , (6) \mathbb{P}(w_p=1)=\sigma\left(\sum_{k=1}^{K}\Delta_{p,k}+\beta\right), \tag*{(6)} P ( w p = 1 ) = σ ( k = 1 ∑ K Δ p , k + β ) , (6)
where Δ p , k \Delta_{p,k} Δ p , k is the score gap on feature k k k (first minus second) and β \beta β is the position coefficient. Hence, the model simply posits that the larger the holistic score, the likelier an item is to win the comparisons it is involved in.
To model citations, the model posits that (conditionally on the overall verdict) each feature is cited independently, with a probability that increases in the margin by which that feature favors the winner (because we ask the LLM to justify its winning-item choice):
P ( c p , k = 1 ∣ w p ) = σ ( ( 2 w p − 1 ) Δ p , k + γ k ) . (7) \mathbb{P}(c_{p,k}=1\mid w_p)=\sigma\left((2w_p-1)\Delta_{p,k}+\gamma_k\right). \tag*{(7)} P ( c p , k = 1 ∣ w p ) = σ ( ( 2 w p − 1 ) Δ p , k + γ k ) . (7)
Here 2 w p − 1 2w_p-1 2 w p − 1 is 1 when the first item won and − 1 -1 − 1 otherwise.
The likelihood of an observed pair is therefore:
P ( w p , c p ) = σ ( ∑ k = 1 K Δ p , k + β ) w p ( 1 − σ ( ∑ k = 1 K Δ p , k + β ) ) 1 − w p ∏ k = 1 K σ ( η p , k ) c p , k ( 1 − σ ( η p , k ) ) 1 − c p , k , (8) \mathbb{P}(\mathbf{w}_p,\mathbf{c}_p)=\sigma\left(\sum_{k=1}^{K}\Delta_{p,k}+\beta\right)^{w_p}\left(1-\sigma\left(\sum_{k=1}^{K}\Delta_{p,k}+\beta\right)\right)^{1-w_p}\prod_{k=1}^{K}\sigma(\eta_{p,k})^{c_{p,k}}\left(1-\sigma(\eta_{p,k})\right)^{1-c_{p,k}}, \tag*{(8)} P ( w p , c p ) = σ ( k = 1 ∑ K Δ p , k + β ) w p ( 1 − σ ( k = 1 ∑ K Δ p , k + β ) ) 1 − w p k = 1 ∏ K σ ( η p , k ) c p , k ( 1 − σ ( η p , k ) ) 1 − c p , k , (8)
with η p , k = ( 2 w p − 1 ) Δ p , k + γ k \eta_{p,k}=(2w_p-1)\Delta_{p,k}+\gamma_k η p , k = ( 2 w p − 1 ) Δ p , k + γ k , and the NLL sums − log -\log − log of (A.3) over pairs, ridge-penalized as in Eq. (3).
The method therefore not only discovers the features and estimates each item’s score on them, but also links these per-feature scores to a holistic verdict, so that the overall judgment decomposes additively into named, interpretable components.
Interpretability. To ensure the model’s identifiability, the fitted scores are mean-zero. If one sets σ k : = std ( S ⋅ , k ) \sigma_k:=\operatorname{std}(S_{\cdot,k}) σ k := std ( S ⋅ , k ) , S ~ i , k : = S i , k σ k \widetilde{S}_{i,k}:=\frac{S_{i,k}}{\sigma_k} S i , k := σ k S i , k and Δ ~ p , k : = Δ p , k σ k \widetilde{\Delta}_{p,k}:=\frac{\Delta_{p,k}}{\sigma_k} Δ p , k := σ k Δ p , k , Equation (A.1) becomes:
P ( w p = 1 ) = σ ( ∑ k = 1 K σ k Δ ~ p , k + β ) , (9) \mathbb{P}(w_p=1)=\sigma\left(\sum_{k=1}^{K}\sigma_k\widetilde{\Delta}_{p,k}+\beta\right), \tag*{(9)} P ( w p = 1 ) = σ ( k = 1 ∑ K σ k Δ p , k + β ) , (9)
with each S ~ i , k \widetilde{S}_{i,k} S i , k having mean zero and variance 1. One can hence interpret σ k \sigma_k σ k as the judge’s perceived importance of feature k k k in the overall verdict, yielding interesting interpretability insights about what each judge rewards.
Application. We demonstrate the method on two essay corpora with human holistic scores: (i) n = 100 n = 100 n = 100 essays from ELLIPSE ([24 ] ), a corpus of English-language-learner writing from grades 8–12 scored for overall language proficiency (as well as six analytic dimensions), and (ii) n = 100 n = 100 n = 100 essays from ASAP 2.0 ([25 ] ), a corpus of persuasive essays from grades 6–10 scored for overall writing quality. Each corpus is compared along a d = 40 d = 40 d = 40 -regular graph. To assess whether the discovered constructs and their importance depend on the judge, we run the pipeline with four judges from four model families: GPT-5.4 MINI, CLAUDE HAIKU 4.5, GEMINI 3.6 FLASH, and DEEPSEEK-V4-FLASH. Canonicalization is performed by GPT-5.4 (HIGH REASONING) in every case, and every feature passes through reliability gates analogous to those of § 3 . To compare the judges’ feature sets, we match features across judges semantically with GPT-5.4 (HIGH REASONING) using Prompt 3; Figure 4 shows the resulting shared constructs.
Figure 4. Which features matter for each judge on each dataset. Radial axis is σ k \sigma_k σ k (with estimation noise removed) on each judge’s reliable features (ρ > 0.5 \rho> 0.5 ρ > 0.5 , κ < 100 \kappa< 100 κ < 100 ). Bars are 95% delta-method intervals from the sandwich covariance. Axes are cross-judge clusters formed by meaning (GPT-5.4 (HIGH REASONING)), restricted to constructs at least three of the four judges discovered (see Figure 6 and Figure 7 for the full plots). An open marker denotes a construct absent from that judge’s reliable basis (never surfaced, or removed by the identifiability screen), not a zero estimate.
Figure 6. Denoised per-feature spread by judge on ELLIPSE. Axes are cross-judge constructs formed by meaning (GPT-5.4 (HIGH REASONING)); annotations give how many of the four judges discovered each construct. The radial coordinate is the denoised spread σ k \sigma_k σ k of each judge’s feature mapped to that construct, on per-judge standardized reliable feature sets (ρ > 0.5 \rho> 0.5 ρ > 0.5 , κ < 100 \kappa< 100 κ < 100 ); bars are 95% delta-method intervals from the sandwich covariance. An open marker denotes a construct absent from that judge’s reliable basis (never surfaced, or removed by the identifiability screen), not a zero estimate.
Figure 7. Denoised per-feature spread by judge on ASAP 2.0. Same construction, quantities, and conventions as Figure 6 , on the ASAP 2.0 corpus.
Results. Figure 4 shows that the judges agree on which constructs structure their verdicts far more than on how much each one separates essays. On ELLIPSE, every construct shared by three judges or more is lexical, mechanical, or organizational, consistent with what varies in L2 learner writing; on ASAP 2.0, the four-judge constructs are discourse-level (thesis control, organizational coherence, argument development, sentence-level control). Since the same pipeline, pairs, and prompts (up to corpus framing) produce both bases, this reorganization is attributable to the corpora rather than to any judge’s habits: the method recovers what the task actually varies on. Within the shared constructs, magnitudes disagree: GPT-5.4 MINI is the most compressed judge throughout (smallest σ k \sigma_k σ k on 5 of 7 ELLIPSE and 4 of 8 ASAP 2.0 consensus axes), thesis control on ASAP 2.0 is the one construct on which all four judges essentially coincide (σ k ∈ [ 0.94 , 1.01 ] \sigma_k \in[0.94, 1.01] σ k ∈ [ 0.94 , 1.01 ] ), and sentence-level control is the most contested (0.82 to 2.11). Interestingly, the largest spreads on both corpora sit on features surfaced by a single judge : the sign of an idiosyncratic emphasis. Figure 5 reports each judge’s fitted total ∑ k S i , k \sum_k S_{i,k} ∑ k S i , k against the human holistic score: rank agreement is high on both ELLIPSE (Spearman rank correlation r s = 0.89 r_s = 0.89 r s = 0.89 –0.92) and ASAP 2.0 (r s = 0.77 r_s = 0.77 r s = 0.77 –0.86). Each total decomposes exactly into named per-feature scores, so unlike a scalar Bradley–Terry fit, every verdict can be audited feature by feature: the property that motivated the extension.
Figure 5. Per-judge overall score distribution and its agreement (Spearman rank correlation) with human scores.
Details of the extension
Data and comparison design. For each corpus we sample n = 100 n = 100 n = 100 essays, band-balanced on the human holistic score, and build 2,000 pairs as the union of 40 random perfect matchings, so every essay appears in exactly 40 comparisons. The essay sample, the pair list, and the presentation order are generated once and are byte-identical across the four judges of a corpus: within a corpus, only the judge varies. Between corpora, only the corpus framing of the elicitation prompt changes (ELLIPSE frames English-language proficiency of grade 8–12 learners; ASAP 2.0 frames source-based argumentative quality and inlines the source article and assignment verbatim); schema, field order, parser, and all downstream prompts are unchanged, so the ELLIPSE–ASAP 2.0 contrast is a corpus contrast, not a prompt contrast.
Elicitation. One call per pair elicits a forced binary verdict (Prompts 1 and 2) followed by the qualities the winner exhibits more than the loser, each with evidence, a short label, and a one-sentence description. The prompt names no features and contains no worked examples (to avoid priming the constructs under study), forbids bundled qualities (“and”, “/”, “&”), and rules out of scope the topic, the stance, length in itself, and the corpora’s anonymization placeholders. The output schema emits the verdict before the qualities, matching the factorization of the likelihood (A.3). Reasoning modes are disabled on all four judges.
Canonicalization and reliability gates. Per judge, cited phrases with at least 3 citations are grouped by meaning into features by GPT-5.4 (HIGH REASONING), with a catch-all “Other” bucket dropped downstream; group membership is validated programmatically. The model of Equation (A.3) is fitted by joint MLE (L-BFGS, analytic gradients, ridge 10 − 3 10^{-3} 1 0 − 3 ), and each feature passes a reliability screen built on the same principle as § 3 : a condition number κ k \kappa_k κ k of the score block of the Fisher information of model (A.3), and a signal-to-noise ratio ρ k \rho_k ρ k obtained by subtracting the sandwich estimation variance from the observed score spread. The statistics differ from those of the main pipeline because the likelihood does; we retain features with ρ > 0.50 \rho> 0.50 ρ > 0.50 and κ < 100 \kappa< 100 κ < 100 , then refit the model on the surviving features. Across the eight judge–corpus cells, the screen retains between 6 and 16 features.
Cross-judge consolidation. The four judges’ reliable features are grouped into shared constructs by a single GPT-5.4 (HIGH REASONING) call per corpus, with at most one feature per judge and per construct. The prompt requires singletons to remain visible (a construct found by one judge is a finding, not noise) and receives no statistics so semantic grouping cannot be contaminated by how essays scored. Figure 4 displays constructs recovered by at least three judges.
Reported quantity and denoising. The radial axis reports a noise-corrected spread: the observed σ k \sigma_k σ k mixes essay-to-essay variation with estimation error, so we subtract the mean sandwich variance (Appendix § 10 .§ 3 ) of the fitted scores, with delta-method standard errors from the sandwich covariance. Denoising shrinks ELLIPSE spreads by 10.5% on average and reorders 10 of 34 features within their own judge.
Fit quality and position bias. Spearman correlation between the fitted total ∑ k S i , k \sum_k S_{i,k} ∑ k S i , k and the human holistic score is 0.89–0.92 on ELLIPSE and 0.77–0.86 on ASAP 2.0. The fitted position coefficient β \beta β varies in sign and magnitude across judges (∣ β ^ ∣ |\hat{\beta}| ∣ β ^ ∣ up to 2.6) and tracks each judge’s raw first-position win rate on both corpora: position bias is a genuine property that depends on the judge and on the dataset.
Sensitivity and design caveats. We verified the sensitivity of the reliability gates: at ρ > 0.70 \rho> 0.70 ρ > 0.70 , κ < 50 \kappa< 50 κ < 50 the ELLIPSE reliable feature sets shrink from 9/9/9/7 to 6/6/6/7 features (GPT-5.4 MINI, CLAUDE HAIKU 4.5, GEMINI 3.6 FLASH, DEEPSEEK-V4-FLASH) and the constructs shared by three judges or more fall from 7 to 5, so consensus counts depend on the gates and we report the setting used. Because different judges lose different features to κ \kappa κ , the consensus counts understate agreement (e.g., Vocabulary is a 3/4 construct on ELLIPSE only because the screen removed DeepSeek’s). The citation floor interacts similarly: in a preliminary three-judge run, a floor of 5 let thinly spread phrasing make constructs appear less shared than they were, which motivated the minimum-citation threshold of 3.
Model calibration. The extension model assigns two probability forecasts of observable binary events, one per channel. The winner gate, P ( w p = 1 ) = σ ( ∑ k Δ ^ p , k + β ^ ) \mathbb{P}(w_p = 1) = \sigma\left(\sum_k \widehat{\Delta}_{p,k} + \widehat{\beta}\right) P ( w p = 1 ) = σ ( ∑ k Δ p , k + β ) , is the probability that the first-shown essay wins the overall verdict, scored once per comparison; the citation gate, P ( c p , k = 1 ∣ w p ) = σ ( ( 2 w p − 1 ) Δ ^ p , k + γ ^ k ) \mathbb{P}(c_{p,k} = 1 \mid w_p) = \sigma\left((2w_p - 1)\widehat{\Delta}_{p,k} + \widehat{\gamma}_k\right) P ( c p , k = 1 ∣ w p ) = σ ( ( 2 w p − 1 ) Δ p , k + γ k ) , is the probability that feature k k k is cited given the verdict, increasing in the margin by which k k k favors the declared winner, scored on every (comparison, feature) cell of the reliable feature set. We score both as forecasts, out of fold (5-fold cross-validation over the 2,000 comparisons, so that every prediction comes from parameters estimated without it, with 95% bootstrap intervals over comparisons) independently for each of the four judges and on both corpora, so calibration is also a robustness check across judges.
Figure 8 and Figure 10 show reliability diagrams for ELLIPSE and ASAP 2.0: within ten equal-width bins of predicted probability, the observed event rate is plotted against the predicted one, so a perfectly calibrated gate lies on the diagonal. The expected calibration error (ECE) summarizes each panel as the mean absolute gap between the two, weighted by bin occupancy. The winner gate is calibrated for every judge: out-of-fold ECE ranges over 0.012–0.028 on ELLIPSE and 0.010–0.019 on ASAP 2.0, in no case worse than in sample. Its predictions concentrate near the endpoints of the unit interval, so the diagnostic is exercised where it is most demanding: the model is confident on most comparisons and right about how confident it is. The citation gate is accurate in sample (ECE 0.003–0.011) but slightly degrades out of fold (ECE 0.027–0.041), with the gap concentrated in the upper bins, where observed citation rates fall short of the predictions for all four judges and on both corpora.
Figure 8. Out-of-fold reliability diagrams for the winner gate (left; unit: comparison) and the citation gate (right; unit: cell) of the model on ELLIPSE, one row per judge. Units are grouped into ten equal-width bins B 1 , … , B 10 B_1,\ldots,B_{10} B 1 , … , B 10 of predicted probability p ^ i \hat{p}_i p ^ i ; points show the observed event rate y ˉ b = 1 ∣ B b ∣ ∑ i ∈ B b y i \bar{y}_b = \frac{1}{|B_b|}\sum_{i\in B_b} y_i y ˉ b = ∣ B b ∣ 1 ∑ i ∈ B b y i against the mean prediction p ˉ b = 1 ∣ B b ∣ ∑ i ∈ B b p ^ i \bar{p}_b = \frac{1}{|B_b|}\sum_{i\in B_b}\hat{p}_i p ˉ b = ∣ B b ∣ 1 ∑ i ∈ B b p ^ i , with 95% bootstrap intervals (resampling comparisons, resp. cells). Faint line: in-sample fit; dashed: perfect calibration (y ˉ b = p ˉ b \bar{y}_b = \bar{p}_b y ˉ b = p ˉ b ). The expected calibration error is E C E = ∑ b ∣ B b ∣ N ∣ y ˉ b − p ˉ b ∣ \mathrm{ECE} = \sum_b \frac{|B_b|}{N}|\bar{y}_b-\bar{p}_b| ECE = ∑ b N ∣ B b ∣ ∣ y ˉ b − p ˉ b ∣ , with N = ∑ b ∣ B b ∣ N=\sum_b |B_b| N = ∑ b ∣ B b ∣ . Bottom strips: ∣ B b ∣ |B_b| ∣ B b ∣ per bin, log scale.
Figure 10. As Figure 8, on ASAP 2.0.
Figure 9 reports discrimination through the Brier skill score against chance (p = 1 / 2 p = 1/2 p = 1/2 , Brier 0.25) and against the empirical rate estimated on the training folds (the rate at which the first-shown essay wins for the winner gate, the per-feature citation rate for the citation gate). The second reference is the demanding one for the citation channel: it is the maximum-likelihood fit of the gap-independent submodel in which P ( c p , k = 1 ∣ w p ) = σ ( γ k ) \mathbb{P}(c_{p,k} = 1 \mid w_p) = \sigma(\gamma_k) P ( c p , k = 1 ∣ w p ) = σ ( γ k ) , citation as pure per-feature habit, so skill over it is a direct out-of-fold test of whether citations track the winner-oriented gaps.
Figure 9. Out-of-fold Brier skill of the two gates, per judge and corpus, with in-sample values overlaid. With p ^ i \hat{p}_i p ^ i the predicted probability and y i ∈ { 0 , 1 } y_i \in\{0,1\} y i ∈ { 0 , 1 } the gated event, the Brier score is B S = N − 1 ∑ i ( p ^ i − y i ) 2 BS = N^{-1}\sum_i(\hat{p}_i-y_i)^2 B S = N − 1 ∑ i ( p ^ i − y i ) 2 and the skill score against a reference forecast q i q_i q i is B S S = 1 − B S / B S r e f BSS = 1-BS/BS_{\mathrm{ref}} B S S = 1 − B S / B S ref , where B S r e f = N − 1 ∑ i ( q i − y i ) 2 BS_{\mathrm{ref}} = N^{-1}\sum_i(q_i-y_i)^2 B S ref = N − 1 ∑ i ( q i − y i ) 2 ; B S S = 1 BSS = 1 B S S = 1 is perfect, 0 matches the reference, and negative values are worse than it. Two references: chance (q i = 1 2 q_i = \frac{1}{2} q i = 2 1 ) and the empirical event rate of the training fold (q i = y ˉ t r a i n q_i = \bar{y}_{\mathrm{train}} q i = y ˉ train ). Intervals as in Figure 8 .
The winner gate’s skill scores against the two references coincide (slot balance keeps the first-shown rate near 1/2) and are large for every judge (0.80–0.84 on ELLIPSE, 0.72–0.85 on ASAP 2.0). The citation gate scores 0.03–0.23 against the habit-only submodel across judges and corpora, rejecting it throughout: citations do track the gaps, though the contribution is second-order next to the verdict channel and shrinks from its in-sample values in line with the reliability curves.
Prompts. The prompts used are reported below.
Prompt 1: Comparative elicitation – ELLIPSE.
=== SYSTEM ===
You are helping build an interpretable scoring framework by examining how student essays
differ in English language proficiency. The framework works by collecting many pairwise
comparisons: for each pair, a judgment of which essay is stronger overall is recorded
together with the qualities that essay has more of. Aggregated across many pairs, these
reveal the distinct dimensions along which student writing varies. There is no fixed rubric
--- the qualities are recovered from comparisons like this one, not specified in advance.
These are essays written by English language learners in grades 8 to 12, during
standardized writing assessments. They span a wide range of proficiency, from beginning to
quite advanced English writers. Each essay responds to an independent writing prompt that
requires no specialized background knowledge; different essays may respond to different
prompts. Read and weigh the essays as you would when assessing the English writing
proficiency of language learners at this level.
Your task. You will be shown two student essays, Essay A and Essay B. They may respond to
different writing prompts. This pair is one comparison in a larger collection.
First, decide which essay demonstrates stronger English language proficiency overall. This
is a forced choice: pick A or B even when the two are close.
Then identify the qualities that the essay you chose has more of than the other essay.
Report the ones that genuinely stand out to you on reading this pair --- bring up whatever
you find salient, and do not work from a predetermined list. Favor real, noticeable
differences over minor ones, but do not force the list to be short: report as many as
actually distinguish these two essays.
For each quality, first name the concrete, specific feature you actually saw in the essays
--- a short phrase pointing to the specific evidence. Then, grounded in that evidence, name
the quality and describe in one sentence what it captures.
Name only qualities on which the essay you chose is genuinely ahead. If the two essays are
comparable on some aspect of proficiency, leave it out rather than listing it. Do not pad
the list to justify your choice, and do not list a quality on which the other essay is in
fact stronger.
Out of scope --- do not treat these as differences in proficiency:
- the writing prompt an essay was responding to, or the topic or difficulty of that prompt
--- judge proficiency, not the task an essay happened to be assigned;
- the position or opinion an essay takes --- any stance can be expressed proficiently;
- length in itself;
- anonymization placeholders inserted by the corpus builders to remove identifying
information, such as Generic_Name, Generic_City, Generic_School, PROPER_NAME,
LOCATION_NAME, STUDENT_NAME, TEACHER_NAME or SCHOOL_NAME. Ignore them entirely: they are
not the student's words. Never count them as spelling, capitalization, spacing or
word-choice errors --- including when a placeholder is misspelled or run together with an
adjacent word (for example "Generic_Namehad" or "wasGeneric_Name").
How to report. List each quality as exactly one item --- do not combine qualities with
"and", "/", "&", or a comma; split related qualities into separate items. For the evidence
field, give a short phrase naming the concrete feature you saw. For the quality field, give
a concise general noun phrase naming the quality (the kind of short label you might use as
a rubric column). For the description field, give one sentence saying what that quality
captures.
Output only JSON of the form:
{"overall_winner": "A" | "B", "qualities": [{"evidence": "<short phrase>", "quality":
"<concise noun phrase>", "description": "<one sentence>"}, ...]}
=== USER ===
Essay A:
⟨first essay ⟩
Essay B: ⟨second essay ⟩
Following the standard above, first give the essay that demonstrates stronger English language proficiency overall, then list the qualities that essay has more of than the other --- for each, first the concrete feature you saw, then the quality and a one-sentence description. Respond with the JSON object only.
Prompt 2: Comparative elicitation – ASAP 2.0.
=== SYSTEM ===
You are helping build an interpretable scoring framework by examining how student argumentative essays differ in quality. The framework works by collecting many pairwise comparisons: for each pair, a judgment of which essay is stronger overall is recorded together with the qualities that essay has more of. Aggregated across many pairs, these reveal the distinct dimensions along which student writing varies. There is no fixed rubric --- the qualities are recovered from comparisons like this one, not specified in advance.
These are argumentative essays written by students in United States schools, primarily in grades 6, 8, 9 and 10. They span a wide range of writing ability and maturity, from beginning writers to quite proficient ones. Every essay responds to the same source-based argumentation task: the students read the article below and were asked to take and defend a position on it, using details from that article as evidence. Read and weigh the essays as you would when assessing argumentative writing at this level.
--- SOURCE ARTICLE ---
⟨source article, inlined verbatim (1,120 words, “Driverless Cars Are Coming”) ⟩
--- ASSIGNMENT GIVEN TO STUDENTS ---
In the article "Driverless Cars are Coming," the author presents both positive and negative aspects of driverless cars. Using details from the article, create an argument for or against the development of these cars. Be sure to include: your position on driverless cars; appropriate details from the article that support your position; an introduction, a body, and a conclusion to your argumentative essay.
Your task. You will be shown two student essays, Essay A and Essay B, both responding to this assignment. This pair is one comparison in a larger collection.
First, decide which essay is the stronger argumentative essay overall. This is a forced choice: pick A or B even when the two are close.
Then identify the qualities that the essay you chose has more of than the other essay. Report the ones that genuinely stand out to you on reading this pair --- bring up whatever you find salient, and do not work from a predetermined list. Favor real, noticeable differences over minor ones, but do not force the list to be short: report as many as actually distinguish these two essays.
For each quality, first name the concrete, specific feature you actually saw in the essays --- a short phrase pointing to the specific evidence. Then, grounded in that evidence, name the quality and describe in one sentence what it captures.
Name only qualities on which the essay you chose is genuinely ahead. If the two essays are comparable on some aspect of quality, leave it out rather than listing it. Do not pad the list to justify your choice, and do not list a quality on which the other essay is in fact stronger.
Out of scope --- do not treat these as differences in quality:
- the topic itself, or which particular details from the article an essay chose to use,
except insofar as those choices affect the quality of the argument;
- length in itself;
- anonymization placeholders inserted by the corpus builders to remove identifying
information, such as PROPER_NAME, OTHER_PII, STUDENT_NAME, LOCATION_NAME or SCHOOL_NAME.
Ignore them entirely: they are not the student's words. Never count them as spelling,
capitalization, spacing or word-choice errors --- including when a placeholder is
misspelled or run together with an adjacent word.
How to report. List each quality as exactly one item --- do not combine qualities with
"and", "/", "&", or a comma; split related qualities into separate items. For the evidence
field, give a short phrase naming the concrete feature you saw. For the quality field, give
a concise general noun phrase naming the quality (the kind of short label you might use as
a rubric column). For the description field, give one sentence saying what that quality
captures.
Output only JSON of the form:
{"overall_winner": "A" | "B", "qualities": [{"evidence": "<short phrase>", "quality":
"<concise noun phrase>", "description": "<one sentence>"}, ...]}
=== USER ===
Essay A:
⟨first essay⟩
Essay B:
⟨second essay⟩
Following the standard above, first give the essay that is the stronger argumentative essay
overall, then list the qualities that essay has more of than the other --- for each, first
the concrete feature you saw, then the quality and a one-sentence description. Respond with
the JSON object only.
Prompt 3: Cross-judge consolidation prompt.
=== SYSTEM ===
You are a careful taxonomist of writing-assessment constructs. Output only valid JSON
matching the requested schema.
=== USER ===
You are consolidating evaluation features discovered independently by several LLM judges
into a set of shared constructs.
Each judge compared ⟨corpus
framing⟩ and, with no fixed rubric, named the qualities that distinguished the stronger
essay. Each judge's raw phrasings have already been canonicalized separately, so within a
single judge the features below are already distinct from one another. Your job is to
identify, ACROSS judges, which features denote the same underlying construct.
Below is the full list. Each entry has an INDEX in square brackets, the JUDGE that produced
it, its NAME, and a one-sentence DEFINITION.
⟨indexed feature list — one entry per line: [i] (judge) name: definition⟩
Instructions:
- Group entries that denote the SAME underlying evaluative construct, judging by MEANING,
not by surface wording. Different words for the same idea belong in one group; the same
word used for different ideas belongs in different groups.
- Keep genuinely DISTINCT constructs separate, even when related. Two constructs a writing
expert would score on separate rubric lines should be separate groups.
- A group may contain AT MOST ONE feature per judge. The features within a judge were
already canonicalized, so two of them never denote the same construct; if you find yourself
wanting to place two features from one judge together, they belong in different groups.
A feature with no counterpart in any other judge forms a group of ONE. Do not absorb it into a loosely related group to tidy up, and do not create a catch-all: a construct only one judge discovered is a real finding and must stand alone.
Do NOT try to balance the groups. There is no target number of groups and no expectation that every group contains every judge. Groups of one, two, three, and four are all equally valid outcomes; forcing coverage would fabricate agreement that is not there.
Do NOT merge two constructs just because they often co-occur or because strong essays tend to score well on both. Group only by shared MEANING.
The judge labels are present only so you can respect the one-per-judge rule. They carry no information about meaning; never group or split on the basis of which judge produced an entry.
For each group, give:
a canonical NAME: a concise noun phrase naming the single construct the group captures. Name the construct itself, not the union of its members' wordings; avoid conjunctions unless the construct genuinely has no single name.
a one-sentence DEFINITION of that construct.
the list of members, each as its INDEX, its JUDGE, and its NAME copied verbatim.
Rules on membership (these will be checked programmatically):
Every input index must appear in exactly one group, exactly once. Place all of them; drop none; duplicate none.
Do not invent indices that are not in the input.
Each member's "judge" and "name" must be the exact input values at that index --- this is a cross-check, so copy them from the bracketed entry, do not paraphrase.
No group may contain two members with the same judge.
Output only JSON:
{"clusters": [{"name": "<canonical noun phrase>", "definition": "<one sentence>",
"members": [{"index": <int>, "judge": "<verbatim>", "name": "<verbatim>"}, ...]}, ...]}
Statistical Model Theory
The Bradley–Terry with ties literature
Classical tie models. The Bradley–Terry model [17 ] assigns each item a latent merit and models a binary win/loss outcome through the logistic link. Classical extensions accommodate a third outcome per comparison: Glenn and David [26 ] add a discrimination threshold to the Thurstone–Mosteller model, Rao and Kupper [27 ] do the same on the logistic scale, and Davidson [28 ] models the tie probability as proportional to the geometric mean of the two merits, governed by a tie intensity ν \nu ν . Davidson and Beaver [29 ] further incorporate multiplicative within-pair order effects, the classical ancestor of our position coefficient β \beta β ; see Cattelan [30 ] for a survey. Since each of our per-feature observations is trinomial (“i i i wins”, “j j j wins”, “not cited”), our model is naturally read against this literature, and this subsection records the relationship.
Davidson’s model is a gated model. Write π i = e s i \pi_i=e^{s_i} π i = e s i . Davidson’s model puts P ( i wins ) ∝ e s i \mathbb{P}(i\text{ wins})\propto e^{s_i} P ( i wins ) ∝ e s i , P ( j wins ) ∝ e s j \mathbb{P}(j\text{ wins})\propto e^{s_j} P ( j wins ) ∝ e s j , and P ( tie ) ∝ ν e ( s i + s j ) / 2 \mathbb{P}(\text{tie})\propto\nu e^{(s_i+s_j)/2} P ( tie ) ∝ ν e ( s i + s j ) /2 . Dividing the three weights by e ( s i + s j ) / 2 e^{(s_i+s_j)/2} e ( s i + s j ) /2 and writing Δ = s i − s j \Delta=s_i-s_j Δ = s i − s j ,
P ( decisive ) = 2 cosh ( Δ / 2 ) 2 cosh ( Δ / 2 ) + ν = σ ( log ( 2 cosh ( Δ / 2 ) ) − log ν ) , P ( i ∣ decisive ) = σ ( Δ ) . \mathbb{P}(\text{decisive})=\frac{2\cosh(\Delta/2)}{2\cosh(\Delta/2)+\nu}=\sigma\left(\log(2\cosh(\Delta/2))-\log\nu\right), \qquad\mathbb{P}(i\mid\text{decisive})=\sigma(\Delta). P ( decisive ) = 2 cosh ( Δ/2 ) + ν 2 cosh ( Δ/2 ) = σ ( log ( 2 cosh ( Δ/2 )) − log ν ) , P ( i ∣ decisive ) = σ ( Δ ) .
Davidson’s model is exactly the single-feature instance of (B.8) with gate link f ( u ) = log ( 2 cosh ( u / 2 ) ) f(u)=\log(2\cosh(u/2)) f ( u ) = log ( 2 cosh ( u /2 )) , gate intercept γ = − log ν \gamma=-\log\nu γ = − log ν , and no position effect. Our link f = log ( 2 cosh ) f=\log(2\cosh) f = log ( 2 cosh ) is a variant of the same choice, and the general model of Appendix § 10 .5.1 is a strict generalization of Davidson’s. The Rao–Kupper model admits no such factorization: it is the ordered logit with symmetric cutpoints, P ( i ≻ j ) = σ ( Δ − log θ ) \mathbb{P}(i\succ j)=\sigma(\Delta-\log\theta) P ( i ≻ j ) = σ ( Δ − log θ ) and P ( i ≻ j or tie ) = σ ( Δ + log θ ) \mathbb{P}(i\succ j\text{ or tie})=\sigma(\Delta+\log\theta) P ( i ≻ j or tie ) = σ ( Δ + log θ ) , so the threshold couples the two channels and the direction given a decisive outcome is not of logistic form in Δ \Delta Δ .
Why tie models do not transfer off the shelf. Since Davidson’s model is a gated model, a battery of K K K independent Davidson fits, one per feature, lies inside our model class (γ k = − log ν k \gamma_k=-\log\nu_k γ k = − log ν k ) and is the natural classical baseline. Three things separate our model from it.
First, the regime and the semantics of the null outcome: the tie literature was developed where ties are a minority outcome among near-equal players and assert near-equality, whereas non-citation is our dominant outcome and signals lack of salience as much as similarity. The fitted intercepts of Table 4 correspond to tie intensities ν k = e − γ k \nu_k=e^{-\gamma_k} ν k = e − γ k ranging over two orders of magnitude, which is to our knowledge far outside anything that literature contemplates.
Table 4. Features surfaced by our comparative method in the point estimate of Table 1. Of 32 surfaced features, we keep the features that have ρ ≥ ρ min : = 0.75 \rho\ge\rho_{\min} := 0.75 ρ ≥ ρ m i n := 0.75 and κ ≤ κ max : = 10 \kappa\le\kappa_{\max} := 10 κ ≤ κ m a x := 10 , leaving 21 features. Fitted β ^ = + 0.005 \hat{\beta} = +0.005 β ^ = + 0.005 (95% CI [− 0.033 -0.033 − 0.033 , + 0.043 +0.043 + 0.043 ]). CIs for β ^ \hat{\beta} β ^ and γ \gamma γ coefficients were computed using (B.3).
Second, the position effect. The classical route [29 ] multiplies the first-presented merit, π i ↦ e β π i \pi_i\mapsto e^\beta\pi_i π i ↦ e β π i , leaving the tie weight ν π i π j \nu\sqrt{\pi_i\pi_j} ν π i π j unchanged; the model stays log-linear, with
P ( decisive ) = σ ( log 2 cosh ( Δ + β 2 ) + β 2 − log ν ) , P ( i ∣ decisive ) = σ ( Δ + β ) . \mathbb{P}(\text{decisive})=\sigma\left(\log2\cosh\left(\frac{\Delta+\beta}{2}\right)+\frac{\beta}{2}-\log\nu\right), \qquad\mathbb{P}(i\mid\text{decisive})=\sigma(\Delta+\beta). P ( decisive ) = σ ( log 2 cosh ( 2 Δ + β ) + 2 β − log ν ) , P ( i ∣ decisive ) = σ ( Δ + β ) .
The conditional direction is exactly our direction channel, but the gate argument is no longer even in Δ \Delta Δ : presentation order alters the citation probability, and the null outcome is most likely at Δ = − β \Delta=-\beta Δ = − β rather than at a genuine tie. Attaching the order effect symmetrically to both merits changes only the constant β / 2 \beta/2 β /2 , absorbed into ν \nu ν ; the non-evenness remains. Our parity structure (even f f f and Y i j k Y_{ijk} Y ij k , odd Z i j k Z_{ijk} Z ij k defined in Appendix § 10 .§ 5 ) instead keeps the direction channel σ ( Δ + β ) \sigma(\Delta+\beta) σ ( Δ + β ) while confining the position effect to it by construction.
The trade-off is exact: an order effect on the merits is what keeps the trinomial model log-linear, so an order-invariant gate combined with a position-biased direction channel forces the likelihood outside the multinomial-logit family – which is precisely where concavity is lost (the signed curvature term of (B.15)) and why the estimation theory of Appendices § 10 .9–§ 10 .12 cannot be imported from that literature.
Third, the K K K features are estimated jointly with a shared position coefficient, so the joint likelihood is not a product of per-feature tie models, both channels carry general covariates, and the inferential targets of § 3 – per-feature Laplacian condition numbers, reliability coefficients, and misspecification-robust intervals – have no counterpart in the paired-comparisons ties literature.
A discrete-choice reading, and scope beyond LLM judges. The gate/direction factorization can also be read as a two-level discrete choice with an outside option: the judge first “chooses” whether the feature is worth mentioning, then chooses a winner, analogous to a nested logit in which non-citation is the outside nest [31 , 32 ] . It is by this analogy that we elaborated the model in the first place. We do not claim a formal random-utility foundation – our gate replaces the inclusive value by the even link f f f – but the analogy delimits the scope of the framework: it applies to any rater who, per comparison, reports verdicts only on the attributes salient enough to mention.
Estimation theory. Modern non-asymptotic theory for the binary Bradley–Terry model studies the growing-dimension regime: Simons and Yao [33 ] establish consistency and asymptotic normality as n → ∞ n \to\infty n → ∞ in the dense case, Negahban et al. [34 ] and Shah et al. [35 ] analyze spectral and minimax aspects, Chen et al. [36 ] obtain ℓ ∞ \ell_\infty ℓ ∞ guarantees through the leave-one-out technique we adapt, and Yan [37 ] provides the uniform-consistency template our Stage-1 analysis builds on. For tie models, we are not aware of comparable growing-dimension ℓ ∞ \ell_\infty ℓ ∞ guarantees. The technical landscape also differs: Davidson’s model is a log-linear multinomial logit and Rao–Kupper an ordered logit, so both have concave log-likelihoods in the natural parametrization, whereas the free even link and the gate covariates make our likelihood non-concave – the signed curvature term of (B.15) – which is the difficulty the machinery of Appendices B.9–B.12 is built to handle.
Why a shared position bias coefficient?
The direction channel of the model includes a position coefficient β \beta β : for a pair p p p on feature k k k , given citation, the probability that the first-presented item wins is d p , k = σ ( Δ p , k + β ) d_{p,k} = \sigma(\Delta_{p,k} + \beta) d p , k = σ ( Δ p , k + β ) , where pairs are written in presentation order, so that Δ p , k = s i , k − s j , k \Delta_{p,k} = s_{i,k} - s_{j,k} Δ p , k = s i , k − s j , k with i i i the item shown first. The coefficient β \beta β is shared across features. A natural alternative is a per-feature coefficient β k \beta_k β k ; we first explain why this choice has almost no effect on the estimated scores, then verify it empirically.
Setup: how β \beta β enters the score equations. Fix a feature k k k (we drop it from the notation). The direction log-likelihood is
ℓ ( s , β ) = ∑ p : c p = 1 [ w p log σ ( Δ p + β ) + ( 1 − w p ) log ( 1 − σ ( Δ p + β ) ) ] \ell(s,\beta) = \sum_{p:c_p=1} \left[w_p \log\sigma(\Delta_p+\beta) + (1-w_p)\log\left(1-\sigma(\Delta_p+\beta)\right)\right] ℓ ( s , β ) = p : c p = 1 ∑ [ w p log σ ( Δ p + β ) + ( 1 − w p ) log ( 1 − σ ( Δ p + β ) ) ]
where w p = 1 w_p=1 w p = 1 if the first-presented item of pair p p p won. Consider an item i i i , and let P i P_i P i be the pairs containing i i i . In each p ∈ P i p \in P_i p ∈ P i , the gap Δ p \Delta_p Δ p contains s i s_i s i with sign
ε p ( i ) = { + 1 if i was presented first in p , − 1 if i was presented second, \varepsilon_p(i) =
\begin{cases}
+1 & \text{if } i\text{ was presented first in }p,\\
-1 & \text{if } i\text{ was presented second,}
\end{cases} ε p ( i ) = { + 1 − 1 if i was presented first in p , if i was presented second,
The score equation for s i s_i s i is therefore
∂ ℓ ∂ s i = ∑ p ∈ P i ε p ( i ) ( w p − d p ) = 0 , d p = σ ( Δ p + β ) . (10) \frac{\partial\ell}{\partial s_i} = \sum_{p\in P_i}\varepsilon_p(i)(w_p-d_p)=0,\qquad d_p=\sigma(\Delta_p+\beta). \tag*{(10)} ∂ s i ∂ ℓ = p ∈ P i ∑ ε p ( i ) ( w p − d p ) = 0 , d p = σ ( Δ p + β ) . (10)
This is the standard Bradley–Terry balance condition (observed wins minus expected wins, counted from i i i ’s side of each pair) except that every expected win d p d_p d p is tilted by the common offset β \beta β .
First-order effect of changing β \beta β . Suppose β \beta β moves by an amount δ \delta δ (for instance, from the shared estimate β ^ \hat{\beta} β ^ to a per-feature estimate β ^ k \hat{\beta}_k β ^ k , so δ = β ^ k − β ^ \delta=\hat{\beta}_k-\hat{\beta} δ = β ^ k − β ^ ). Each expected win moves by d p = w θ , p δ d_p=w_{\theta,p}\delta d p = w θ , p δ with w θ , p = d p ( 1 − d p ) w_{\theta,p}=d_p(1-d_p) w θ , p = d p ( 1 − d p ) , so the score equation (B.1) is thrown off balance by
− δ ∑ p ∈ P i ε p ( i ) w θ , p . -\delta\sum_{p\in P_i}\varepsilon_p(i)w_{\theta,p}. − δ p ∈ P i ∑ ε p ( i ) w θ , p .
To restore balance, s i s_i s i must adjust. Differentiating (B.1) in s i s_i s i (and neglecting the adjustment of the other items’ scores, which enters only through the off-diagonal Laplacian terms), the required adjustment is, to first order,
Δ s ^ i ≈ − δ ∑ p ∈ P i ε p ( i ) w θ , p ∑ p ∈ P i w θ , p . (11) \Delta\hat{s}_i \approx-\delta\frac{\sum_{p\in P_i}\varepsilon_p(i)w_{\theta,p}}{\sum_{p\in P_i}w_{\theta,p}}. \tag*{(11)} Δ s ^ i ≈ − δ ∑ p ∈ P i w θ , p ∑ p ∈ P i ε p ( i ) w θ , p . (11)
We write ≈ \approx ≈ because modifying β \beta β also modifies the other s j s_j s j , which in turn modify s i s_i s i : we neglect these second-order perturbations. The denominator is a sum of positive weights; the numerator is the same weights, but signed by presentation order. The score shift is thus δ \delta δ times a weighted average of ± 1 \pm1 ± 1 ’s: + 1 +1 + 1 for each pair where i i i was shown first, − 1 -1 − 1 where it was shown second.
Cancellation under the balanced design. Equation (B.2) exposes exactly what the design controls. If the weights were equal, the numerator would be n f i r s t ( i ) − n s e c o n d ( i ) n_{\mathrm{first}}(i)-n_{\mathrm{second}}(i) n first ( i ) − n second ( i ) : the imbalance in how often item i i i appeared in each presentation slot. Our comparison graph assigns presentation order uniformly at random, independently of the items; with m m m comparisons per item, each item is first Binomial ( m , 1 2 ) \operatorname{Binomial}(m,\frac{1}{2}) Binomial ( m , 2 1 ) times, so the slot fraction deviates from 1 2 \frac{1}{2} 2 1 by O p ( m − 1 / 2 ) O_p(m^{-1/2}) O p ( m − 1/2 ) (for m = 40 m=40 m = 40 : standard deviation 0.079, matching the realized imbalance). Hence Δ s ^ i = O p ( δ / m ) \Delta\widehat{s}_i=O_p(\delta/\sqrt{m}) Δ s i = O p ( δ / m ) : the effect of any change in β \beta β on any score is reduced by the square root of the number of comparisons, because presentation order is randomized.
Two remarks. First, the cancellation is not exact: under randomization the two presentation orders of a pair carry weights σ ′ ( Δ + β ) \sigma'(\Delta+\beta) σ ′ ( Δ + β ) and σ ′ ( Δ − β ) \sigma'(\Delta-\beta) σ ′ ( Δ − β ) , whose signed average is 1 2 ( σ ′ ( Δ + β ) − σ ′ ( Δ − β ) ) = β σ ′ ′ ( Δ ) + O ( β 3 ) \frac{1}{2}\bigl(\sigma'(\Delta+\beta)-\sigma'(\Delta-\beta)\bigr)=\beta\sigma''(\Delta)+O(\beta^3) 2 1 ( σ ′ ( Δ + β ) − σ ′ ( Δ − β ) ) = β σ ′′ ( Δ ) + O ( β 3 ) ; the numerator of (B.2) thus retains a deterministic residual of order O ( β ) O(\beta) O ( β ) per pair, on top of the O p ( m − 1 / 2 ) O_p(m^{-1/2}) O p ( m − 1/2 ) stochastic imbalance. Both are negligible here (β ^ \widehat{\beta} β and ∣ σ ′ ′ ∣ |\sigma''| ∣ σ ′′ ∣ are small), but the orthogonality is first-order, not exact. Second, the citation channel contains no position term at all (the gate is σ ( f ( Δ ) + γ k ) \sigma(f(\Delta)+\gamma_k) σ ( f ( Δ ) + γ k ) ), so the information that citations carry about score magnitudes is untouched by β \beta β under either specification.
Empirical check. When we refit the data of Section 4 with per-feature β k \beta_k β k , the positional bias estimates differ significantly across features, yet the scores are essentially unchanged : the median absolute shift is 1% of the per-feature score spread (90th percentile: 5%). The residual movement is attributable to the imperfect slot balance: across features, the score shift correlates with ∣ β ^ k ∣ |\widehat{\beta}_k| ∣ β k ∣ at r = 0.87 r=0.87 r = 0.87 , as the cancellation argument predicts.
We therefore retain the shared β \beta β in the main model; the per-feature β ^ k \widehat{\beta}_k β k remain interpretable as an audit of the judge (features differ in their susceptibility to position bias) but the design prevents this heterogeneity from reaching the scores.
Uncertainty quantification
Confidence intervals
The estimator θ ^ \widehat{\theta} θ is an M-estimator, which lets us adopt the weaker, standard view from robust statistics [38 , 39 ] to account for potential model misspecifications in our estimates. θ ^ \widehat{\theta} θ estimates the pseudo-true parameter θ ∗ \theta^* θ ∗ (which is the Kullback–Leibler projection of the judge’s actual comparison distribution onto our model class). In the regime where the item set is held fixed and the number N N N of edges of the comparison graph grows, the usual M-estimation argument yields
N ( θ ^ − θ ∗ ) → d N → ∞ N ( 0 , H − 1 G H − 1 ) , H = E [ ∇ θ 2 L ( θ ∗ ) ] , G = E [ ∇ θ L ( θ ∗ ) ∇ θ L ( θ ∗ ) T ] . (12) \sqrt{N}(\widehat{\theta}-\theta^*) \xrightarrow{d}_{N\to\infty} \mathcal{N}(0,H^{-1}GH^{-1}), \qquad H=\mathbb{E}\left[\nabla_{\theta}^{2}\mathcal{L}(\theta^*)\right], \qquad G=\mathbb{E}\left[\nabla_{\theta}\mathcal{L}(\theta^*)\nabla_{\theta}\mathcal{L}(\theta^*)^{\mathsf{T}}\right]. \tag*{(12)} N ( θ − θ ∗ ) d N → ∞ N ( 0 , H − 1 G H − 1 ) , H = E [ ∇ θ 2 L ( θ ∗ ) ] , G = E [ ∇ θ L ( θ ∗ ) ∇ θ L ( θ ∗ ) T ] . (12)
estimated by their empirical counterparts H ^ \widehat{H} H (average Hessian) and G ^ \widehat{G} G (average outer product of per-comparison gradients) at θ ^ \widehat{\theta} θ . The resulting sandwich covariance Σ ^ s a n d = 1 N H ^ − 1 G ^ H ^ − 1 \widehat{\Sigma}^{\mathrm{sand}}=\frac{1}{N}\widehat{H}^{-1}\widehat{G}\widehat{H}^{-1} Σ sand = N 1 H − 1 G H − 1 coincides with the inverse Fisher information when the model is correct (by the information-matrix equality G = H G=H G = H ), and remains an asymptotically valid covariance for the projection θ ∗ \theta^* θ ∗ when it is not. Hence, the confidence intervals on scores are statements about this projection, namely the score configuration that best summarizes the judge’s comparisons within our model class.
Reliability diagnostics
Since citation is sporadic, there may be features that are not cited enough or whose winners are inconsistent so that the scores are not precisely pinned down by the data. One advantage of our method is that it rests on a statistical model: we can hence inspect the condition numbers of blocks of the Hessian to diagnose poorly identified features [18 ] . Indeed, for each feature k k k , the unpenalized score block of the Fisher information is a weighted Laplacian L k L_k L k . By computing:
κ k = λ max ( L k ) λ 2 ( L k ) (13) \kappa_k = \frac{\lambda_{\max}(L_k)}{\lambda_2(L_k)} \tag*{(13)} κ k = λ 2 ( L k ) λ m a x ( L k ) (13)
and keeping features that have κ ≤ κ max \kappa\le\kappa_{\max} κ ≤ κ m a x (where κ max \kappa_{\max} κ m a x is specified a priori) we can filter out poorly identified features. Features that have κ > κ max \kappa> \kappa_{\max} κ > κ m a x are called unidentified .
Moreover, an interesting quantity to compute is the spread of the scores, as it tells us how markedly items differ along each feature in the dataset. However, a high spread could also be due to noisy estimates. We can hence denoise the estimated spreads of the scores by subtracting the analytic estimation variance:
ν k : = 1 n Tr ( Σ ^ k s a n d ) (14) \nu_k := \frac{1}{n}\operatorname{Tr}\left(\widehat{\Sigma}^{\mathrm{sand}}_k\right) \tag*{(14)} ν k := n 1 Tr ( Σ k sand ) (14)
where Σ ^ k s a n d \widehat{\Sigma}^{\mathrm{sand}}_k Σ k sand is the k k k -th score block of the sandwich covariance. Thus ρ k : = Var ( s ^ k ) − ν k Var ( s ^ k ) \rho_k := \frac{\operatorname{Var}(\hat{s}_k)-\nu_k}{\operatorname{Var}(\hat{s}_k)} ρ k := Var ( s ^ k ) Var ( s ^ k ) − ν k is the fraction of observed score spread attributable to genuine between-item signal. Keeping features with ρ k > ρ min \rho_k > \rho_{\min} ρ k > ρ m i n (where ρ min \rho_{\min} ρ m i n is specified a priori) filters out unreliable features.
Having fitted the model a first time, spotted unidentified and unreliable features, and possibly chosen features that we want to drop or fuse, we can then refit the model and obtain scores on reliable (and chosen) features.
Model, likelihood, score, and Hessian
Model
We study the model where there is only one feature (case K = 1 K = 1 K = 1 ). There are n + 1 n + 1 n + 1 items indexed by { 0 , 1 , … , n } \{0,1,\ldots,n\} { 0 , 1 , … , n } . For each unordered pair { i , j } \{i,j\} { i , j } with $0 \le i < j \le n$ we observe m i j m_{ij} m ij comparisons, indexed k = 1 , … , m i j k = 1,\ldots,m_{ij} k = 1 , … , m ij . Each comparison yields a trinomial outcome
r i j k ∈ { 0 , + 1 , − 1 } : 0 = “pair not cited” , + 1 = “i wins” , − 1 = “j wins” . r_{ijk} \in\{0,+1,-1\}:
\qquad0 = \text{“pair not cited”},\quad+1 = \text{“i wins”},\quad-1 = \text{“j wins”}. r ij k ∈ { 0 , + 1 , − 1 } : 0 = “pair not cited” , + 1 = “i wins” , − 1 = “j wins” .
We encode it by a citation indicator and a direction indicator,
c i j k : = 1 { r i j k ≠ 0 } ∈ { 0 , 1 } , a i j k 1 : = 1 { r i j k = + 1 } , a i j k − 1 : = 1 { r i j k = − 1 } , c_{ijk} := \mathbb{1}_{\{r_{ijk}\ne0\}} \in\{0,1\},\qquad
a^1_{ijk} := \mathbb{1}_{\{r_{ijk}=+1\}},\qquad
a^{-1}_{ijk} := \mathbb{1}_{\{r_{ijk}=-1\}}, c ij k := 1 { r ij k = 0 } ∈ { 0 , 1 } , a ij k 1 := 1 { r ij k = + 1 } , a ij k − 1 := 1 { r ij k = − 1 } ,
so that a i j k 1 + a i j k − 1 = c i j k a^1_{ijk} + a^{-1}_{ijk} = c_{ijk} a ij k 1 + a ij k − 1 = c ij k .
Each item i i i carries a latent merit s i ∈ R s_i \in\mathbb{R} s i ∈ R ; we write s = ( s 0 , … , s n ) T ∈ R n + 1 \mathbf{s} = (s_0,\ldots,s_n)^{\mathsf{T}} \in\mathbb{R}^{n+1} s = ( s 0 , … , s n ) T ∈ R n + 1 for the merit vector (suppressing the dependence on n n n when clear) and Δ i j : = s i − s j \Delta_{ij} := s_i - s_j Δ ij := s i − s j for the gap. There are two fixed-dimension structural parameters: the gate coefficient γ ∈ R p γ \gamma\in\mathbb{R}^{p_\gamma} γ ∈ R p γ and the direction (position-bias) coefficient β ∈ R p β \beta\in\mathbb{R}^{p_\beta} β ∈ R p β . Each comparison carries covariates Y i j k ∈ R p γ \mathbf{Y}_{ijk} \in\mathbb{R}^{p_\gamma} Y ij k ∈ R p γ (gate) and Z i j k ∈ R p β \mathbf{Z}_{ijk} \in\mathbb{R}^{p_\beta} Z ij k ∈ R p β (direction), with parities
Y j i k = Y i j k (even) , Z j i k = − Z i j k (odd) , Δ j i = − Δ i j . (15) \mathbf{Y}_{jik} = \mathbf{Y}_{ijk}\ \text{(even)},\qquad
\mathbf{Z}_{jik} = -\mathbf{Z}_{ijk}\ \text{(odd)},\qquad
\Delta_{ji} = -\Delta_{ij}. \tag*{(15)} Y j ik = Y ij k (even) , Z j ik = − Z ij k (odd) , Δ j i = − Δ ij . (15)
Let σ ( x ) = e x / ( 1 + e x ) \sigma(x) = e^x/(1 + e^x) σ ( x ) = e x / ( 1 + e x ) and let f : R → R f:\mathbb{R}\to\mathbb{R} f : R → R be a fixed even link. Define the two linear predictors and their sigmoids,
η i j k : = f ( Δ i j ) + Y i j k T γ , g i j k : = σ ( η i j k ) ; θ i j k : = Δ i j + Z i j k T β , d i j k : = σ ( θ i j k ) . (16) \eta_{ijk} := f(\Delta_{ij}) + \mathbf{Y}_{ijk}^{\mathsf{T}}\gamma,\qquad
g_{ijk} := \sigma(\eta_{ijk});\qquad
\theta_{ijk} := \Delta_{ij} + \mathbf{Z}_{ijk}^{\mathsf{T}}\beta,\qquad
d_{ijk} := \sigma(\theta_{ijk}). \tag*{(16)} η ij k := f ( Δ ij ) + Y ij k T γ , g ij k := σ ( η ij k ) ; θ ij k := Δ ij + Z ij k T β , d ij k := σ ( θ ij k ) . (16)
The model is, conditionally independently across all ( i , j , k ) (i,j,k) ( i , j , k ) ,
P ( r i j k = 0 ) = 1 − g i j k , P ( r i j k = + 1 ) = g i j k d i j k , P ( r i j k = − 1 ) = g i j k ( 1 − d i j k ) . (17) \mathbb{P}(r_{ijk}=0) = 1 - g_{ijk},\qquad
\mathbb{P}(r_{ijk}=+1) = g_{ijk}d_{ijk},\qquad
\mathbb{P}(r_{ijk}=-1) = g_{ijk}(1-d_{ijk}). \tag*{(17)} P ( r ij k = 0 ) = 1 − g ij k , P ( r ij k = + 1 ) = g ij k d ij k , P ( r ij k = − 1 ) = g ij k ( 1 − d ij k ) . (17)
Equivalently c i j k ∼ Bernoulli ( g i j k ) c_{ijk} \sim\operatorname{Bernoulli}(g_{ijk}) c ij k ∼ Bernoulli ( g ij k ) , and conditionally on c i j k = 1 c_{ijk}=1 c ij k = 1 , a i j k 1 ∼ Bernoulli ( d i j k ) a^1_{ijk} \sim\operatorname{Bernoulli}(d_{ijk}) a ij k 1 ∼ Bernoulli ( d ij k ) . The parities (B.6) make (B.8) invariant under i ↔ j i \leftrightarrow j i ↔ j (the gate is invariant; the direction flips), since f f f is even while Δ \Delta Δ and Z Z Z are odd.
Identification (the 1 ⊥ 1^\perp 1 ⊥ gauge). Adding a constant to every s i s_i s i leaves (B.8) unchanged (only gaps enter), so we impose the gauge s ∈ 1 ⊥ : = { x ∈ R n + 1 : 1 T x = 0 } \mathbf{s} \in1^\perp:= \{\mathbf{x} \in\mathbb{R}^{n+1}: \mathbf{1}^{\mathsf{T}}\mathbf{x}=0\} s ∈ 1 ⊥ := { x ∈ R n + 1 : 1 T x = 0 } throughout. This choice is natural here: every merit-direction score and Hessian block below is a linear combination of the centered vectors ω i j : = e i − e j ∈ 1 ⊥ \boldsymbol{\omega}_{ij} := e_i - e_j \in1^\perp ω ij := e i − e j ∈ 1 ⊥ , so the gauge is preserved automatically by the estimating equations and by the gradient iterations of Appendix B.11, with no explicit projection.
Log-likelihood
From (B.8), the per-comparison log-likelihood is
ℓ i j k = c i j k log g i j k + ( 1 − c i j k ) log ( 1 − g i j k ) + a i j k 1 log d i j k + a i j k − 1 log ( 1 − d i j k ) , \ell_{ijk}=c_{ijk}\log g_{ijk}+(1-c_{ijk})\log(1-g_{ijk})+a^{1}_{ijk}\log d_{ijk}+a^{-1}_{ijk}\log(1-d_{ijk}), ℓ ij k = c ij k log g ij k + ( 1 − c ij k ) log ( 1 − g ij k ) + a ij k 1 log d ij k + a ij k − 1 log ( 1 − d ij k ) ,
and the total is
ℓ ( s , γ , β ) = ∑ 0 ≤ i < j ≤ n ∑ k = 1 m i j ℓ i j k . (18) \ell(s,\gamma,\beta)=\sum_{0\leq i<j\leq n}\sum_{k=1}^{m_{ij}}\ell_{ijk}. \tag*{(18)} ℓ ( s , γ , β ) = 0 ≤ i < j ≤ n ∑ k = 1 ∑ m ij ℓ ij k . (18)
Scores
Using σ ′ = σ ( 1 − σ ) \sigma'=\sigma(1-\sigma) σ ′ = σ ( 1 − σ ) and the chain rule through (B.7) (∂ η i j k ∂ Δ i j = f ′ ( Δ i j ) \frac{\partial\eta_{ijk}}{\partial\Delta_{ij}}=f'(\Delta_{ij}) ∂ Δ ij ∂ η ij k = f ′ ( Δ ij ) , ∂ θ i j k ∂ Δ i j = 1 \frac{\partial\theta_{ijk}}{\partial\Delta_{ij}}=1 ∂ Δ ij ∂ θ ij k = 1 ), the per-comparison scores are
∂ ℓ i j k ∂ η i j k = c i j k − g i j k , ∂ ℓ i j k ∂ θ i j k = a i j k 1 − c i j k d i j k . \frac{\partial\ell_{ijk}}{\partial\eta_{ijk}}=c_{ijk}-g_{ijk},\qquad\frac{\partial\ell_{ijk}}{\partial\theta_{ijk}}=a^{1}_{ijk}-c_{ijk}d_{ijk}. ∂ η ij k ∂ ℓ ij k = c ij k − g ij k , ∂ θ ij k ∂ ℓ ij k = a ij k 1 − c ij k d ij k .
The three score blocks are
∇ s ℓ = ∑ i < j ∑ k [ ( c i j k − g i j k ) f ′ ( Δ i j ) + ( a i j k 1 − c i j k d i j k ) ] ω i j ∈ R n + 1 , (19) \nabla_s\ell=\sum_{i<j}\sum_k\left[(c_{ijk}-g_{ijk})f'(\Delta_{ij})+(a^{1}_{ijk}-c_{ijk}d_{ijk})\right]\omega_{ij}\in\mathbb{R}^{n+1}, \tag*{(19)} ∇ s ℓ = i < j ∑ k ∑ [ ( c ij k − g ij k ) f ′ ( Δ ij ) + ( a ij k 1 − c ij k d ij k ) ] ω ij ∈ R n + 1 , (19)
∇ γ ℓ = ∑ i < j ∑ k ( c i j k − g i j k ) Y i j k ∈ R p γ , (20) \nabla_\gamma\ell=\sum_{i<j}\sum_k(c_{ijk}-g_{ijk})Y_{ijk}\in\mathbb{R}^{p_\gamma}, \tag*{(20)} ∇ γ ℓ = i < j ∑ k ∑ ( c ij k − g ij k ) Y ij k ∈ R p γ , (20)
∇ β ℓ = ∑ i < j ∑ k ( a i j k 1 − c i j k d i j k ) Z i j k ∈ R p β . (21) \nabla_\beta\ell=\sum_{i<j}\sum_k(a^{1}_{ijk}-c_{ijk}d_{ijk})Z_{ijk}\in\mathbb{R}^{p_\beta}. \tag*{(21)} ∇ β ℓ = i < j ∑ k ∑ ( a ij k 1 − c ij k d ij k ) Z ij k ∈ R p β . (21)
Define the per-coordinate merit score
G a ( s , γ , β ) : = − ∂ ℓ ∂ s a = − ∑ j ≠ a ∑ k ψ a j k , ψ a j k : = ( c a j k − g a j k ) f ′ ( Δ a j ) + ( a a j k 1 − c a j k d a j k ) . (22) G_a(s,\gamma,\beta):=-\frac{\partial\ell}{\partial s_a}=-\sum_{j\ne a}\sum_k\psi_{ajk},\qquad\psi_{ajk}:=(c_{ajk}-g_{ajk})f'(\Delta_{aj})+(a^{1}_{ajk}-c_{ajk}d_{ajk}). \tag*{(22)} G a ( s , γ , β ) := − ∂ s a ∂ ℓ = − j = a ∑ k ∑ ψ aj k , ψ aj k := ( c aj k − g aj k ) f ′ ( Δ aj ) + ( a aj k 1 − c aj k d aj k ) . (22)
Collecting the coordinates, the merit score map is
G ( s , γ , β ) : = ∇ s ( − ℓ ) = ( G 0 , … , G n ) ⊤ = − ∑ i < j ∑ k [ ( c i j k − g i j k ) f ′ ( Δ i j ) + ( a i j k 1 − c i j k d i j k ) ] ω i j , (23) G(s,\gamma,\beta):=\nabla_s(-\ell)=(G_0,\ldots,G_n)^\top=-\sum_{i<j}\sum_k\left[(c_{ijk}-g_{ijk})f'(\Delta_{ij})+(a^{1}_{ijk}-c_{ijk}d_{ijk})\right]\omega_{ij}, \tag*{(23)} G ( s , γ , β ) := ∇ s ( − ℓ ) = ( G 0 , … , G n ) ⊤ = − i < j ∑ k ∑ [ ( c ij k − g ij k ) f ′ ( Δ ij ) + ( a ij k 1 − c ij k d ij k ) ] ω ij , (23)
which takes values in 1 ⊥ \mathbf{1}^{\perp} 1 ⊥ because every ω i j = e i − e j \omega_{ij}=e_i-e_j ω ij = e i − e j is centered. When ( γ , β ) (\gamma,\beta) ( γ , β ) are held fixed we write G γ , β ( s ) : = G ( s , γ , β ) G_{\gamma,\beta}(s):=G(s,\gamma,\beta) G γ , β ( s ) := G ( s , γ , β ) , viewed as a map R n + 1 → 1 ⊥ ⊂ R n + 1 \mathbb{R}^{n+1}\to\mathbf{1}^{\perp}\subset\mathbb{R}^{n+1} R n + 1 → 1 ⊥ ⊂ R n + 1 . The MLE solves G ( s , γ , β ) = 0 G(s,\gamma,\beta)=0 G ( s , γ , β ) = 0 on 1 ⊥ \mathbf{1}^{\perp} 1 ⊥ together with (B.11) and (B.12).
Hessian
Set w η , i j k : = g i j k ( 1 − g i j k ) w_{\eta,ijk}:=g_{ijk}(1-g_{ijk}) w η , ij k := g ij k ( 1 − g ij k ) and w θ , i j k : = d i j k ( 1 − d i j k ) w_{\theta,ijk}:=d_{ijk}(1-d_{ijk}) w θ , ij k := d ij k ( 1 − d ij k ) . Differentiating once more (θ \theta θ is linear so ∂ 2 θ ∂ Δ 2 = 0 \frac{\partial^2\theta}{\partial\Delta^2}=0 ∂ Δ 2 ∂ 2 θ = 0 , while ∂ 2 η ∂ Δ 2 = f ′ ′ \frac{\partial^2\eta}{\partial\Delta^2}=f'' ∂ Δ 2 ∂ 2 η = f ′′ ) gives the per-comparison merit curvature
W i j k : = − ∂ 2 ℓ i j k ∂ Δ 2 = w η , i j k f ′ ( Δ i j ) 2 + c i j k w θ , i j k − ( c i j k − g i j k ) f ′ ′ ( Δ i j ) , (24) W_{ijk}:=-\frac{\partial^2\ell_{ijk}}{\partial\Delta^2}=w_{\eta,ijk}f'(\Delta_{ij})^2+c_{ijk}w_{\theta,ijk}-\bigl(c_{ijk}-g_{ijk}\bigr)f''(\Delta_{ij}), \tag*{(24)} W ij k := − ∂ Δ 2 ∂ 2 ℓ ij k = w η , ij k f ′ ( Δ ij ) 2 + c ij k w θ , ij k − ( c ij k − g ij k ) f ′′ ( Δ ij ) , (24)
and the full Hessian of − ℓ -\ell − ℓ has the block structure (all sums ∑ i < j ∑ k \sum_{i<j}\sum_k ∑ i < j ∑ k )
J ( s , γ , β ) : = ∇ s 2 ( − ℓ ) = ∑ W i j k ω i j ω i j ⊤ (full merit block, ( n + 1 ) × ( n + 1 ) ) (25) J(s,\gamma,\beta):=\nabla_s^2(-\ell)=\sum W_{ijk}\,\omega_{ij}\omega_{ij}^{\top}\qquad\text{(full merit block, }(n+1)\times(n+1)\text{)} \tag*{(25)} J ( s , γ , β ) := ∇ s 2 ( − ℓ ) = ∑ W ij k ω ij ω ij ⊤ (full merit block, ( n + 1 ) × ( n + 1 ) ) (25)
A γ γ : = ∇ γ γ 2 ( − ℓ ) = ∑ w η , i j k Y i j k Y i j k ⊤ , (26) A_{\gamma\gamma}:=\nabla_{\gamma\gamma}^2(-\ell)=\sum w_{\eta,ijk}Y_{ijk}Y_{ijk}^{\top}, \tag*{(26)} A γ γ := ∇ γ γ 2 ( − ℓ ) = ∑ w η , ij k Y ij k Y ij k ⊤ , (26)
A β β : = ∇ β β 2 ( − ℓ ) = ∑ c i j k w θ , i j k Z i j k Z i j k ⊤ , (27) A_{\beta\beta}:=\nabla_{\beta\beta}^2(-\ell)=\sum c_{ijk}w_{\theta,ijk}Z_{ijk}Z_{ijk}^{\top}, \tag*{(27)} A β β := ∇ β β 2 ( − ℓ ) = ∑ c ij k w θ , ij k Z ij k Z ij k ⊤ , (27)
B γ : = ∇ s γ 2 ( − ℓ ) = ∑ w η , i j k f ′ ( Δ i j ) ω i j Y i j k ⊤ . (28) B_{\gamma}:=\nabla_{s\gamma}^2(-\ell)=\sum w_{\eta,ijk}f'(\Delta_{ij})\omega_{ij}Y_{ijk}^{\top}. \tag*{(28)} B γ := ∇ s γ 2 ( − ℓ ) = ∑ w η , ij k f ′ ( Δ ij ) ω ij Y ij k ⊤ . (28)
B β : = ∇ s β 2 ( − ℓ ) = ∑ c i j k w θ , i j k ω i j Z i j k T , (29) B_{\beta} := \nabla^{2}_{s_{\beta}}(-\ell) = \sum c_{ijk} w_{\theta,ijk}\omega_{ij} Z_{ijk}^{\mathsf{T}}, \tag*{(29)} B β := ∇ s β 2 ( − ℓ ) = ∑ c ij k w θ , ij k ω ij Z ij k T , (29)
∇ γ β 2 ( − ℓ ) = 0. (30) \nabla^{2}_{\gamma\beta}(-\ell) = 0. \tag*{(30)} ∇ γ β 2 ( − ℓ ) = 0. (30)
Blocks (B.17)–(B.18) are positive semidefinite; the cross block (B.21) vanishes because, given the predictors, the two channels share no randomness. The merit block (B.16) is not PSD pathwise: the saboteur in (B.15) is signed, so a single W i j k W_{ijk} W ij k can be negative, e.g. on a cited near-tie, where f ′ ≈ 0 f' \approx0 f ′ ≈ 0 and c = 1 c = 1 c = 1 , W ≈ w θ − ( 1 − g ) f ′ ′ W \approx w_{\theta} - (1-g)f'' W ≈ w θ − ( 1 − g ) f ′′ . This non-convexity is the central difficulty. The two off-diagonal blocks B γ , B β B_{\gamma}, B_{\beta} B γ , B β are named here for later use (Lemma 8 and Corollary 1).
Notation. Following the template conventions, bold lowercase letters denote vectors ( s , u , ω i j ) (s,u,\omega_{ij}) ( s , u , ω ij ) and bold uppercase letters denote matrices ( J , A , B , V , Σ ) (J,A,B,V,\Sigma) ( J , A , B , V , Σ ) ; in keeping with the Bradley–Terry literature we also write in bold uppercase the per-comparison covariate vectors Y i j k ∈ R p γ Y_{ijk} \in\mathbb{R}^{p_{\gamma}} Y ij k ∈ R p γ , Z i j k ∈ R p β Z_{ijk} \in\mathbb{R}^{p_{\beta}} Z ij k ∈ R p β and the vector-valued score maps G , D , T G,D,T G , D , T . Calligraphic letters denote balls ( B ) (\mathcal{B}) ( B ) , events ( E ) (\mathcal{E}) ( E ) , σ \sigma σ -algebras ( F ) (\mathcal{F}) ( F ) and nets ( N ) (\mathcal{N}) ( N ) . Starred quantities ( s ∗ , γ ∗ , β ∗ , g i j k ∗ , … ) (s^{*},\gamma^{*},\beta^{*},g^{*}_{ijk},\ldots) ( s ∗ , γ ∗ , β ∗ , g ij k ∗ , … ) are evaluated at the truth. Universal constants may change from line to line.
Norms. For x ∈ R d x \in\mathbb{R}^{d} x ∈ R d , ∥ x ∥ ∞ = max a ∣ x a ∣ \lVert x\rVert_{\infty} = \max_{a}|x_{a}| ∥ x ∥ ∞ = max a ∣ x a ∣ and ∥ x ∥ 2 = ( ∑ a x a 2 ) 1 / 2 \lVert x\rVert_{2} = \left(\sum_{a}x_{a}^{2}\right)^{1/2} ∥ x ∥ 2 = ( ∑ a x a 2 ) 1/2 . For a matrix A = ( A a b ) A=(A_{ab}) A = ( A ab ) we write ∥ A ∥ max = max a , b ∣ A a b ∣ \lVert A\rVert_{\max} = \max_{a,b}|A_{ab}| ∥ A ∥ m a x = max a , b ∣ A ab ∣ (entrywise) and use triple bars for the induced operator norms: ∥ ∥ ∥ A ∥ ∥ ∥ 2 \lVert\!\lVert\!\lVert A\rVert\!\rVert\!\rVert_{2} ∥ ∥ ∥ A ∥ ∥ ∥ 2 is the spectral norm (induced by ∥ ⋅ ∥ 2 \lVert\cdot\rVert_{2} ∥ ⋅ ∥ 2 ) and ∥ ∥ ∥ A ∥ ∥ ∥ ∞ = max a ∑ b ∣ A a b ∣ \lVert\!\lVert\!\lVert A\rVert\!\rVert\!\rVert_{\infty} = \max_{a}\sum_{b}|A_{ab}| ∥ ∥ ∥ A ∥ ∥ ∥ ∞ = max a ∑ b ∣ A ab ∣ is the maximum absolute row sum (induced by ∥ ⋅ ∥ ∞ \lVert\cdot\rVert_{\infty} ∥ ⋅ ∥ ∞ ). For symmetric A A A , ∥ ∥ ∥ A ∥ ∥ ∥ 2 ≤ ∥ ∥ ∥ A ∥ ∥ ∥ ∞ \lVert\!\lVert\!\lVert A\rVert\!\rVert\!\rVert_{2} \le\lVert\!\lVert\!\lVert A\rVert\!\rVert\!\rVert_{\infty} ∥ ∥ ∥ A ∥ ∥ ∥ 2 ≤ ∥ ∥ ∥ A ∥ ∥ ∥ ∞ . For symmetric A A A acting on the centered subspace, λ min , ⊥ ( A ) \lambda_{\min,\perp}(A) λ m i n , ⊥ ( A ) is the smallest eigenvalue restricted to 1 ⊥ = { x : 1 T x = 0 } \mathbf{1}^{\perp} = \{x : \mathbf{1}^{\mathsf{T}}x = 0\} 1 ⊥ = { x : 1 T x = 0 } .
Restricted spectral norm. For a symmetric matrix M M M that maps 1 ⊥ \mathbf{1}^{\perp} 1 ⊥ into itself, write
∥ ∥ ∥ M ∥ ∥ ∥ 2 , ⊥ : = sup x ∈ 1 ⊥ ∥ x ∥ 2 = 1 ∥ M x ∥ 2 \lVert\!\lVert\!\lVert M\rVert\!\rVert\!\rVert_{2,\perp}
:= \sup_{\substack{x\in\mathbf{1}^{\perp}\\ \lVert x\rVert_{2}=1}}\lVert Mx\rVert_{2} ∥ ∥ ∥ M ∥ ∥ ∥ 2 , ⊥ := x ∈ 1 ⊥ ∥ x ∥ 2 = 1 sup ∥ M x ∥ 2
for its spectral norm restricted to the centered subspace. In particular, if M = I − η N M = I-\eta N M = I − η N with N N N symmetric and PSD on 1 ⊥ \mathbf{1}^{\perp} 1 ⊥ and 0 ≤ η λ ≤ 1 0 \le\eta\lambda\le1 0 ≤ η λ ≤ 1 for every eigenvalue λ \lambda λ of N ∣ 1 ⊥ N|_{\mathbf{1}^{\perp}} N ∣ 1 ⊥ , then
∥ ∥ ∥ I − η N ∥ ∥ ∥ 2 , ⊥ = 1 − η λ min , ⊥ ( N ) . (31) \lVert\!\lVert\!\lVert I-\eta N\rVert\!\rVert\!\rVert_{2,\perp}
= 1-\eta\lambda_{\min,\perp}(N). \tag*{(31)} ∥ ∥ ∥ I − η N ∥ ∥ ∥ 2 , ⊥ = 1 − η λ m i n , ⊥ ( N ) . (31)
Balls. For x ∈ R d x \in\mathbb{R}^{d} x ∈ R d and r > 0 r>0 r > 0 , B 2 ( x , r ) B_{2}(x,r) B 2 ( x , r ) and B ∞ ( x , r ) B_{\infty}(x,r) B ∞ ( x , r ) denote the open ℓ 2 \ell_{2} ℓ 2 and ℓ ∞ \ell_{\infty} ℓ ∞ balls of radius r r r centered at x x x , and B ‾ 2 , B ‾ ∞ \overline{B}_{2},\overline{B}_{\infty} B 2 , B ∞ their closures. For the fixed-dimensional structural parameters we write B ( x , r ) \mathcal{B}(x,r) B ( x , r ) for the open ℓ 2 \ell_{2} ℓ 2 ball.
Sigmoid bounds. For all x x x : 0 < σ ′ ( x ) = σ ( 1 − σ ) ≤ 1 4 0 < \sigma'(x) = \sigma(1-\sigma) \le\frac{1}{4} 0 < σ ′ ( x ) = σ ( 1 − σ ) ≤ 4 1 , ∣ σ ′ ′ ∣ ≤ 1 4 |\sigma''| \le\frac{1}{4} ∣ σ ′′ ∣ ≤ 4 1 , and d d x [ σ ( 1 − σ ) ] = σ ( 1 − σ ) ( 1 − 2 σ ) \frac{d}{dx}[\sigma(1-\sigma)] = \sigma(1-\sigma)(1-2\sigma) d x d [ σ ( 1 − σ )] = σ ( 1 − σ ) ( 1 − 2 σ ) has absolute value ≤ 1 4 \le\frac{1}{4} ≤ 4 1 .
We import the following standard tools.
Fact 1 (Vector Bernstein). For independent mean-zero x ℓ ∈ R d x_{\ell} \in\mathbb{R}^{d} x ℓ ∈ R d with ∥ x ℓ ∥ 2 ≤ R \lVert x_{\ell}\rVert_{2} \le R ∥ x ℓ ∥ 2 ≤ R and ∑ ℓ E ∥ x ℓ ∥ 2 2 ≤ ν \sum_{\ell}\mathbb{E}\lVert x_{\ell}\rVert_{2}^{2} \le\nu ∑ ℓ E ∥ x ℓ ∥ 2 2 ≤ ν ,
P ( ∥ ∑ ℓ x ℓ ∥ 2 ≥ t ) ≤ 2 ( d + 1 ) exp ( − t 2 2 ν + 2 R t / 3 ) . \mathbb{P}\left(\left\lVert\sum_{\ell}x_{\ell}\right\rVert_{2}\ge t\right)
\le2(d+1)\exp\left(-\frac{t^{2}}{2\nu+2Rt/3}\right). P ( ℓ ∑ x ℓ 2 ≥ t ) ≤ 2 ( d + 1 ) exp ( − 2 ν + 2 R t /3 t 2 ) .
Fact 2 (Matrix Bernstein). For independent symmetric mean-zero X ℓ ∈ R d × d X_{\ell} \in\mathbb{R}^{d\times d} X ℓ ∈ R d × d with ∥ ∥ ∥ X ℓ ∥ ∥ ∥ 2 ≤ R \lVert\!\lVert\!\lVert X_{\ell}\rVert\!\rVert\!\rVert_{2} \le R ∥ ∥ ∥ X ℓ ∥ ∥ ∥ 2 ≤ R and \left\lVert\!\left\lVert\!\left\lVert\sum_{\ell}\mathbb{E}X_{\ell}^{2}\right\rVert\!\right\!\right\rVert_{2} \le\nu ,
\mathbb{P}\left(\left\lVert\!\left\lVert\!\left\lVert\sum_{\ell}X_{\ell}\right\rVert\!\right\!\right\rVert_{2}\ge t\right)
\le2d\exp\left(-\frac{t^{2}}{2\nu+2Rt/3}\right).
Fact 3 (Hoeffding). For independent x ℓ ∈ [ α ℓ , β ℓ ] x_{\ell} \in[\alpha_{\ell},\beta_{\ell}] x ℓ ∈ [ α ℓ , β ℓ ] ,
P ( ∣ ∑ ℓ ( x ℓ − E x ℓ ) ∣ ≥ t ) ≤ 2 exp ( − 2 t 2 ∑ ℓ ( β ℓ − α ℓ ) 2 ) . \mathbb{P}\left(\left|\sum_{\ell}(x_{\ell}-\mathbb{E}x_{\ell})\right|\ge t\right)
\le2\exp\left(-\frac{2t^{2}}{\sum_{\ell}(\beta_{\ell}-\alpha_{\ell})^{2}}\right). P ( ℓ ∑ ( x ℓ − E x ℓ ) ≥ t ) ≤ 2 exp ( − ∑ ℓ ( β ℓ − α ℓ ) 2 2 t 2 ) .
Fact 4 (Degree-theoretic existence; Ortega–Rheinboldt 1970). If Ω ⊂ R p \Omega\subset\mathbb{R}^{p} Ω ⊂ R p is open and bounded, F : Ω ‾ → R p F:\overline{\Omega}\to\mathbb{R}^{p} F : Ω → R p is continuous, and there is x 0 ∈ Ω x_{0}\in\Omega x 0 ∈ Ω with ( x − x 0 ) T F ( x ) ≥ 0 (x-x_{0})^{\mathsf{T}}F(x)\ge0 ( x − x 0 ) T F ( x ) ≥ 0 for all x ∈ ∂ Ω x\in\partial\Omega x ∈ ∂ Ω , then F F F has a zero in Ω ‾ \overline{\Omega} Ω .
Fact 5 (Gershgorin disc theorem). If A = ( A a b ) a , b = 1 p A=(A_{ab})_{a,b=1}^{p} A = ( A ab ) a , b = 1 p is a p × p p\times p p × p matrix, then
spec ( A ) ⊆ ⋃ a = 1 p { z ∈ C : ∣ z − A a a ∣ ≤ ∑ b ≠ a ∣ A a b ∣ } . \operatorname{spec}(A)\subseteq\bigcup_{a=1}^{p}\left\{z\in\mathbb{C}:|z-A_{aa}|\le\sum_{b\ne a}|A_{ab}|\right\}. spec ( A ) ⊆ a = 1 ⋃ p ⎩ ⎨ ⎧ z ∈ C : ∣ z − A aa ∣ ≤ b = a ∑ ∣ A ab ∣ ⎭ ⎬ ⎫ . In particular, λ max ( A ) ≤ max a ( A a a + ∑ b ≠ a ∣ A a b ∣ ) \lambda_{\max}(A)\le\max_{a}\left(A_{aa}+\sum_{b\ne a}|A_{ab}|\right) λ m a x ( A ) ≤ max a ( A aa + ∑ b = a ∣ A ab ∣ ) when A A A is symmetric.
Fact 6 (Schur complement criterion for positive definiteness; Haynsworth 1968). Let M M M be a symmetric matrix in block form
M = ( P Q Q T R ) , M=\begin{pmatrix}P&Q\\Q^{\mathsf{T}}&R\end{pmatrix}, M = ( P Q T Q R ) , with P P P symmetric and invertible, and let M / P : = R − Q T P − 1 Q M/P:=R-Q^{\mathsf{T}}P^{-1}Q M / P := R − Q T P − 1 Q denote the Schur complement of P P P in M M M . Then the inertias add,
In ( M ) = In ( P ) + In ( M / P ) , \operatorname{In}(M)=\operatorname{In}(P)+\operatorname{In}(M/P), In ( M ) = In ( P ) + In ( M / P ) , where In ( ⋅ ) = ( n + , n − , n 0 ) \operatorname{In}(\cdot)=(n_{+},n_{-},n_{0}) In ( ⋅ ) = ( n + , n − , n 0 ) counts the positive, negative and zero eigenvalues. In particular, if P ≻ 0 P\succ0 P ≻ 0 then
M ≻ 0 ⟺ M / P ≻ 0. M\succ0\quad\Longleftrightarrow\quad M/P\succ0. M ≻ 0 ⟺ M / P ≻ 0.
Assumptions and the population curvature floor
Assumption 1 (Dense bounded design). 1 ≤ m i j ≤ m ∗ 1\le m_{ij}\le m^{*} 1 ≤ m ij ≤ m ∗ for all 0 ≤ i < j ≤ n 0\le i<j\le n 0 ≤ i < j ≤ n , with m ∗ m^{*} m ∗ a fixed constant.
Assumption 2 (Compactness). There exist absolute constants C s , κ > 0 C_s,\kappa>0 C s , κ > 0 such that, for all n n n ,
∥ s ∗ ∥ ∞ ≤ C s , sup 0 ≤ i , j ≤ n , k ∥ Y i j k ∥ 2 ≤ κ , sup 0 ≤ i , j ≤ n , k ∥ Z i j k ∥ 2 ≤ κ . \lVert s^{*}\rVert_{\infty}\le C_s,\qquad
\sup_{0\le i,j\le n,\,k}\lVert Y_{ijk}\rVert_{2}\le\kappa,\qquad
\sup_{0\le i,j\le n,\,k}\lVert Z_{ijk}\rVert_{2}\le\kappa. ∥ s ∗ ∥ ∞ ≤ C s , 0 ≤ i , j ≤ n , k sup ∥ Y ij k ∥ 2 ≤ κ , 0 ≤ i , j ≤ n , k sup ∥ Z ij k ∥ 2 ≤ κ .
Assumption 3 (Smooth even gate). f f f is even and C 3 C^{3} C 3 , so that
F 1 : = sup ∣ u ∣ ≤ 2 C s + 1 ∣ f ′ ( u ) ∣ , F 2 : = sup ∣ u ∣ ≤ 2 C s + 1 ∣ f ′ ′ ( u ) ∣ , F 3 : = sup ∣ u ∣ ≤ 2 C s + 1 ∣ f ′ ′ ′ ( u ) ∣ F_{1}:=\sup_{|u|\le2C_s+1}|f'(u)|,\qquad
F_{2}:=\sup_{|u|\le2C_s+1}|f''(u)|,\qquad
F_{3}:=\sup_{|u|\le2C_s+1}|f'''(u)| F 1 := ∣ u ∣ ≤ 2 C s + 1 sup ∣ f ′ ( u ) ∣ , F 2 := ∣ u ∣ ≤ 2 C s + 1 sup ∣ f ′′ ( u ) ∣ , F 3 := ∣ u ∣ ≤ 2 C s + 1 sup ∣ f ′′′ ( u ) ∣
are finite.
Remark 1 (Working region). The supremum range ∣ u ∣ ≤ 2 C s + 1 |u|\le2C_s+1 ∣ u ∣ ≤ 2 C s + 1 in A3 is a fixed interval strictly containing the range [ − 2 C s , 2 C s ] [-2C_s,2C_s] [ − 2 C s , 2 C s ] of the population gaps. Throughout, the working region is the set { s : ∥ s ∥ ∞ ≤ C s + 1 2 } \{s:\lVert s\rVert_{\infty}\le C_s+\frac{1}{2}\} { s : ∥ s ∥ ∞ ≤ C s + 2 1 } : every merit vector s s s that arises below lies within o ( 1 ) o(1) o ( 1 ) of s ∗ s^{*} s ∗ , hence lies in the working region for n n n large; its gaps then obey ∣ Δ i j ∣ ≤ 2 C s + 1 |\Delta_{ij}|\le2C_s+1 ∣ Δ ij ∣ ≤ 2 C s + 1 , so the bounds F 1 , F 2 , F 3 F_1,F_2,F_3 F 1 , F 2 , F 3 apply to every gap encountered. The Hessian-geometry lemmas of Appendix B.10 are stated for ∥ s ∥ ∞ ≤ C s + 1 2 \lVert s\rVert_{\infty}\le C_s+\frac{1}{2} ∥ s ∥ ∞ ≤ C s + 2 1 accordingly.
Under A1–A3 the predictors η i j k ∗ , θ i j k ∗ \eta^{*}_{ijk},\theta^{*}_{ijk} η ij k ∗ , θ ij k ∗ lie in a fixed compact interval, so there are absolute constants 0 < g ≤ g ‾ < 1 0<g\le\overline{g}<1 0 < g ≤ g < 1 with g ≤ g i j k ∗ ≤ g ‾ g\le g^{*}_{ijk}\le\overline{g} g ≤ g ij k ∗ ≤ g , and likewise for d i j k ∗ d^{*}_{ijk} d ij k ∗ ; hence w η , i j k ∗ , w θ , i j k ∗ ∈ [ w ‾ , 1 4 ] w^{*}_{\eta,ijk},w^{*}_{\theta,ijk}\in[\underline{w},\frac{1}{4}] w η , ij k ∗ , w θ , ij k ∗ ∈ [ w , 4 1 ] for an absolute w ‾ > 0 \underline{w}>0 w > 0 . Define the population merit weight
w ‾ i j k : = w η , i j k ∗ f ′ ( Δ i j ∗ ) 2 + g i j k ∗ w θ , i j k ∗ ( ≥ 0 ) . (32) \overline{w}_{ijk}:=w^{*}_{\eta,ijk}f'(\Delta^{*}_{ij})^{2}+g^{*}_{ijk}w^{*}_{\theta,ijk}\quad(\ge0). \tag*{(32)} w ij k := w η , ij k ∗ f ′ ( Δ ij ∗ ) 2 + g ij k ∗ w θ , ij k ∗ ( ≥ 0 ) . (32)
Lemma 1 (Curvature floor). Under A2–A3 there are absolute constants 0 < b 0 ≤ b 1 0<b_{0}\le b_{1} 0 < b 0 ≤ b 1 with b 0 ≤ w ‾ i j k ≤ b 1 b_{0}\le\overline{w}_{ijk}\le b_{1} b 0 ≤ w ij k ≤ b 1 for all i , j , k i,j,k i , j , k . One may take b 0 = g w ‾ b_{0}=g\underline{w} b 0 = g w and b 1 = 1 4 ( F 1 2 + 1 ) b_{1}=\frac{1}{4}(F_{1}^{2}+1) b 1 = 4 1 ( F 1 2 + 1 ) .
Proof. The upper bound is immediate: w η ∗ ≤ 1 4 w^{*}_{\eta}\le\frac{1}{4} w η ∗ ≤ 4 1 , f ′ ( Δ ∗ ) 2 ≤ F 1 2 f'(\Delta^{*})^{2}\le F_{1}^{2} f ′ ( Δ ∗ ) 2 ≤ F 1 2 , g ∗ ≤ 1 g^{*}\le1 g ∗ ≤ 1 , w θ ∗ ≤ 1 4 w^{*}_{\theta}\le\frac{1}{4} w θ ∗ ≤ 4 1 , so w ‾ i j k ≤ 1 4 F 1 2 + 1 4 \overline{w}_{ijk}\le\frac{1}{4}F_{1}^{2}+\frac{1}{4} w ij k ≤ 4 1 F 1 2 + 4 1 . For the lower bound, both terms in (B.23) are nonnegative, so it suffices to bound the second:
w ‾ i j k ≥ g i j k ∗ w θ , i j k ∗ ≥ g w ‾ = : b 0 > 0. \overline{w}_{ijk}\ge g^{*}_{ijk}w^{*}_{\theta,ijk}\ge g\underline{w}=:b_{0}>0. w ij k ≥ g ij k ∗ w θ , ij k ∗ ≥ g w =: b 0 > 0.
The point is that at a tie (Δ i j ∗ = 0 \Delta^{*}_{ij}=0 Δ ij ∗ = 0 ) the gate channel vanishes, since f f f even implies f ′ ( 0 ) = 0 f'(0)=0 f ′ ( 0 ) = 0 ; the floor is then carried entirely by the direction channel g ∗ w θ ∗ g^{*}w^{*}_{\theta} g ∗ w θ ∗ , which is positive precisely because tied pairs are still cited with probability ≥ g \ge g ≥ g under compactness. □ \square □
:::
Remark 2. If ∥ s ∗ ∥ ∞ \|s^*\|_\infty ∥ s ∗ ∥ ∞ is allowed to grow, g → 0 g \to0 g → 0 and b 0 → 0 b_0 \to0 b 0 → 0 , which is the regime where the rates stated below degrade.
Assumption 4 (Gate design). There is a constant c 0 > 0 c_0 > 0 c 0 > 0 such that
λ min ( ∑ i < j ∑ k w η , i j k ∗ Y i j k Y i j k T ) ≥ c 0 n 2 . \lambda_{\min}\left(\sum_{i<j}\sum_k w^*_{\eta,ijk}Y_{ijk}Y_{ijk}^{\mathsf T}\right) \ge c_0n^2. λ m i n ( i < j ∑ k ∑ w η , ij k ∗ Y ij k Y ij k T ) ≥ c 0 n 2 .
Assumption 5 (Direction design). There is a constant c 0 > 0 c_0 > 0 c 0 > 0 such that
λ min ( ∑ i < j ∑ k g i j k ∗ w θ , i j k ∗ Z i j k Z i j k T ) ≥ c 0 n 2 . \lambda_{\min}\left(\sum_{i<j}\sum_k g^*_{ijk}w^*_{\theta,ijk}Z_{ijk}Z_{ijk}^{\mathsf T}\right) \ge c_0n^2. λ m i n ( i < j ∑ k ∑ g ij k ∗ w θ , ij k ∗ Z ij k Z ij k T ) ≥ c 0 n 2 .
Remark 3. A4 and A5 posit constants that may differ; without loss of generality we use a single c 0 c_0 c 0 for both, namely the smaller of the two. A further assumption, on the efficient information for the structural block, is stated as A6 in Appendix B.12, where the relevant Schur complement is introduced; it implies A4–A5 (Remark 6), which we nevertheless retain for their interpretability.
The population Hessian floor
At the truth, decompose the realized merit Hessian J ∗ : = J ( s ∗ , γ ∗ , β ∗ ) J^* := J(s^*,\gamma^*,\beta^*) J ∗ := J ( s ∗ , γ ∗ , β ∗ ) as
J ∗ = V + E , V : = E [ J ∗ ] = ∑ i < j ∑ k w ˉ i j k ω i j ω i j T , E : = ∑ i < j ∑ k ξ i j k ω i j ω i j T , (33) J^* = V + E,\qquad V := \mathbb{E}[J^*] = \sum_{i<j}\sum_k \bar{w}_{ijk}\omega_{ij}\omega_{ij}^{\mathsf T},\qquad E := \sum_{i<j}\sum_k \xi_{ijk}\omega_{ij}\omega_{ij}^{\mathsf T}, \tag*{(33)} J ∗ = V + E , V := E [ J ∗ ] = i < j ∑ k ∑ w ˉ ij k ω ij ω ij T , E := i < j ∑ k ∑ ξ ij k ω ij ω ij T , (33)
where, using E [ c i j k ] = g i j k ∗ \mathbb{E}[c_{ijk}] = g^*_{ijk} E [ c ij k ] = g ij k ∗ and w θ ∗ w^*_\theta w θ ∗ deterministic given s ∗ , β ∗ s^*,\beta^* s ∗ , β ∗ ,
ξ i j k : = W i j k ∗ − w ˉ i j k = ( c i j k − g i j k ∗ ) ( w θ , i j k ∗ − f ′ ′ ( Δ i j ∗ ) ) , E [ ξ i j k ] = 0 , ∣ ξ i j k ∣ ≤ B : = 1 4 + F 2 . (34) \xi_{ijk} := W^*_{ijk}-\bar{w}_{ijk}=(c_{ijk}-g^*_{ijk})\left(w^*_{\theta,ijk}-f''(\Delta^*_{ij})\right),\qquad\mathbb{E}[\xi_{ijk}]=0,\qquad|\xi_{ijk}|\le B:=\frac{1}{4}+F_2. \tag*{(34)} ξ ij k := W ij k ∗ − w ˉ ij k = ( c ij k − g ij k ∗ ) ( w θ , ij k ∗ − f ′′ ( Δ ij ∗ ) ) , E [ ξ ij k ] = 0 , ∣ ξ ij k ∣ ≤ B := 4 1 + F 2 . (34)
V V V is the expected (population) Hessian; E E E is the mean-zero “saboteur”. The symbol ξ i j k \xi_{ijk} ξ ij k denotes this curvature fluctuation throughout; the unrelated leave-one-out score fluctuation introduced in (B.30) is written χ i m k \chi_{imk} χ imk .
Lemma 2 (Population floor). Under A1–A3 , λ min , ⊥ ( V ) ≥ n b 0 \lambda_{\min,\perp}(V) \ge nb_0 λ m i n , ⊥ ( V ) ≥ n b 0 , with b 0 b_0 b 0 as in Lemma 1.
Proof. V V V is the Laplacian of the comparison graph with edge weights ∑ k w ˉ i j k \sum_k\bar{w}_{ijk} ∑ k w ˉ ij k . For centered x x x (1 T x = 0 \mathbf{1}^{\mathsf T}x=0 1 T x = 0 ),
x T V x = ∑ i < j ∑ k w ˉ i j k ( x i − x j ) 2 ≥ b 0 ∑ i < j ( x i − x j ) 2 = b 0 x T L x , x^{\mathsf T}Vx=\sum_{i<j}\sum_k\bar{w}_{ijk}(x_i-x_j)^2\ge b_0\sum_{i<j}(x_i-x_j)^2=b_0x^{\mathsf T}Lx, x T V x = i < j ∑ k ∑ w ˉ ij k ( x i − x j ) 2 ≥ b 0 i < j ∑ ( x i − x j ) 2 = b 0 x T Lx ,
using w ˉ i j k ≥ b 0 \bar{w}_{ijk}\ge b_0 w ˉ ij k ≥ b 0 (Lemma 1) and m i j ≥ 1 m_{ij}\ge1 m ij ≥ 1 (A1), where L L L is the unweighted Laplacian of K n + 1 K_{n+1} K n + 1 (the complete graph on { 0 , … , n } \{0,\ldots,n\} { 0 , … , n } ). Since λ min , ⊥ ( L ) = λ 2 ( L ) = n + 1 \lambda_{\min,\perp}(L)=\lambda_2(L)=n+1 λ m i n , ⊥ ( L ) = λ 2 ( L ) = n + 1 for K n + 1 K_{n+1} K n + 1 , x T V x ≥ n b 0 ∥ x ∥ 2 2 x^{\mathsf T}Vx\ge nb_0\|x\|_2^2 x T V x ≥ n b 0 ∥ x ∥ 2 2 . Minimizing over centered unit x x x gives the claim.
The leave-one-out construction
The ℓ ∞ \ell_\infty ℓ ∞ analysis of Appendix B.11 compares each gradient iterate with a leave-one-out (LOO) proxy, defined here once and for all. Fix an item m ∈ { 0 , 1 , … , n } m\in\{0,1,\ldots,n\} m ∈ { 0 , 1 , … , n } , and let F − m \mathcal{F}_{-m} F − m be the σ \sigma σ -algebra generated by the outcomes of all comparisons not involving item m m m ,
F − m : = σ ( { c i j k , a i j k 1 : 0 ≤ i < j ≤ n , m ∉ { i , j } , k } ) . \mathcal{F}_{-m}:=\sigma\left(\{c_{ijk},a^1_{ijk}:0\le i<j\le n,\ m\notin\{i,j\},\ k\}\right). F − m := σ ( { c ij k , a ij k 1 : 0 ≤ i < j ≤ n , m ∈ / { i , j } , k } ) .
The LOO objective replaces every comparison involving m m m by its expectation under the truth, leaving the remaining comparisons untouched:
ℓ ( m ) ( s , γ , β ) = ∑ i < j i ≠ m , j ≠ m ∑ k ℓ i j k ( s , γ , β ) + ∑ i ≠ m ∑ k ℓ ˉ i m k ( s , γ , β ) , (35) \ell^{(m)}(s,\gamma,\beta)=\sum_{\substack{i<j\\i\ne m,\ j\ne m}}\sum_k\ell_{ijk}(s,\gamma,\beta)+\sum_{i\ne m}\sum_k\bar{\ell}_{imk}(s,\gamma,\beta), \tag*{(35)} ℓ ( m ) ( s , γ , β ) = i < j i = m , j = m ∑ k ∑ ℓ ij k ( s , γ , β ) + i = m ∑ k ∑ ℓ ˉ imk ( s , γ , β ) , (35)
where
ℓ ˉ i m k ( s , γ , β ) : = g i m k ∗ log g i m k + ( 1 − g i m k ∗ ) log ( 1 − g i m k ) + g i m k ∗ [ d i m k ∗ log d i m k + ( 1 − d i m k ∗ ) log ( 1 − d i m k ) ] . \bar{\ell}_{imk}(s,\gamma,\beta):=g^*_{imk}\log g_{imk}+(1-g^*_{imk})\log(1-g_{imk})+g^*_{imk}\left[d^*_{imk}\log d_{imk}+(1-d^*_{imk})\log(1-d_{imk})\right]. ℓ ˉ imk ( s , γ , β ) := g imk ∗ log g imk + ( 1 − g imk ∗ ) log ( 1 − g imk ) + g imk ∗ [ d imk ∗ log d imk + ( 1 − d imk ∗ ) log ( 1 − d imk ) ] .
Its merit gradient G γ , β ( m ) : = ∇ s ( − ℓ ( m ) ) G_{\gamma,\beta}^{(m)} := \nabla_s(-\ell^{(m)}) G γ , β ( m ) := ∇ s ( − ℓ ( m ) ) is
G γ , β ( m ) ( s ) = − ∑ i < j i ≠ m , j ≠ m ∑ k [ ( c i j k − g i j k ) f ′ ( Δ i j ) + ( a i j k 1 − c i j k d i j k ) ] ω i j − ∑ i ≠ m ∑ k [ ( g i m k ∗ − g i m k ) f ′ ( Δ i m ) + g i m k ∗ ( d i m k ∗ − d i m k ) ] ω i m . (36) G_{\gamma,\beta}^{(m)}(s) = - \sum_{\substack{i<j\\ i\ne m,\ j\ne m}} \sum_k \left[(c_{ijk}-g_{ijk})f'(\Delta_{ij})+(a_{ijk}^{1}-c_{ijk}d_{ijk})\right]\omega_{ij}
-\sum_{i\ne m}\sum_k \left[(g_{imk}^{*}-g_{imk})f'(\Delta_{im})+g_{imk}^{*}(d_{imk}^{*}-d_{imk})\right]\omega_{im}. \tag*{(36)} G γ , β ( m ) ( s ) = − i < j i = m , j = m ∑ k ∑ [ ( c ij k − g ij k ) f ′ ( Δ ij ) + ( a ij k 1 − c ij k d ij k ) ] ω ij − i = m ∑ k ∑ [ ( g imk ∗ − g imk ) f ′ ( Δ im ) + g imk ∗ ( d imk ∗ − d imk ) ] ω im . (36)
and its merit Hessian J ( m ) : = ∇ s 2 ( − ℓ ( m ) ) \mathcal{J}^{(m)} := \nabla_s^2(-\ell^{(m)}) J ( m ) := ∇ s 2 ( − ℓ ( m ) ) is the graph Laplacian
J ( m ) ( s , γ , β ) = ∑ i < j W i j ( m ) ω i j ω i j T , W i j ( m ) = ∑ k W i j k ( m ) . (37) \mathcal{J}^{(m)}(s,\gamma,\beta)=\sum_{i<j}W_{ij}^{(m)}\omega_{ij}\omega_{ij}^{\mathsf T},\qquad W_{ij}^{(m)}=\sum_k W_{ijk}^{(m)}. \tag*{(37)} J ( m ) ( s , γ , β ) = i < j ∑ W ij ( m ) ω ij ω ij T , W ij ( m ) = k ∑ W ij k ( m ) . (37)
with edge weights
W i j k ( m ) = { W i j k , m ∉ { i , j } , w η , i m k f ′ ( Δ i m ) 2 + g i m k ∗ w θ , i m k − ( g i m k ∗ − g i m k ) f ′ ′ ( Δ i m ) , m ∈ { i , j } . (38) W_{ijk}^{(m)}=
\begin{cases}
W_{ijk}, & m\notin\{i,j\},\\
w_{\eta,imk}f'(\Delta_{im})^2+g_{imk}^{*}w_{\theta,imk}-(g_{imk}^{*}-g_{imk})f''(\Delta_{im}), & m\in\{i,j\}.
\end{cases} \tag*{(38)} W ij k ( m ) = { W ij k , w η , imk f ′ ( Δ im ) 2 + g imk ∗ w θ , imk − ( g imk ∗ − g imk ) f ′′ ( Δ im ) , m ∈ / { i , j } , m ∈ { i , j } . (38)
Remark 4 (Symmetry of the weights). Every per-comparison curvature weight is symmetric under i ↔ j i\leftrightarrow j i ↔ j : indeed g j i k = g i j k g_{jik}=g_{ijk} g j ik = g ij k and w η , j i k = w η , i j k w_{\eta,jik}=w_{\eta,ijk} w η , j ik = w η , ij k (the gate predictor is even), w θ , j i k = w θ , i j k w_{\theta,jik}=w_{\theta,ijk} w θ , j ik = w θ , ij k (since d j i k = 1 − d i j k d_{jik}=1-d_{ijk} d j ik = 1 − d ij k ), f ′ ( Δ j i ) 2 = f ′ ( Δ i j ) 2 f'(\Delta_{ji})^2=f'(\Delta_{ij})^2 f ′ ( Δ j i ) 2 = f ′ ( Δ ij ) 2 and f ′ ′ ( Δ j i ) = f ′ ′ ( Δ i j ) f''(\Delta_{ji})=f''(\Delta_{ij}) f ′′ ( Δ j i ) = f ′′ ( Δ ij ) (f ′ f' f ′ odd, f ′ ′ f'' f ′′ even), and c j i k = c i j k c_{jik}=c_{ijk} c j ik = c ij k ; hence W j i k = W i j k W_{jik}=W_{ijk} W j ik = W ij k and likewise W m i k ( m ) = W i m k ( m ) W_{mik}^{(m)}=W_{imk}^{(m)} W mik ( m ) = W imk ( m ) (using g m i k ∗ = g i m k ∗ g_{mik}^{*}=g_{imk}^{*} g mik ∗ = g imk ∗ ). The score summands are instead antisymmetric, ψ j i k = − ψ i j k \psi_{jik}=-\psi_{ijk} ψ j ik = − ψ ij k , matching ω j i = − ω i j \omega_{ji}=-\omega_{ij} ω j i = − ω ij . We use the two orderings of a pair interchangeably below.
Like G G G and J J J , the objects G γ , β ( m ) G_{\gamma,\beta}^{(m)} G γ , β ( m ) and J ( m ) \mathcal{J}^{(m)} J ( m ) take values in / act on 1 ⊥ 1^\perp 1 ⊥ (every ω i m \omega_{im} ω im is centered). Two structural facts are used repeatedly.
Measurability. The construction (B.26) strips out all of item m m m ’s randomness: in the m m m -edge terms ℓ ˉ i m k \bar{\ell}_{imk} ℓ ˉ imk the outcomes c i m k , a i m k 1 c_{imk},a_{imk}^{1} c imk , a imk 1 have been replaced by their population expectations g i m k ∗ g_{imk}^{*} g imk ∗ and g i m k ∗ d i m k ∗ g_{imk}^{*}d_{imk}^{*} g imk ∗ d imk ∗ , so ℓ ˉ i m k \bar{\ell}_{imk} ℓ ˉ imk is a non-random function of ( s , γ , β ) (s,\gamma,\beta) ( s , γ , β ) (the covariates and the population quantities are fixed under A1–A3). The remaining terms depend on the data only through the outcomes of comparisons disjoint from m m m , i.e. the generators of F − m \mathcal{F}_{-m} F − m . Hence ℓ ( m ) ( ⋅ , γ , β ) \ell^{(m)}(\cdot,\gamma,\beta) ℓ ( m ) ( ⋅ , γ , β ) , G γ , β ( m ) G_{\gamma,\beta}^{(m)} G γ , β ( m ) , and J ( m ) \mathcal{J}^{(m)} J ( m ) are F − m \mathcal{F}_{-m} F − m -measurable as random functions of s s s (a non-random function being measurable with respect to any σ \sigma σ -algebra). Consequently any iterative scheme driven by G γ , β ( m ) G_{\gamma,\beta}^{(m)} G γ , β ( m ) and started at the deterministic point s ∗ s^{*} s ∗ produces F − m \mathcal{F}_{-m} F − m -measurable iterates; in particular the LOO trajectory { s t ( m ) } t \{s_t^{(m)}\}_t { s t ( m ) } t of Algorithm 2 below is F − m \mathcal{F}_{-m} F − m -measurable at every step.
The score discrepancy. Subtracting (B.14) from (B.27), all pairs disjoint from m m m cancel and the m m m -edges combine into
G γ , β ( m ) ( s ) − G γ , β ( s ) = D γ , β ( m ) ( s ) : = ∑ i ≠ m ∑ k χ i m k ω i m , (39) G_{\gamma,\beta}^{(m)}(s)-G_{\gamma,\beta}(s)=D_{\gamma,\beta}^{(m)}(s):=\sum_{i\ne m}\sum_k\chi_{imk}\omega_{im}, \tag*{(39)} G γ , β ( m ) ( s ) − G γ , β ( s ) = D γ , β ( m ) ( s ) := i = m ∑ k ∑ χ imk ω im , (39)
χ i m k : = ( c i m k − g i m k ∗ ) ( f ′ ( Δ i m ) − d i m k ) + ( a i m k 1 − g i m k ∗ d i m k ∗ ) . \chi_{imk}:=(c_{imk}-g_{imk}^{*})(f'(\Delta_{im})-d_{imk})+(a_{imk}^{1}-g_{imk}^{*}d_{imk}^{*}). χ imk := ( c imk − g imk ∗ ) ( f ′ ( Δ im ) − d imk ) + ( a imk 1 − g imk ∗ d imk ∗ ) .
Note that D γ , β ( m ) D_{\gamma,\beta}^{(m)} D γ , β ( m ) depends on the structural parameters only through d i m k = σ ( Δ i m + Z i m k T β ) d_{imk}=\sigma(\Delta_{im}+Z_{imk}^{\mathsf T}\beta) d imk = σ ( Δ im + Z imk T β ) , i.e. on β \beta β and not on γ \gamma γ .
Concentration
Lemma 3 (Score bounds). Under A1–A3, for every c > 0 c>0 c > 0 , with probability at least 1 − O ( n − c ) 1-O(n^{-c}) 1 − O ( n − c ) ,
(merit) max 0 ≤ a ≤ n ∣ G a ( s ∗ , γ ∗ , β ∗ ) ∣ ≤ C ψ n log n , C ψ : = 2 ( F 1 + 1 ) ( c + 1 ) m ∗ , (40) \text{(merit)}\qquad\max_{0\le a\le n}|G_a(s^{*},\gamma^{*},\beta^{*})|\le C_{\psi}\sqrt{n\log n},\qquad C_{\psi}:=2(F_1+1)\sqrt{(c+1)m^{*}}, \tag*{(40)} (merit) 0 ≤ a ≤ n max ∣ G a ( s ∗ , γ ∗ , β ∗ ) ∣ ≤ C ψ n log n , C ψ := 2 ( F 1 + 1 ) ( c + 1 ) m ∗ , (40) (structural) ∥ ∇ γ ℓ ( s ∗ , γ ∗ , β ∗ ) ∥ 2 ≤ C κ n log n , ∥ ∇ β ℓ ( s ∗ , γ ∗ , β ∗ ) ∥ 2 ≤ C κ n log n , (41) \text{(structural)}\qquad\|\nabla_{\gamma}\ell(s^{*},\gamma^{*},\beta^{*})\|_2\le C_{\kappa}n\sqrt{\log n},\qquad\|\nabla_{\beta}\ell(s^{*},\gamma^{*},\beta^{*})\|_2\le C_{\kappa}n\sqrt{\log n}, \tag*{(41)} (structural) ∥ ∇ γ ℓ ( s ∗ , γ ∗ , β ∗ ) ∥ 2 ≤ C κ n log n , ∥ ∇ β ℓ ( s ∗ , γ ∗ , β ∗ ) ∥ 2 ≤ C κ n log n , (41) where C κ : = κ 2 m ∗ ( c + 1 ) max { p γ , p β } C_{\kappa}:=\kappa\sqrt{2m^{*}(c+1)}\max\{p_{\gamma},p_{\beta}\} C κ := κ 2 m ∗ ( c + 1 ) max { p γ , p β } .
Proof. Merit. Fix a a a . By (B.13), G a = − ∑ j ≠ a ∑ k ψ a j k G_a=-\sum_{j\ne a}\sum_k\psi_{ajk} G a = − ∑ j = a ∑ k ψ aj k at the truth, a sum of m a : = ∑ j ≠ a m a j ∈ [ n , n m ∗ ] m_a:=\sum_{j\ne a}m_{aj}\in[n,nm^*] m a := ∑ j = a m aj ∈ [ n , n m ∗ ] independent terms. Each ψ a j k \psi_{ajk} ψ aj k is a bounded function of ( c a j k , a a j k 1 ) (c_{ajk},a^1_{ajk}) ( c aj k , a aj k 1 ) with ∣ ψ a j k ∣ ≤ F 1 + 1 |\psi_{ajk}|\le F_1+1 ∣ ψ aj k ∣ ≤ F 1 + 1 and is mean-zero: E [ ( c − g ∗ ) f ′ ] = 0 \mathbb{E}[(c-g^*)f']=0 E [( c − g ∗ ) f ′ ] = 0 since E c = g ∗ \mathbb{E}c=g^* E c = g ∗ , and E [ a 1 − c d ∗ ] = E [ E [ a 1 ∣ c ] − c d ∗ ] = E [ c d ∗ − c d ∗ ] = 0 \mathbb{E}[a^1-cd^*]=\mathbb{E}[\mathbb{E}[a^1\mid c]-cd^*]=\mathbb{E}[cd^*-cd^*]=0 E [ a 1 − c d ∗ ] = E [ E [ a 1 ∣ c ] − c d ∗ ] = E [ c d ∗ − c d ∗ ] = 0 . Hoeffding (Fact 3) with per-term range 2 ( F 1 + 1 ) 2(F_1+1) 2 ( F 1 + 1 ) gives
P ( ∣ G a ∣ ≥ t ) ≤ 2 exp ( − 2 t 2 m a ( 2 ( F 1 + 1 ) ) 2 ) = 2 exp ( − t 2 2 m a ( F 1 + 1 ) 2 ) . \mathbb{P}(|G_a|\ge t)\le2\exp\left(-\frac{2t^2}{m_a(2(F_1+1))^2}\right)=2\exp\left(-\frac{t^2}{2m_a(F_1+1)^2}\right). P ( ∣ G a ∣ ≥ t ) ≤ 2 exp ( − m a ( 2 ( F 1 + 1 ) ) 2 2 t 2 ) = 2 exp ( − 2 m a ( F 1 + 1 ) 2 t 2 ) .
Take t = C ψ n log n t=C_\psi\sqrt{n\log n} t = C ψ n log n with C ψ = 2 ( F 1 + 1 ) ( c + 1 ) m ∗ C_\psi=2(F_1+1)\sqrt{(c+1)m^*} C ψ = 2 ( F 1 + 1 ) ( c + 1 ) m ∗ . Since m a ≤ n m ∗ m_a\le nm^* m a ≤ n m ∗ , the exponent is
− C ψ 2 n log n 2 m a ( F 1 + 1 ) 2 ≤ − 4 ( F 1 + 1 ) 2 ( c + 1 ) m ∗ n log n 2 n m ∗ ( F 1 + 1 ) 2 = − 2 ( c + 1 ) log n , -\frac{C_\psi^2n\log n}{2m_a(F_1+1)^2}\le-\frac{4(F_1+1)^2(c+1)m^*n\log n}{2nm^*(F_1+1)^2}=-2(c+1)\log n, − 2 m a ( F 1 + 1 ) 2 C ψ 2 n log n ≤ − 2 n m ∗ ( F 1 + 1 ) 2 4 ( F 1 + 1 ) 2 ( c + 1 ) m ∗ n log n = − 2 ( c + 1 ) log n ,
so P ( ∣ G a ∣ ≥ t ) ≤ 2 n − 2 ( c + 1 ) \mathbb{P}(|G_a|\ge t)\le2n^{-2(c+1)} P ( ∣ G a ∣ ≥ t ) ≤ 2 n − 2 ( c + 1 ) . A union bound over a = 0 , … , n a=0,\ldots,n a = 0 , … , n gives the merit claim with probability ≥ 1 − 2 ( n + 1 ) n − 2 ( c + 1 ) = 1 − O ( n − ( 2 c + 1 ) ) = 1 − O ( n − c ) \ge1-2(n+1)n^{-2(c+1)}=1-O(n^{-(2c+1)})=1-O(n^{-c}) ≥ 1 − 2 ( n + 1 ) n − 2 ( c + 1 ) = 1 − O ( n − ( 2 c + 1 ) ) = 1 − O ( n − c ) .
Structural. Each coordinate of ∇ γ ℓ \nabla_\gamma\ell ∇ γ ℓ (and of ∇ β ℓ \nabla_\beta\ell ∇ β ℓ ) is a sum of M : = ∑ i < j m i j ≤ ( n + 1 2 ) m ∗ M:=\sum_{i<j}m_{ij}\le\binom{n+1}{2}m^* M := ∑ i < j m ij ≤ ( 2 n + 1 ) m ∗ independent mean-zero terms; the l l l -th coordinate of ∇ γ ℓ \nabla_\gamma\ell ∇ γ ℓ is ∑ i < j , k ( c i j k − g i j k ∗ ) Y i j k , l \sum_{i<j,k}(c_{ijk}-g^*_{ijk})Y_{ijk,l} ∑ i < j , k ( c ij k − g ij k ∗ ) Y ij k , l , with each term bounded in absolute value by κ \kappa κ since ∣ c − g ∗ ∣ ≤ 1 |c-g^*|\le1 ∣ c − g ∗ ∣ ≤ 1 and ∣ Y i j k , l ∣ ≤ ∥ Y i j k ∥ 2 ≤ κ |Y_{ijk,l}|\le\|Y_{ijk}\|_2\le\kappa ∣ Y ij k , l ∣ ≤ ∥ Y ij k ∥ 2 ≤ κ (and likewise for β \beta β ). Hoeffding with per-term range 2 κ 2\kappa 2 κ and t = C κ ′ M log n t=C'_\kappa\sqrt{M\log n} t = C κ ′ M log n , C κ ′ : = κ 2 ( c + 1 ) C'_\kappa:=\kappa\sqrt{2(c+1)} C κ ′ := κ 2 ( c + 1 ) , gives, for each fixed coordinate,
P ( ∣ coord ∣ ≥ t ) ≤ 2 exp ( − 2 t 2 M ( 2 κ ) 2 ) = 2 exp ( − ( c + 1 ) log n ) = 2 n − ( c + 1 ) . \mathbb{P}(|\operatorname{coord}|\ge t)\le2\exp\left(-\frac{2t^2}{M(2\kappa)^2}\right)=2\exp(-(c+1)\log n)=2n^{-(c+1)}. P ( ∣ coord ∣ ≥ t ) ≤ 2 exp ( − M ( 2 κ ) 2 2 t 2 ) = 2 exp ( − ( c + 1 ) log n ) = 2 n − ( c + 1 ) .
A union bound over the fixed p γ + p β p_\gamma+p_\beta p γ + p β coordinates yields max { ∥ ∇ γ ℓ ∥ ∞ , ∥ ∇ β ℓ ∥ ∞ } ≤ C κ ′ M log n \max\{\|\nabla_\gamma\ell\|_\infty,\|\nabla_\beta\ell\|_\infty\}\le C'_\kappa\sqrt{M\log n} max { ∥ ∇ γ ℓ ∥ ∞ , ∥ ∇ β ℓ ∥ ∞ } ≤ C κ ′ M log n with probability 1 − O ( n − ( c + 1 ) ) = 1 − O ( n − c ) 1-O(n^{-(c+1)})=1-O(n^{-c}) 1 − O ( n − ( c + 1 ) ) = 1 − O ( n − c ) . Since p γ , p β p_\gamma,p_\beta p γ , p β are fixed and M ≤ ( n + 1 2 ) m ∗ ≤ n 2 m ∗ M\le\binom{n+1}{2}m^*\le n^2m^* M ≤ ( 2 n + 1 ) m ∗ ≤ n 2 m ∗ (using ( n + 1 2 ) ≤ n 2 \binom{n+1}{2}\le n^2 ( 2 n + 1 ) ≤ n 2 for n ≥ 1 n\ge1 n ≥ 1 ), so M ≤ n m ∗ \sqrt{M}\le n\sqrt{m^*} M ≤ n m ∗ ,
∥ ∇ γ ℓ ∥ 2 ≤ p γ C κ ′ M log n ≤ κ 2 p γ m ∗ ( c + 1 ) n log n ≤ C κ n log n , \|\nabla_\gamma\ell\|_2\le\sqrt{p_\gamma}C'_\kappa\sqrt{M\log n}\le\kappa\sqrt{2p_\gamma m^*(c+1)}n\sqrt{\log n}\le C_\kappa n\sqrt{\log n}, ∥ ∇ γ ℓ ∥ 2 ≤ p γ C κ ′ M log n ≤ κ 2 p γ m ∗ ( c + 1 ) n log n ≤ C κ n log n ,
with C κ : = κ 2 m ∗ ( c + 1 ) max { p γ , p β } C_\kappa:=\kappa\sqrt{2m^*(c+1)}\max\{p_\gamma,p_\beta\} C κ := κ 2 m ∗ ( c + 1 ) max { p γ , p β } , and identically for ∇ β ℓ \nabla_\beta\ell ∇ β ℓ . □
Lemma 4 (Spectral control of the saboteur). Under A1–A3, for every c > 0 c>0 c > 0 there is an absolute constant C E C_E C E such that the event
E s a b : = { ∥ E ∥ 2 ≤ C E n log n } \mathcal{E}_{\mathrm{sab}}:=\{\|E\|_2\le C_E\sqrt{n\log n}\} E sab := { ∥ E ∥ 2 ≤ C E n log n } has probability at least 1 − O ( n − c ) 1-O(n^{-c}) 1 − O ( n − c ) .
Proof. Write E = ∑ i < j , k X i j k E=\sum_{i<j,k}X_{ijk} E = ∑ i < j , k X ij k with X i j k = ξ i j k ω i j ω i j T X_{ijk}=\xi_{ijk}\omega_{ij}\omega_{ij}^{\mathsf T} X ij k = ξ ij k ω ij ω ij T , independent, symmetric, mean-zero (E ξ i j k = 0 \mathbb{E}\xi_{ijk}=0 E ξ ij k = 0 by (B.25)). Since ∥ ω i j ∥ 2 2 = 2 \|\omega_{ij}\|_2^2=2 ∥ ω ij ∥ 2 2 = 2 and ∣ ξ i j k ∣ ≤ B |\xi_{ijk}|\le B ∣ ξ ij k ∣ ≤ B , ∥ X i j k ∥ 2 ≤ 2 B = : R \|X_{ijk}\|_2\le2B=:R ∥ X ij k ∥ 2 ≤ 2 B =: R . Using ( ω i j ω i j T ) 2 = 2 ω i j ω i j T (\omega_{ij}\omega_{ij}^{\mathsf T})^2=2\omega_{ij}\omega_{ij}^{\mathsf T} ( ω ij ω ij T ) 2 = 2 ω ij ω ij T and E [ ξ i j k 2 ] ≤ B 2 \mathbb{E}[\xi_{ijk}^2]\le B^2 E [ ξ ij k 2 ] ≤ B 2 ,
∑ i < j , k E X i j k 2 = ∑ i < j , k 2 E [ ξ i j k 2 ] ω i j ω i j T ⪯ 2 B 2 m ∗ ∑ i < j ω i j ω i j T , \sum_{i<j,k}\mathbb{E}X_{ijk}^2=\sum_{i<j,k}2\mathbb{E}[\xi_{ijk}^2]\omega_{ij}\omega_{ij}^{\mathsf T}\preceq2B^2m^*\sum_{i<j}\omega_{ij}\omega_{ij}^{\mathsf T}, i < j , k ∑ E X ij k 2 = i < j , k ∑ 2 E [ ξ ij k 2 ] ω ij ω ij T ⪯ 2 B 2 m ∗ i < j ∑ ω ij ω ij T ,
and the right-hand side of the last display is 2 B 2 m ∗ 2B^2m^* 2 B 2 m ∗ times the Laplacian of K n + 1 K_{n+1} K n + 1 , of spectral norm n + 1 n+1 n + 1 , so ∥ ∑ i < j , k E X i j k 2 ∥ 2 ≤ 2 B 2 m ∗ ( n + 1 ) = : ν \left\|\sum_{i<j,k}\mathbb{E}X_{ijk}^2\right\|_2\le2B^2m^*(n+1)=:\nu ∑ i < j , k E X ij k 2 2 ≤ 2 B 2 m ∗ ( n + 1 ) =: ν . With t = C E n log n t=C_E\sqrt{n\log n} t = C E n log n , the matrix-Bernstein (Fact 2) exponent satisfies
− t 2 2 ν + 2 R t / 3 ∼ n → ∞ − C E 2 4 B 2 m ∗ log n , -\frac{t^2}{2\nu+2Rt/3}\underset{n\to\infty}{\sim}-\frac{C_E^2}{4B^2m^*}\log n, − 2 ν + 2 R t /3 t 2 n → ∞ ∼ − 4 B 2 m ∗ C E 2 log n ,
so P ( ∥ E ∥ 2 ≥ t ) ≤ 2 ( n + 1 ) n − C E 2 / ( 4 B 2 m ∗ ) = O ( n − c ) \mathbb{P}(\|E\|_2\ge t)\le2(n+1)n^{-C_E^2/(4B^2m^*)}=O(n^{-c}) P ( ∥ E ∥ 2 ≥ t ) ≤ 2 ( n + 1 ) n − C E 2 / ( 4 B 2 m ∗ ) = O ( n − c ) for C E C_E C E large enough. □
Lemma 5 (Leave-one-out perturbation bound). Assume A1–A3, and fix m ∈ { 0 , 1 , … , n } m \in\{0,1,\ldots,n\} m ∈ { 0 , 1 , … , n } , a pair ( γ , β ) (\gamma,\beta) ( γ , β ) , and an F − m \mathcal{F}_{-m} F − m -measurable random point s ∈ R n + 1 s \in\mathbb{R}^{n+1} s ∈ R n + 1 . With D γ , β ( m ) D_{\gamma,\beta}^{(m)} D γ , β ( m ) and χ i m k \chi_{imk} χ imk as in (B.30), for every c > 0 c > 0 c > 0 , with K : = 2 ( F 1 + 2 ) 2 ( c + 1 ) m ∗ K := 2(F_1+2)\sqrt{2(c+1)m^*} K := 2 ( F 1 + 2 ) 2 ( c + 1 ) m ∗ , there is n 0 = n 0 ( c ) n_0=n_0(c) n 0 = n 0 ( c ) such that for all n ≥ n 0 n \ge n_0 n ≥ n 0 ,
P ( { ∥ D γ , β ( m ) ( s ) ∥ 2 > K n log n } ∩ { ∥ s ∥ ∞ ≤ C s + 1 2 } ) = O ( n − c ) . \mathbb{P}\left(\left\{\left\|D_{\gamma,\beta}^{(m)}(s)\right\|_2 > K\sqrt{n\log n}\right\}\cap\left\{\|s\|_\infty\le C_s+\frac{1}{2}\right\}\right)=O(n^{-c}). P ( { D γ , β ( m ) ( s ) 2 > K n log n } ∩ { ∥ s ∥ ∞ ≤ C s + 2 1 } ) = O ( n − c ) . In particular, if ∥ s ∥ ∞ ≤ C s + 1 2 \|s\|_\infty\le C_s+\frac{1}{2} ∥ s ∥ ∞ ≤ C s + 2 1 almost surely, then ∥ D γ , β ( m ) ( s ) ∥ 2 ≤ K n log n \left\|D_{\gamma,\beta}^{(m)}(s)\right\|_2 \le K\sqrt{n\log n} D γ , β ( m ) ( s ) 2 ≤ K n log n with probability at least 1 − O ( n − c ) 1-O(n^{-c}) 1 − O ( n − c ) .
Proof. Condition on F − m \mathcal{F}_{-m} F − m . By hypothesis s s s is then deterministic, hence so are f ′ ( Δ i m ) f'(\Delta_{im}) f ′ ( Δ im ) and d i m k d_{imk} d imk , and the confinement event { ∥ s ∥ ∞ ≤ C s + 1 2 } \{\|s\|_\infty\le C_s+\frac{1}{2}\} { ∥ s ∥ ∞ ≤ C s + 2 1 } is F − m \mathcal{F}_{-m} F − m -measurable. On the F − m \mathcal{F}_{-m} F − m -atoms where confinement fails, the conditional probability of the displayed intersection is zero, so it suffices to prove the conditional bound on the atoms where ∥ s ∥ ∞ ≤ C s + 1 2 \|s\|_\infty\le C_s+\frac{1}{2} ∥ s ∥ ∞ ≤ C s + 2 1 ; there, every gap satisfies ∣ Δ i m ∣ ≤ 2 C s + 1 |\Delta_{im}| \le2C_s+1 ∣ Δ im ∣ ≤ 2 C s + 1 , so the bounds of A3 apply. The pairs { ( c i m k , a i m k 1 ) } i ≠ m , k \{(c_{imk},a^1_{imk})\}_{i\ne m,k} {( c imk , a imk 1 ) } i = m , k are independent across ( i , k ) (i,k) ( i , k ) with the law (B.8) and independent of F − m \mathcal{F}_{-m} F − m . Set X i m k : = χ i m k ω i m ∈ R n + 1 X_{imk}:=\chi_{imk}\omega_{im}\in\mathbb{R}^{n+1} X imk := χ imk ω im ∈ R n + 1 and verify vector Bernstein (Fact 1) for { X i m k } i ≠ m , k \{X_{imk}\}_{i\ne m,k} { X imk } i = m , k under the conditional law.
(a) Mean-zero. E [ c i m k − g i m k ∗ ] = 0 \mathbb{E}[c_{imk}-g^*_{imk}]=0 E [ c imk − g imk ∗ ] = 0 and E [ a i m k 1 − g i m k ∗ d i m k ∗ ] = 0 \mathbb{E}[a^1_{imk}-g^*_{imk}d^*_{imk}]=0 E [ a imk 1 − g imk ∗ d imk ∗ ] = 0 . Since f ′ ( Δ i m ) − d i m k f'(\Delta_{im})-d_{imk} f ′ ( Δ im ) − d imk is deterministic under the conditioning, E [ χ i m k ∣ F − m ] = 0 \mathbb{E}[\chi_{imk}\mid\mathcal{F}_{-m}]=0 E [ χ imk ∣ F − m ] = 0 , hence E [ X i m k ∣ F − m ] = 0 \mathbb{E}[X_{imk}\mid\mathcal{F}_{-m}]=0 E [ X imk ∣ F − m ] = 0 .
(b) Uniform bound. ∣ c − g ∗ ∣ ≤ 1 |c-g^*|\le1 ∣ c − g ∗ ∣ ≤ 1 , ∣ a 1 − g ∗ d ∗ ∣ ≤ 1 |a^1-g^*d^*|\le1 ∣ a 1 − g ∗ d ∗ ∣ ≤ 1 , ∣ f ′ ∣ ≤ F 1 |f'|\le F_1 ∣ f ′ ∣ ≤ F 1 (A3, on the confinement atoms), d ∈ [ 0 , 1 ] d\in[0,1] d ∈ [ 0 , 1 ] , so ∣ χ i m k ∣ ≤ ( F 1 + 1 ) + 1 = F 1 + 2 |\chi_{imk}|\le(F_1+1)+1=F_1+2 ∣ χ imk ∣ ≤ ( F 1 + 1 ) + 1 = F 1 + 2 . As ∥ ω i m ∥ 2 = 2 \|\omega_{im}\|_2=\sqrt{2} ∥ ω im ∥ 2 = 2 , ∥ X i m k ∥ 2 = 2 ∣ χ i m k ∣ ≤ 2 ( F 1 + 2 ) = : R \|X_{imk}\|_2=\sqrt{2}|\chi_{imk}|\le\sqrt{2}(F_1+2)=:R ∥ X imk ∥ 2 = 2 ∣ χ imk ∣ ≤ 2 ( F 1 + 2 ) =: R .
(c) Total second moment. There are n n n indices i ≠ m i\ne m i = m in { 0 , … , n } \{0,\ldots,n\} { 0 , … , n } , each with ≤ m ∗ \le m^* ≤ m ∗ comparisons, so
∑ i ≠ m , k E [ ∥ X i m k ∥ 2 2 ∣ F − m ] = ∑ i ≠ m , k 2 E [ χ i m k 2 ∣ F − m ] ≤ 2 ( F 1 + 2 ) 2 m ∗ n = : ν . \sum_{i\ne m,k}\mathbb{E}\left[\|X_{imk}\|_2^2\mid\mathcal{F}_{-m}\right]=\sum_{i\ne m,k}2\mathbb{E}\left[\chi_{imk}^2\mid\mathcal{F}_{-m}\right]\le2(F_1+2)^2m^*n=:\nu. i = m , k ∑ E [ ∥ X imk ∥ 2 2 ∣ F − m ] = i = m , k ∑ 2 E [ χ imk 2 ∣ F − m ] ≤ 2 ( F 1 + 2 ) 2 m ∗ n =: ν .
By vector Bernstein in dimension d = n + 1 d=n+1 d = n + 1 , for any t ≥ 0 t\ge0 t ≥ 0 ,
P ( ∥ D γ , β ( m ) ( s ) ∥ 2 ≥ t ∣ F − m ) ≤ 2 ( n + 2 ) exp ( − t 2 2 ν + 2 R t / 3 ) . \mathbb{P}\left(\left\|D_{\gamma,\beta}^{(m)}(s)\right\|_2\ge t\mid\mathcal{F}_{-m}\right)\le2(n+2)\exp\left(-\frac{t^2}{2\nu+2Rt/3}\right). P ( D γ , β ( m ) ( s ) 2 ≥ t ∣ F − m ) ≤ 2 ( n + 2 ) exp ( − 2 ν + 2 R t /3 t 2 ) .
Take t = K n log n t=K\sqrt{n\log n} t = K n log n with K = 2 ( F 1 + 2 ) 2 ( c + 1 ) m ∗ K=2(F_1+2)\sqrt{2(c+1)m^*} K = 2 ( F 1 + 2 ) 2 ( c + 1 ) m ∗ . Then
R t 3 ν = 2 ( F 1 + 2 ) K n log n 3 ⋅ 2 ( F 1 + 2 ) 2 m ∗ n = K 3 2 ( F 1 + 2 ) m ∗ log n n ≤ 1 \frac{Rt}{3\nu}=\frac{\sqrt{2}(F_1+2)K\sqrt{n\log n}}{3\cdot2(F_1+2)^2m^*n}=\frac{K}{3\sqrt{2}(F_1+2)m^*}\sqrt{\frac{\log n}{n}}\le1 3 ν R t = 3 ⋅ 2 ( F 1 + 2 ) 2 m ∗ n 2 ( F 1 + 2 ) K n log n = 3 2 ( F 1 + 2 ) m ∗ K n log n ≤ 1
for n ≥ n 0 n\ge n_0 n ≥ n 0 , so 2 ν + 2 R t / 3 ≤ 4 ν 2\nu+2Rt/3\le4\nu 2 ν + 2 R t /3 ≤ 4 ν and
P ( ∥ D γ , β ( m ) ( s ) ∥ 2 ≥ t ∣ F − m ) ≤ 2 ( n + 2 ) exp ( − t 2 4 ν ) = 2 ( n + 2 ) exp ( − K 2 n log n 8 ( F 1 + 2 ) 2 m ∗ n ) . \mathbb{P}\left(\left\|D_{\gamma,\beta}^{(m)}(s)\right\|_2\ge t\mid\mathcal{F}_{-m}\right)\le2(n+2)\exp\left(-\frac{t^2}{4\nu}\right)=2(n+2)\exp\left(-\frac{K^2n\log n}{8(F_1+2)^2m^*n}\right). P ( D γ , β ( m ) ( s ) 2 ≥ t ∣ F − m ) ≤ 2 ( n + 2 ) exp ( − 4 ν t 2 ) = 2 ( n + 2 ) exp ( − 8 ( F 1 + 2 ) 2 m ∗ n K 2 n log n ) .
With K 2 = 8 ( c + 1 ) ( F 1 + 2 ) 2 m ∗ K^2=8(c+1)(F_1+2)^2m^* K 2 = 8 ( c + 1 ) ( F 1 + 2 ) 2 m ∗ the exponent is − ( c + 1 ) log n -(c+1)\log n − ( c + 1 ) log n , so the conditional probability is at most 2 ( n + 2 ) n − ( c + 1 ) = O ( n − c ) 2(n+2)n^{-(c+1)}=O(n^{-c}) 2 ( n + 2 ) n − ( c + 1 ) = O ( n − c ) on the confinement atoms. Taking expectations over F − m \mathcal{F}_{-m} F − m completes the proof. □
Hessian estimates
Lemma 6 (Hessian Lipschitz bound in s s s ). Under A1–A3, for all x , y x,y x , y with ∥ x ∥ ∞ , ∥ y ∥ ∞ ≤ C s + 1 2 \lVert x\rVert_{\infty},\lVert y\rVert_{\infty}\le C_s+\frac{1}{2} ∥ x ∥ ∞ , ∥ y ∥ ∞ ≤ C s + 2 1 and any fixed γ , β , \gamma,\beta, γ , β ,
∥ J ( x , γ , β ) − J ( y , γ , β ) ∥ ∞ ≤ 4 L f ( max a m a ) ∥ x − y ∥ ∞ ≤ 4 L f m ∗ n ∥ x − y ∥ ∞ , \lVert J(x,\gamma,\beta)-J(y,\gamma,\beta)\rVert_{\infty}\le4L_f\left(\max_a m_a\right)\lVert x-y\rVert_{\infty}\le4L_fm^{*}n\lVert x-y\rVert_{\infty}, ∥ J ( x , γ , β ) − J ( y , γ , β ) ∥ ∞ ≤ 4 L f ( a max m a ) ∥ x − y ∥ ∞ ≤ 4 L f m ∗ n ∥ x − y ∥ ∞ , where L f = 1 4 F 1 3 + 3 4 F 1 F 2 + F 3 + 1 4 L_f=\frac{1}{4}F_1^3+\frac{3}{4}F_1F_2+F_3+\frac{1}{4} L f = 4 1 F 1 3 + 4 3 F 1 F 2 + F 3 + 4 1 and m a : = ∑ b ≠ a m a b m_a:=\sum_{b\ne a}m_{ab} m a := ∑ b = a m ab . The same bound holds verbatim for the leave-one-out Hessian J ( m ) J^{(m)} J ( m ) of (B.28).
Proof. Since J J J depends on s s s only through the gaps, it suffices to bound the scalar derivative W i j k ′ ( Δ ) : = d W i j k d Δ W'_{ijk}(\Delta):=\frac{dW_{ijk}}{d\Delta} W ij k ′ ( Δ ) := d Δ d W ij k from (B.15). Using
d g d Δ = w η f ′ , d w η d Δ = w η ( 1 − 2 g ) f ′ , d d d Δ = w θ , d w θ d Δ = w θ ( 1 − 2 d ) , d c d Δ = 0 , \frac{dg}{d\Delta}=w_\eta f',\qquad\frac{dw_\eta}{d\Delta}=w_\eta(1-2g)f',\qquad\frac{dd}{d\Delta}=w_\theta,\qquad\frac{dw_\theta}{d\Delta}=w_\theta(1-2d),\qquad\frac{dc}{d\Delta}=0, d Δ d g = w η f ′ , d Δ d w η = w η ( 1 − 2 g ) f ′ , d Δ dd = w θ , d Δ d w θ = w θ ( 1 − 2 d ) , d Δ d c = 0 ,
the three terms of W W W differentiate to
d d Δ [ w η f ′ 2 ] = w η ( 1 − 2 g ) f ′ 3 + 2 w η f ′ f ′ ′ , \frac{d}{d\Delta}\left[w_\eta f'^2\right]=w_\eta(1-2g)f'^3+2w_\eta f'f'', d Δ d [ w η f ′2 ] = w η ( 1 − 2 g ) f ′3 + 2 w η f ′ f ′′ ,
d d Δ [ c w θ ] = c w θ ( 1 − 2 d ) , \frac{d}{d\Delta}\left[cw_\theta\right]=cw_\theta(1-2d), d Δ d [ c w θ ] = c w θ ( 1 − 2 d ) ,
d d Δ [ − ( c − g ) f ′ ′ ] = w η f ′ f ′ ′ − ( c − g ) f ′ ′ ′ . \frac{d}{d\Delta}\left[-(c-g)f''\right]=w_\eta f'f''-(c-g)f'''. d Δ d [ − ( c − g ) f ′′ ] = w η f ′ f ′′ − ( c − g ) f ′′′ .
Adding (the two w η f ′ f ′ ′ w_\eta f'f'' w η f ′ f ′′ terms combine to 3 w η f ′ f ′ ′ 3w_\eta f'f'' 3 w η f ′ f ′′ ),
W ′ ( Δ ) = w η ( 1 − 2 g ) f ′ 3 + 3 w η f ′ f ′ ′ + c w θ ( 1 − 2 d ) − ( c − g ) f ′ ′ ′ . W'(\Delta)=w_\eta(1-2g)f'^3+3w_\eta f'f''+cw_\theta(1-2d)-(c-g)f'''. W ′ ( Δ ) = w η ( 1 − 2 g ) f ′3 + 3 w η f ′ f ′′ + c w θ ( 1 − 2 d ) − ( c − g ) f ′′′ .
Bounding with w η , w θ ≤ 1 4 w_\eta,w_\theta\le\frac{1}{4} w η , w θ ≤ 4 1 , ∣ 1 − 2 g ∣ , ∣ 1 − 2 d ∣ ≤ 1 \lvert1-2g\rvert,\lvert1-2d\rvert\le1 ∣ 1 − 2 g ∣ , ∣ 1 − 2 d ∣ ≤ 1 , c ∈ [ 0 , 1 ] c\in[0,1] c ∈ [ 0 , 1 ] , ∣ c − g ∣ ≤ 1 \lvert c-g\rvert\le1 ∣ c − g ∣ ≤ 1 , and ∣ f ′ ∣ ≤ F 1 , ∣ f ′ ′ ∣ ≤ F 2 , ∣ f ′ ′ ′ ∣ ≤ F 3 \lvert f'\rvert\le F_1,\lvert f''\rvert\le F_2,\lvert f'''\rvert\le F_3 ∣ f ′ ∣ ≤ F 1 , ∣ f ′′ ∣ ≤ F 2 , ∣ f ′′′ ∣ ≤ F 3 (valid since ∥ x ∥ ∞ , ∥ y ∥ ∞ ≤ C s + 1 2 \lVert x\rVert_\infty,\lVert y\rVert_\infty\le C_s+\frac{1}{2} ∥ x ∥ ∞ , ∥ y ∥ ∞ ≤ C s + 2 1 gives gaps of magnitude at most 2 C s + 1 2C_s+1 2 C s + 1 , the range of A3) gives ∣ W ′ ( Δ ) ∣ ≤ 1 4 F 1 3 + 3 4 F 1 F 2 + 1 4 + F 3 = L f \lvert W'(\Delta)\rvert\le\frac{1}{4}F_1^3+\frac{3}{4}F_1F_2+\frac{1}{4}+F_3=L_f ∣ W ′ ( Δ )∣ ≤ 4 1 F 1 3 + 4 3 F 1 F 2 + 4 1 + F 3 = L f . To assemble: J J J is a Laplacian with edge weights W i j = ∑ k W i j k W_{ij}=\sum_k W_{ijk} W ij = ∑ k W ij k ; set δ W a b = W a b ( x ) − W a b ( y ) \delta W_{ab}=W_{ab}(x)-W_{ab}(y) δ W ab = W ab ( x ) − W ab ( y ) . Row a a a of J ( x ) − J ( y ) J(x)-J(y) J ( x ) − J ( y ) has diagonal ∑ b ≠ a δ W a b \sum_{b\ne a}\delta W_{ab} ∑ b = a δ W ab and off-diagonals − δ W a b -\delta W_{ab} − δ W ab , so
∥ ( J ( x ) − J ( y ) ) a ⋅ ∥ 1 = ∣ ∑ b ≠ a δ W a b ∣ + ∑ b ≠ a ∣ δ W a b ∣ ≤ 2 ∑ b ≠ a ∣ δ W a b ∣ . \lVert(J(x)-J(y))_{a\cdot}\rVert_1=\left|\sum_{b\ne a}\delta W_{ab}\right|+\sum_{b\ne a}|\delta W_{ab}|\le2\sum_{b\ne a}|\delta W_{ab}|. ∥( J ( x ) − J ( y ) ) a ⋅ ∥ 1 = b = a ∑ δ W ab + b = a ∑ ∣ δ W ab ∣ ≤ 2 b = a ∑ ∣ δ W ab ∣.
By the mean value theorem and ∣ Δ a b x − Δ a b y ∣ = ∣ ( x a − y a ) − ( x b − y b ) ∣ ≤ 2 ∥ x − y ∥ ∞ \lvert\Delta^x_{ab}-\Delta^y_{ab}\rvert=\lvert(x_a-y_a)-(x_b-y_b)\rvert\le2\lVert x-y\rVert_\infty ∣ Δ ab x − Δ ab y ∣ = ∣( x a − y a ) − ( x b − y b )∣ ≤ 2 ∥ x − y ∥ ∞ , each comparison contributes ∣ W a b k ′ ∣ ∣ Δ a b x − Δ a b y ∣ ≤ 2 L f ∥ x − y ∥ ∞ \lvert W'_{abk}\rvert\lvert\Delta^x_{ab}-\Delta^y_{ab}\rvert\le2L_f\lVert x-y\rVert_\infty ∣ W abk ′ ∣ ∣ Δ ab x − Δ ab y ∣ ≤ 2 L f ∥ x − y ∥ ∞ , so ∣ δ W a b ∣ ≤ m a b L f ⋅ 2 ∥ x − y ∥ ∞ \lvert\delta W_{ab}\rvert\le m_{ab}L_f\cdot2\lVert x-y\rVert_\infty ∣ δ W ab ∣ ≤ m ab L f ⋅ 2 ∥ x − y ∥ ∞ . Summing over b b b ,
∥ ( J ( x ) − J ( y ) ) a ⋅ ∥ 1 ≤ 4 L f ( ∑ b ≠ a m a b ) ∥ x − y ∥ ∞ = 4 L f m a ∥ x − y ∥ ∞ , \lVert(J(x)-J(y))_{a\cdot}\rVert_1\le4L_f\left(\sum_{b\ne a}m_{ab}\right)\lVert x-y\rVert_\infty=4L_fm_a\lVert x-y\rVert_\infty, ∥( J ( x ) − J ( y ) ) a ⋅ ∥ 1 ≤ 4 L f b = a ∑ m ab ∥ x − y ∥ ∞ = 4 L f m a ∥ x − y ∥ ∞ ,
and maximizing over a a a and using m a ≤ m ∗ n m_a\le m^{*}n m a ≤ m ∗ n (A1) gives the claim. For J ( m ) J^{(m)} J ( m ) the m m m -edge weight W i m k ( m ) W^{(m)}_{imk} W imk ( m ) has, by the same calculation,
∣ d W i m k ( m ) d Δ ∣ = ∣ w η ( 1 − 2 g ) f ′ 3 + 3 w η f ′ f ′ ′ + g ∗ w θ ( 1 − 2 d ) − ( g ∗ − g ) f ′ ′ ′ ∣ ≤ L f \left|\frac{dW^{(m)}_{imk}}{d\Delta}\right|=\left|w_\eta(1-2g)f'^3+3w_\eta f'f''+g^{*}w_\theta(1-2d)-(g^{*}-g)f'''\right|\le L_f d Δ d W imk ( m ) = w η ( 1 − 2 g ) f ′3 + 3 w η f ′ f ′′ + g ∗ w θ ( 1 − 2 d ) − ( g ∗ − g ) f ′′′ ≤ L f
(now g ∗ ≤ 1 g^{*}\le1 g ∗ ≤ 1 replaces c ∈ [ 0 , 1 ] c\in[0,1] c ∈ [ 0 , 1 ] and ∣ g ∗ − g ∣ ≤ 1 \lvert g^{*}-g\rvert\le1 ∣ g ∗ − g ∣ ≤ 1 replaces ∣ c − g ∣ ≤ 1 \lvert c-g\rvert\le1 ∣ c − g ∣ ≤ 1 ), so the identical bound holds. ◻
Lemma 7 (Hessian Lipschitz bound in ( γ , β ) (\gamma,\beta) ( γ , β ) ). *Under A1–A3, there is an absolute constant L ζ L_\zeta L ζ such that for all s s s with ∥ s ∥ ∞ ≤ C s + 1 2 \lVert s\rVert_\infty\le C_s+\frac{1}{2} ∥ s ∥ ∞ ≤ C s + 2 1 and all γ , γ ′ ∈ R p γ \gamma,\gamma'\in\mathbb{R}^{p_\gamma} γ , γ ′ ∈ R p γ , β , β ′ ∈ R p β , \beta,\beta'\in\mathbb{R}^{p_\beta}, β , β ′ ∈ R p β ,
∥ J ( s , γ , β ) − J ( s , γ ′ , β ′ ) ∥ ∞ ≤ L ζ n ( ∥ γ − γ ′ ∥ 2 + ∥ β − β ′ ∥ 2 ) . \lVert J(s,\gamma,\beta)-J(s,\gamma',\beta')\rVert_\infty\le L_\zeta n\left(\lVert\gamma-\gamma'\rVert_2+\lVert\beta-\beta'\rVert_2\right). ∥ J ( s , γ , β ) − J ( s , γ ′ , β ′ ) ∥ ∞ ≤ L ζ n ( ∥ γ − γ ′ ∥ 2 + ∥ β − β ′ ∥ 2 ) . The same bound holds verbatim for the leave-one-out Hessian J ( m ) J^{(m)} J ( m ) of (B.28).
Proof. As in Lemma 6, J J J is a Laplacian with edge weights W i j = ∑ k W i j k W_{ij} = \sum_k W_{ijk} W ij = ∑ k W ij k , and the row-sum norm is controlled by ∑ b ≠ a , k ∣ W a b k ( γ , β ) − W a b k ( γ ′ , β ′ ) ∣ \sum_{b\ne a,k} \lvert W_{abk}(\gamma,\beta)-W_{abk}(\gamma',\beta')\rvert ∑ b = a , k ∣ W abk ( γ , β ) − W abk ( γ ′ , β ′ )∣ up to the factor 2 from the diagonal-plus-off-diagonal structure. It therefore suffices to bound the per-comparison increment of W i j k W_{ijk} W ij k in ( γ , β ) (\gamma,\beta) ( γ , β ) at fixed s s s .
Recall W i j k = w η f ′ ( Δ i j ) 2 + c w θ − ( c − g ) f ′ ′ ( Δ i j ) W_{ijk}=w_\eta f'(\Delta_{ij})^2+cw_\theta-(c-g)f''(\Delta_{ij}) W ij k = w η f ′ ( Δ ij ) 2 + c w θ − ( c − g ) f ′′ ( Δ ij ) with g = σ ( f ( Δ i j ) + Y i j k T γ ) g=\sigma(f(\Delta_{ij})+Y_{ijk}^{\mathsf T}\gamma) g = σ ( f ( Δ ij ) + Y ij k T γ ) , w η = g ( 1 − g ) w_\eta=g(1-g) w η = g ( 1 − g ) , d = σ ( Δ i j + Z i j k T β ) d=\sigma(\Delta_{ij}+Z_{ijk}^{\mathsf T}\beta) d = σ ( Δ ij + Z ij k T β ) , w θ = d ( 1 − d ) w_\theta=d(1-d) w θ = d ( 1 − d ) . Differentiating through the predictors η = f + Y T γ \eta=f+Y^{\mathsf T}\gamma η = f + Y T γ and θ = Δ + Z T β \theta=\Delta+Z^{\mathsf T}\beta θ = Δ + Z T β :
∇ γ g = w η Y , ∇ γ w η = w η ( 1 − 2 g ) Y , ∇ β d = w θ Z , ∇ β w θ = w θ ( 1 − 2 d ) Z , \nabla_\gamma g=w_\eta Y,\qquad\nabla_\gamma w_\eta=w_\eta(1-2g)Y,\qquad\nabla_\beta d=w_\theta Z,\qquad\nabla_\beta w_\theta=w_\theta(1-2d)Z, ∇ γ g = w η Y , ∇ γ w η = w η ( 1 − 2 g ) Y , ∇ β d = w θ Z , ∇ β w θ = w θ ( 1 − 2 d ) Z ,
while ∇ γ Δ = ∇ β Δ = 0 \nabla_\gamma\Delta=\nabla_\beta\Delta=0 ∇ γ Δ = ∇ β Δ = 0 at fixed s s s and ∇ β g = ∇ γ d = 0 \nabla_\beta g=\nabla_\gamma d=0 ∇ β g = ∇ γ d = 0 . Hence
∇ γ W i j k = [ w η ( 1 − 2 g ) f ′ ( Δ i j ) 2 + f ′ ′ ( Δ i j ) w η ] Y i j k , \nabla_\gamma W_{ijk}=\left[w_\eta(1-2g)f'(\Delta_{ij})^2+f''(\Delta_{ij})w_\eta\right]Y_{ijk}, ∇ γ W ij k = [ w η ( 1 − 2 g ) f ′ ( Δ ij ) 2 + f ′′ ( Δ ij ) w η ] Y ij k ,
∇ β W i j k = c w θ ( 1 − 2 d ) Z i j k . \nabla_\beta W_{ijk}=cw_\theta(1-2d)Z_{ijk}. ∇ β W ij k = c w θ ( 1 − 2 d ) Z ij k .
Using w η , w θ ≤ 1 4 w_\eta,w_\theta\le\frac{1}{4} w η , w θ ≤ 4 1 , ∣ 1 − 2 g ∣ , ∣ 1 − 2 d ∣ ≤ 1 \lvert1-2g\rvert,\lvert1-2d\rvert\le1 ∣ 1 − 2 g ∣ , ∣ 1 − 2 d ∣ ≤ 1 , c ∈ [ 0 , 1 ] c\in[0,1] c ∈ [ 0 , 1 ] , ∣ f ′ ∣ ≤ F 1 \lvert f'\rvert\le F_1 ∣ f ′ ∣ ≤ F 1 , ∣ f ′ ′ ∣ ≤ F 2 \lvert f''\rvert\le F_2 ∣ f ′′ ∣ ≤ F 2 on the working region (A3), and ∥ Y i j k ∥ 2 , ∥ Z i j k ∥ 2 ≤ κ \lVert Y_{ijk}\rVert_2,\lVert Z_{ijk}\rVert_2\le\kappa ∥ Y ij k ∥ 2 , ∥ Z ij k ∥ 2 ≤ κ (A2),
∥ ∇ γ W i j k ∥ 2 ≤ 1 4 ( F 1 2 + F 2 ) κ , ∥ ∇ β W i j k ∥ 2 ≤ 1 4 κ . \lVert\nabla_\gamma W_{ijk}\rVert_2\le\frac{1}{4}(F_1^2+F_2)\kappa,\qquad\lVert\nabla_\beta W_{ijk}\rVert_2\le\frac{1}{4}\kappa. ∥ ∇ γ W ij k ∥ 2 ≤ 4 1 ( F 1 2 + F 2 ) κ , ∥ ∇ β W ij k ∥ 2 ≤ 4 1 κ .
By the mean value theorem along the segment from ( γ ′ , β ′ ) (\gamma',\beta') ( γ ′ , β ′ ) to ( γ , β ) (\gamma,\beta) ( γ , β ) ,
∣ W a b k ( γ , β ) − W a b k ( γ ′ , β ′ ) ∣ ≤ 1 4 ( F 1 2 + F 2 ) κ ∥ γ − γ ′ ∥ 2 + 1 4 κ ∥ β − β ′ ∥ 2 . \lvert W_{abk}(\gamma,\beta)-W_{abk}(\gamma',\beta')\rvert\le\frac{1}{4}(F_1^2+F_2)\kappa\lVert\gamma-\gamma'\rVert_2+\frac{1}{4}\kappa\lVert\beta-\beta'\rVert_2. ∣ W abk ( γ , β ) − W abk ( γ ′ , β ′ )∣ ≤ 4 1 ( F 1 2 + F 2 ) κ ∥ γ − γ ′ ∥ 2 + 4 1 κ ∥ β − β ′ ∥ 2 .
Summing over the ≤ m ∗ \le m^* ≤ m ∗ comparisons per pair and over b ≠ a b\ne a b = a , applying the factor 2 from the Laplacian row structure, and maximizing over a a a ,
∥ J ( s , γ , β ) − J ( s , γ ′ , β ′ ) ∥ ∞ ≤ 1 2 m ∗ κ n ( ( F 1 2 + F 2 ) ∥ γ − γ ′ ∥ 2 + ∥ β − β ′ ∥ 2 ) ≤ L ζ n ( ∥ γ − γ ′ ∥ 2 + ∥ β − β ′ ∥ 2 ) , \begin{aligned}
\lVert J(s,\gamma,\beta)-J(s,\gamma',\beta')\rVert_\infty\le\frac{1}{2}m^*\kappa n\left((F_1^2+F_2)\lVert\gamma-\gamma'\rVert_2+\lVert\beta-\beta'\rVert_2\right) \\
&\le L_\zeta n\left(\lVert\gamma-\gamma'\rVert_2+\lVert\beta-\beta'\rVert_2\right),
\end{aligned} ∥ J ( s , γ , β ) − J ( s , γ ′ , β ′ ) ∥ ∞ ≤ 2 1 m ∗ κn ( ( F 1 2 + F 2 ) ∥ γ − γ ′ ∥ 2 + ∥ β − β ′ ∥ 2 ) ≤ L ζ n ( ∥ γ − γ ′ ∥ 2 + ∥ β − β ′ ∥ 2 ) ,
with L ζ : = 1 2 m ∗ κ max { F 1 2 + F 2 , 1 } L_\zeta:=\frac{1}{2}m^*\kappa\max\{F_1^2+F_2,1\} L ζ := 2 1 m ∗ κ max { F 1 2 + F 2 , 1 } , an absolute constant under A1–A3.
For J ( m ) J^{(m)} J ( m ) , the m m m -edge weight W i m k ( m ) W_{imk}^{(m)} W imk ( m ) of (B.29) has, by the same differentiation (the saboteur − ( g i m k ∗ − g i m k ) f ′ ′ ( Δ i m ) -(g^*_{imk}-g_{imk})f''(\Delta_{im}) − ( g imk ∗ − g imk ) f ′′ ( Δ im ) contributing + ∇ γ g i m k f ′ ′ ( Δ i m ) = w η , i m k f ′ ′ ( Δ i m ) Y i m k +\nabla_\gamma g_{imk}f''(\Delta_{im})=w_{\eta,imk}f''(\Delta_{im})Y_{imk} + ∇ γ g imk f ′′ ( Δ im ) = w η , imk f ′′ ( Δ im ) Y imk , exactly as − ( c i j k − g i j k ) f ′ ′ -(c_{ijk}-g_{ijk})f'' − ( c ij k − g ij k ) f ′′ does above),
∇ γ W i m k ( m ) = [ w η , i m k ( 1 − 2 g i m k ) f ′ ( Δ i m ) 2 + w η , i m k f ′ ′ ( Δ i m ) ] Y i m k , \nabla_\gamma W_{imk}^{(m)}=\left[w_{\eta,imk}(1-2g_{imk})f'(\Delta_{im})^2+w_{\eta,imk}f''(\Delta_{im})\right]Y_{imk}, ∇ γ W imk ( m ) = [ w η , imk ( 1 − 2 g imk ) f ′ ( Δ im ) 2 + w η , imk f ′′ ( Δ im ) ] Y imk ,
∇ β W i m k ( m ) = g i m k ∗ w θ , i m k ( 1 − 2 d i m k ) Z i m k , \nabla_\beta W_{imk}^{(m)}=g^*_{imk}w_{\theta,imk}(1-2d_{imk})Z_{imk}, ∇ β W imk ( m ) = g imk ∗ w θ , imk ( 1 − 2 d imk ) Z imk ,
so with g i m k ∗ ≤ 1 g^*_{imk}\le1 g imk ∗ ≤ 1 replacing c i j k ∈ [ 0 , 1 ] c_{ijk}\in[0,1] c ij k ∈ [ 0 , 1 ] the bounds ∥ ∇ γ W i m k ( m ) ∥ 2 ≤ 1 4 ( F 1 2 + F 2 ) κ \lVert\nabla_\gamma W_{imk}^{(m)}\rVert_2\le\frac{1}{4}(F_1^2+F_2)\kappa ∥ ∇ γ W imk ( m ) ∥ 2 ≤ 4 1 ( F 1 2 + F 2 ) κ and ∥ ∇ β W i m k ( m ) ∥ 2 ≤ 1 4 κ \lVert\nabla_\beta W_{imk}^{(m)}\rVert_2\le\frac{1}{4}\kappa ∥ ∇ β W imk ( m ) ∥ 2 ≤ 4 1 κ are unchanged, and the row-sum assembly gives the identical bound. □
Lemma 8 (Cross-Jacobian bound). Assume A1–A3 and n ≥ 2 n\ge2 n ≥ 2 . There is an absolute constant C M C_M C M such that, for all ( γ , β ) ∈ R p γ × R p β (\gamma,\beta)\in\mathbb{R}^{p_\gamma}\times\mathbb{R}^{p_\beta} ( γ , β ) ∈ R p γ × R p β and all s s s with ∥ s ∥ ∞ ≤ C s + 1 2 \lVert s\rVert_\infty\le C_s+\frac{1}{2} ∥ s ∥ ∞ ≤ C s + 2 1 ,
∥ ∂ G γ , β ( s ) ∂ ( γ , β ) ∥ 2 ≤ C M n 3 / 2 . \left\lVert\frac{\partial G_{\gamma,\beta}(s)}{\partial(\gamma,\beta)}\right\rVert_2\le C_M n^{3/2}. ∂ ( γ , β ) ∂ G γ , β ( s ) 2 ≤ C M n 3/2 . The same bound holds verbatim for the leave-one-out cross-Jacobian ∂ G γ , β ( m ) ( s ) / ∂ ( γ , β ) \partial G_{\gamma,\beta}^{(m)}(s)/\partial(\gamma,\beta) ∂ G γ , β ( m ) ( s ) / ∂ ( γ , β ) , with G γ , β ( m ) G_{\gamma,\beta}^{(m)} G γ , β ( m ) as in (B.27).
Proof. Since G γ , β = ∇ s ( − ℓ ) G_{\gamma,\beta}=\nabla_s(-\ell) G γ , β = ∇ s ( − ℓ ) (the merit score (B.14)),
∂ G γ , β ∂ ( γ , β ) = ∇ s , ( γ , β ) 2 ( − ℓ ) = [ B γ ∣ B β ] , \frac{\partial G_{\gamma,\beta}}{\partial(\gamma,\beta)}=\nabla^2_{s,(\gamma,\beta)}(-\ell)=[B_\gamma\mid B_\beta], ∂ ( γ , β ) ∂ G γ , β = ∇ s , ( γ , β ) 2 ( − ℓ ) = [ B γ ∣ B β ] , with B γ , B β B_\gamma,B_\beta B γ , B β as in (B.19)–(B.20). We bound ∥ B γ ∥ 2 \lVert B_\gamma\rVert_2 ∥ B γ ∥ 2 and ∥ B β ∥ 2 \lVert B_\beta\rVert_2 ∥ B β ∥ 2 directly.
For B γ B_\gamma B γ , take unit u ∈ 1 ⊥ ⊂ R n + 1 u \in1^\perp\subset\mathbb{R}^{n+1} u ∈ 1 ⊥ ⊂ R n + 1 and unit v ∈ R p γ v \in\mathbb{R}^{p_\gamma} v ∈ R p γ (the columns of B γ B_\gamma B γ lie in 1 ⊥ 1^\perp 1 ⊥ , so this restriction loses nothing):
u ⊤ B γ v = ∑ i < j , k w η , i j k f ′ ( Δ i j ) ( u i − u j ) ( Y i j k ⊤ v ) . u^\top B_\gamma v = \sum_{i<j,k} w_{\eta,ijk}f'(\Delta_{ij})(u_i-u_j)(Y_{ijk}^\top v). u ⊤ B γ v = i < j , k ∑ w η , ij k f ′ ( Δ ij ) ( u i − u j ) ( Y ij k ⊤ v ) .
By Cauchy–Schwarz,
∣ u ⊤ B γ v ∣ ≤ ( ∑ i < j , k ( w η , i j k f ′ ( Δ i j ) ) 2 ( u i − u j ) 2 ⏟ = : T 1 ) 1 / 2 ( ∑ i < j , k ( Y i j k ⊤ v ) 2 ⏟ = : T 2 ) 1 / 2 . \lvert u^\top B_\gamma v\rvert\leq\left(\underbrace{\sum_{i<j,k}(w_{\eta,ijk}f'(\Delta_{ij}))^2(u_i-u_j)^2}_{=:T_1}\right)^{1/2}\left(\underbrace{\sum_{i<j,k}(Y_{ijk}^\top v)^2}_{=:T_2}\right)^{1/2}. ∣ u ⊤ B γ v ∣ ≤ =: T 1 i < j , k ∑ ( w η , ij k f ′ ( Δ ij ) ) 2 ( u i − u j ) 2 1/2 =: T 2 i < j , k ∑ ( Y ij k ⊤ v ) 2 1/2 .
For T 1 T_1 T 1 : ∣ w η , i j k f ′ ( Δ i j ) ∣ ≤ 1 4 F 1 \lvert w_{\eta,ijk}f'(\Delta_{ij})\rvert\leq\frac{1}{4}F_1 ∣ w η , ij k f ′ ( Δ ij )∣ ≤ 4 1 F 1 , and summing the ≤ m ∗ \leq m^* ≤ m ∗ comparisons per pair, ∑ i < j , k ( u i − u j ) 2 ≤ m ∗ ∑ i < j ( u i − u j ) 2 ≤ m ∗ ( n + 1 ) \sum_{i<j,k}(u_i-u_j)^2 \leq m^*\sum_{i<j}(u_i-u_j)^2 \leq m^*(n+1) ∑ i < j , k ( u i − u j ) 2 ≤ m ∗ ∑ i < j ( u i − u j ) 2 ≤ m ∗ ( n + 1 ) , so T 1 ≤ 1 4 F 1 m ∗ ( n + 1 ) T_1 \leq\frac{1}{4}F_1\sqrt{m^*(n+1)} T 1 ≤ 4 1 F 1 m ∗ ( n + 1 ) . For T 2 T_2 T 2 : ( Y i j k ⊤ v ) 2 ≤ κ 2 (Y_{ijk}^\top v)^2 \leq\kappa^2 ( Y ij k ⊤ v ) 2 ≤ κ 2 and there are ≤ ( n + 1 2 ) m ∗ \leq\binom{n+1}{2}m^* ≤ ( 2 n + 1 ) m ∗ terms, so T 2 ≤ κ ( n + 1 2 ) m ∗ ≤ 1 2 κ m ∗ ( n + 1 ) T_2 \leq\kappa\sqrt{\binom{n+1}{2}m^*} \leq\frac{1}{\sqrt{2}}\kappa\sqrt{m^*}(n+1) T 2 ≤ κ ( 2 n + 1 ) m ∗ ≤ 2 1 κ m ∗ ( n + 1 ) . Taking the supremum over u , v u,v u , v ,
∥ B γ ∥ 2 ≤ T 1 T 2 ≤ 1 4 2 F 1 κ m ∗ ( n + 1 ) 3 / 2 . \lVert B_\gamma\rVert_2 \leq T_1T_2 \leq\frac{1}{4\sqrt{2}}F_1\kappa m^*(n+1)^{3/2}. ∥ B γ ∥ 2 ≤ T 1 T 2 ≤ 4 2 1 F 1 κ m ∗ ( n + 1 ) 3/2 .
We bound B β B_\beta B β identically, with ∣ c i j k w θ , i j k ∣ ≤ 1 4 \lvert c_{ijk}w_{\theta,ijk}\rvert\leq\frac{1}{4} ∣ c ij k w θ , ij k ∣ ≤ 4 1 (so F 1 F_1 F 1 is replaced by 1 1 1 ) and Z i j k Z_{ijk} Z ij k in place of Y i j k Y_{ijk} Y ij k :
∥ B β ∥ 2 ≤ 1 4 2 κ m ∗ ( n + 1 ) 3 / 2 . \lVert B_\beta\rVert_2 \leq\frac{1}{4\sqrt{2}}\kappa m^*(n+1)^{3/2}. ∥ B β ∥ 2 ≤ 4 2 1 κ m ∗ ( n + 1 ) 3/2 .
Using ∥ [ B γ ∣ B β ] ∥ 2 ≤ ∥ B γ ∥ 2 + ∥ B β ∥ 2 \lVert[B_\gamma\mid B_\beta]\rVert_2 \leq\lVert B_\gamma\rVert_2+\lVert B_\beta\rVert_2 ∥[ B γ ∣ B β ] ∥ 2 ≤ ∥ B γ ∥ 2 + ∥ B β ∥ 2 and ( n + 1 ) 3 / 2 ≤ 2 n 3 / 2 (n+1)^{3/2}\leq2n^{3/2} ( n + 1 ) 3/2 ≤ 2 n 3/2 for n ≥ 2 n\geq2 n ≥ 2 ,
∥ ∂ G γ , β ( s ) ∂ ( γ , β ) ∥ 2 ≤ 1 2 2 ( F 1 + 1 ) κ m ∗ n 3 / 2 = C M n 3 / 2 , C M : = 1 2 2 ( F 1 + 1 ) κ m ∗ . \left\lVert\frac{\partial G_{\gamma,\beta}(s)}{\partial(\gamma,\beta)}\right\rVert_2
\leq\frac{1}{2\sqrt{2}}(F_1+1)\kappa m^*n^{3/2}=C_Mn^{3/2},
\qquad C_M:=\frac{1}{2\sqrt{2}}(F_1+1)\kappa m^*. ∂ ( γ , β ) ∂ G γ , β ( s ) 2 ≤ 2 2 1 ( F 1 + 1 ) κ m ∗ n 3/2 = C M n 3/2 , C M := 2 2 1 ( F 1 + 1 ) κ m ∗ .
For the leave-one-out variant, differentiating the m m m -edge terms of (B.27) at fixed s s s (using ∇ γ ( g i m k ∗ − g i m k ) = − w η , i m k Y i m k \nabla_\gamma(g^*_{imk}-g_{imk})=-w_{\eta,imk}Y_{imk} ∇ γ ( g imk ∗ − g imk ) = − w η , imk Y imk and ∇ β ( d i m k ∗ − d i m k ) = − w θ , i m k Z i m k \nabla_\beta(d^*_{imk}-d_{imk})=-w_{\theta,imk}Z_{imk} ∇ β ( d imk ∗ − d imk ) = − w θ , imk Z imk , the starred quantities being constants) gives
B γ ( m ) : = ∂ γ G γ , β ( m ) = ∑ i < j m ∉ { i , j } ∑ k w η , i j k f ′ ( Δ i j ) ω i j Y i j k ⊤ + ∑ i ≠ m ∑ k w η , i m k f ′ ( Δ i m ) ω i m Y i m k ⊤ , B_\gamma^{(m)}:=\partial_\gamma G_{\gamma,\beta}^{(m)}
=\sum_{\substack{i<j\\m\notin\{i,j\}}}\sum_k w_{\eta,ijk}f'(\Delta_{ij})\omega_{ij}Y_{ijk}^{\top}
+\sum_{i\ne m}\sum_k w_{\eta,imk}f'(\Delta_{im})\omega_{im}Y_{imk}^{\top}, B γ ( m ) := ∂ γ G γ , β ( m ) = i < j m ∈ / { i , j } ∑ k ∑ w η , ij k f ′ ( Δ ij ) ω ij Y ij k ⊤ + i = m ∑ k ∑ w η , imk f ′ ( Δ im ) ω im Y imk ⊤ ,
and B β ( m ) B_\beta^{(m)} B β ( m ) analogously with g i m k ∗ w θ , i m k g^*_{imk}w_{\theta,imk} g imk ∗ w θ , imk in place of c i j k w θ , i j k c_{ijk}w_{\theta,ijk} c ij k w θ , ij k . Every column again lies in 1 ⊥ 1^\perp 1 ⊥ , each coefficient obeys the same bounds (∣ w η , i m k f ′ ( Δ i m ) ∣ ≤ 1 4 F 1 \lvert w_{\eta,imk}f'(\Delta_{im})\rvert\leq\frac{1}{4}F_1 ∣ w η , imk f ′ ( Δ im )∣ ≤ 4 1 F 1 , ∣ g i m k ∗ w θ , i m k ∣ ≤ 1 4 \lvert g^*_{imk}w_{\theta,imk}\rvert\leq\frac{1}{4} ∣ g imk ∗ w θ , imk ∣ ≤ 4 1 , with g i m k ∗ ≤ 1 g^*_{imk}\leq1 g imk ∗ ≤ 1 replacing c i j k ∈ [ 0 , 1 ] c_{ijk}\in[0,1] c ij k ∈ [ 0 , 1 ] ), and the sums range over each unordered pair exactly once with ≤ m ∗ \leq m^* ≤ m ∗ comparisons, so the Cauchy–Schwarz argument applies verbatim and yields the identical bound C M n 3 / 2 C_Mn^{3/2} C M n 3/2 .
Lemma 9 (Cross-block Lipschitz bound). *Assume A1–A3 and n ≥ 2 n\geq2 n ≥ 2 . Write B ( s , γ , β ) : = [ B γ ∣ B β ] B(s,\gamma,\beta):=[B_\gamma\mid B_\beta] B ( s , γ , β ) := [ B γ ∣ B β ] for the merit–structural cross block of (B.19)–(B.20). There is an absolute constant L B L_B L B such that, for all s , s ′ s,s' s , s ′ with ∥ s ∥ ∞ , ∥ s ′ ∥ ∞ ≤ C s + 1 2 \lVert s\rVert_\infty,\lVert s'\rVert_\infty\leq C_s+\frac{1}{2} ∥ s ∥ ∞ , ∥ s ′ ∥ ∞ ≤ C s + 2 1 and all ζ = ( γ , β ) \zeta=(\gamma,\beta) ζ = ( γ , β ) , ζ ′ = ( γ ′ , β ′ ) \zeta'=(\gamma',\beta') ζ ′ = ( γ ′ , β ′ ) ,
∥ B ( s , ζ ) − B ( s ′ , ζ ′ ) ∥ 2 ≤ L B n 3 / 2 ( ∥ s − s ′ ∥ ∞ + ∥ ζ − ζ ′ ∥ 2 ) . \lVert B(s,\zeta)-B(s',\zeta')\rVert_2\leq L_Bn^{3/2}\left(\lVert s-s'\rVert_\infty+\lVert\zeta-\zeta'\rVert_2\right). ∥ B ( s , ζ ) − B ( s ′ , ζ ′ ) ∥ 2 ≤ L B n 3/2 ( ∥ s − s ′ ∥ ∞ + ∥ ζ − ζ ′ ∥ 2 ) .
Proof. Both blocks share the structure ∑ i < j ∑ k ϕ i j k ω i j v i j k ⊤ \sum_{i<j}\sum_k\phi_{ijk}\omega_{ij}v_{ijk}^{\top} ∑ i < j ∑ k ϕ ij k ω ij v ij k ⊤ with a scalar per-comparison coefficient ϕ i j k \phi_{ijk} ϕ ij k and a fixed covariate v i j k v_{ijk} v ij k : for B γ B_\gamma B γ , ϕ i j k = w η , i j k f ′ ( Δ i j ) \phi_{ijk}=w_{\eta,ijk}f'(\Delta_{ij}) ϕ ij k = w η , ij k f ′ ( Δ ij ) and v i j k = Y i j k v_{ijk}=Y_{ijk} v ij k = Y ij k ; for B β B_\beta B β , ϕ i j k = c i j k w θ , i j k \phi_{ijk}=c_{ijk}w_{\theta,ijk} ϕ ij k = c ij k w θ , ij k and v i j k = Z i j k v_{ijk}=Z_{ijk} v ij k = Z ij k . The difference of two such blocks has the same structure with coefficient δ i j k : = ϕ i j k ( s , ζ ) − ϕ i j k ( s ′ , ζ ′ ) \delta_{ijk}:=\phi_{ijk}(s,\zeta)-\phi_{ijk}(s',\zeta') δ ij k := ϕ ij k ( s , ζ ) − ϕ ij k ( s ′ , ζ ′ ) , so the Cauchy–Schwarz factorization of Lemma 8 applies verbatim: for unit u ∈ 1 ⊥ u\in1^\perp u ∈ 1 ⊥ and unit v v v ,
∣ u ⊤ ( B γ ( s , ζ ) − B γ ( s ′ , ζ ′ ) ) v ∣ ≤ ( ∑ i < j , k δ i j k 2 ( u i − u j ) 2 ) 1 / 2 ( ∑ i < j , k ( Y i j k ⊤ v ) 2 ) 1 / 2 ≤ max i < j , k ∣ δ i j k ∣ ⋅ 1 2 κ m ∗ ( n + 1 ) 3 / 2 . \begin{aligned}
\left\lvert u^\top\left(B_\gamma(s,\zeta)-B_\gamma(s',\zeta')\right)v\right\rvert
\leq\left(\sum_{i<j,k}\delta_{ijk}^2(u_i-u_j)^2\right)^{1/2}
\left(\sum_{i<j,k}(Y_{ijk}^\top v)^2\right)^{1/2} \\
&\leq\max_{i<j,k}\lvert\delta_{ijk}\rvert\cdot\frac{1}{\sqrt{2}}\kappa m^*(n+1)^{3/2}.
\end{aligned} u ⊤ ( B γ ( s , ζ ) − B γ ( s ′ , ζ ′ ) ) v ≤ i < j , k ∑ δ ij k 2 ( u i − u j ) 2 1/2 i < j , k ∑ ( Y ij k ⊤ v ) 2 1/2 ≤ i < j , k max ∣ δ ij k ∣ ⋅ 2 1 κ m ∗ ( n + 1 ) 3/2 .
It remains to bound the coefficient increments. For B γ B_\gamma B γ : on the working region,
∣ ∂ ( w η f ′ ) ∂ Δ ∣ = ∣ w η ( 1 − 2 g ) f ′ 2 + w η f ′ ′ ∣ ≤ 1 4 ( F 1 2 + F 2 ) , \left|\frac{\partial(w_\eta f')}{\partial\Delta}\right|=\left|w_\eta(1-2g)f'^2+w_\eta f''\right|\leq\frac{1}{4}(F_1^2+F_2), ∂ Δ ∂ ( w η f ′ ) = w η ( 1 − 2 g ) f ′2 + w η f ′′ ≤ 4 1 ( F 1 2 + F 2 ) ,
∥ ∇ γ ( w η f ′ ) ∥ 2 = ∣ w η ( 1 − 2 g ) f ′ ∣ ∥ Y i j k ∥ 2 ≤ 1 4 F 1 κ , \left\|\nabla_\gamma(w_\eta f')\right\|_2=\left|w_\eta(1-2g)f'\right|\left\|Y_{ijk}\right\|_2\leq\frac{1}{4}F_1\kappa, ∥ ∇ γ ( w η f ′ ) ∥ 2 = ∣ w η ( 1 − 2 g ) f ′ ∣ ∥ Y ij k ∥ 2 ≤ 4 1 F 1 κ ,
and ∇ β ( w η f ′ ) = 0 \nabla_\beta(w_\eta f')=0 ∇ β ( w η f ′ ) = 0 ; with ∣ Δ i j ( s ) − Δ i j ( s ′ ) ∣ ≤ 2 ∥ s − s ′ ∥ ∞ |\Delta_{ij}(s)-\Delta_{ij}(s')|\leq2\|s-s'\|_\infty ∣ Δ ij ( s ) − Δ ij ( s ′ ) ∣ ≤ 2∥ s − s ′ ∥ ∞ and the mean value theorem along the segment (which stays in the working region, the ℓ ∞ \ell_\infty ℓ ∞ ball being convex),
∣ δ i j k ∣ ≤ 1 2 ( F 1 2 + F 2 ) ∥ s − s ′ ∥ ∞ + 1 4 F 1 κ ∥ γ − γ ′ ∥ 2 . |\delta_{ijk}|\leq\frac{1}{2}(F_1^2+F_2)\|s-s'\|_\infty+\frac{1}{4}F_1\kappa\|\gamma-\gamma'\|_2. ∣ δ ij k ∣ ≤ 2 1 ( F 1 2 + F 2 ) ∥ s − s ′ ∥ ∞ + 4 1 F 1 κ ∥ γ − γ ′ ∥ 2 .
For B β B_\beta B β : at fixed data c i j k ∈ { 0 , 1 } c_{ijk}\in\{0,1\} c ij k ∈ { 0 , 1 } , ∣ ∂ ( c w θ ) / ∂ Δ ∣ = ∣ c w θ ( 1 − 2 d ) ∣ ≤ 1 4 \left|\partial(cw_\theta)/\partial\Delta\right|=|cw_\theta(1-2d)|\leq\frac{1}{4} ∣ ∂ ( c w θ ) / ∂ Δ ∣ = ∣ c w θ ( 1 − 2 d ) ∣ ≤ 4 1 and ∥ ∇ β ( c w θ ) ∥ 2 ≤ 1 4 κ \|\nabla_\beta(cw_\theta)\|_2\leq\frac{1}{4}\kappa ∥ ∇ β ( c w θ ) ∥ 2 ≤ 4 1 κ , with ∇ γ ( c w θ ) = 0 \nabla_\gamma(cw_\theta)=0 ∇ γ ( c w θ ) = 0 , giving ∣ δ i j k ∣ ≤ 1 2 ∥ s − s ′ ∥ ∞ + 1 4 κ ∥ β − β ′ ∥ 2 |\delta_{ijk}|\leq\frac{1}{2}\|s-s'\|_\infty+\frac{1}{4}\kappa\|\beta-\beta'\|_2 ∣ δ ij k ∣ ≤ 2 1 ∥ s − s ′ ∥ ∞ + 4 1 κ ∥ β − β ′ ∥ 2 . Combining the two blocks by ∥ [ B γ ∣ B β ] ∥ 2 ≤ ∥ B γ ∥ 2 + ∥ B β ∥ 2 \|[B_\gamma\mid B_\beta]\|_2\leq\|B_\gamma\|_2+\|B_\beta\|_2 ∥ [ B γ ∣ B β ] ∥ 2 ≤ ∥ B γ ∥ 2 + ∥ B β ∥ 2 and using ( n + 1 ) 3 / 2 ≤ 2 n 3 / 2 (n+1)^{3/2}\leq2n^{3/2} ( n + 1 ) 3/2 ≤ 2 n 3/2 for n ≥ 2 n\geq2 n ≥ 2 ,
∥ B ( s , ζ ) − B ( s ′ , ζ ′ ) ∥ 2 ≤ L B n 3 / 2 ( ∥ s − s ′ ∥ ∞ + ∥ ζ − ζ ′ ∥ 2 ) , \|B(s,\zeta)-B(s',\zeta')\|_2\leq L_Bn^{3/2}\left(\|s-s'\|_\infty+\|\zeta-\zeta'\|_2\right), ∥ B ( s , ζ ) − B ( s ′ , ζ ′ ) ∥ 2 ≤ L B n 3/2 ( ∥ s − s ′ ∥ ∞ + ∥ ζ − ζ ′ ∥ 2 ) ,
with L B : = 2 κ m ∗ ( F 1 2 + F 2 + F 1 κ + 1 + κ ) L_B:=\sqrt{2}\kappa m^*(F_1^2+F_2+F_1\kappa+1+\kappa) L B := 2 κ m ∗ ( F 1 2 + F 2 + F 1 κ + 1 + κ ) , an absolute constant under A1–A3.
Lemma 10 (Structural-block Lipschitz bound). Assume A1–A3. Write A : = diag ( A γ γ , A β β ) A:=\operatorname{diag}(A_{\gamma\gamma},A_{\beta\beta}) A := diag ( A γ γ , A β β ) for the block-diagonal structural Hessian of (B.17) and (B.18). There is an absolute constant L A L_A L A such that, for all s , s ′ s,s' s , s ′ with ∥ s ∥ ∞ , ∥ s ′ ∥ ∞ ≤ C s + 1 2 \|s\|_\infty,\|s'\|_\infty\leq C_s+\frac{1}{2} ∥ s ∥ ∞ , ∥ s ′ ∥ ∞ ≤ C s + 2 1 and all ζ = ( γ , β ) \zeta=(\gamma,\beta) ζ = ( γ , β ) , ζ ′ = ( γ ′ , β ′ ) \zeta'=(\gamma',\beta') ζ ′ = ( γ ′ , β ′ ) ,
∥ A ( s , ζ ) − A ( s ′ , ζ ′ ) ∥ 2 ≤ L A n 2 ( ∥ s − s ′ ∥ ∞ + ∥ ζ − ζ ′ ∥ 2 ) . \|A(s,\zeta)-A(s',\zeta')\|_2\leq L_A n^2\left(\|s-s'\|_\infty+\|\zeta-\zeta'\|_2\right). ∥ A ( s , ζ ) − A ( s ′ , ζ ′ ) ∥ 2 ≤ L A n 2 ( ∥ s − s ′ ∥ ∞ + ∥ ζ − ζ ′ ∥ 2 ) .
Proof. A A A is block-diagonal, so it suffices to bound each block. For the gate block, each weight w η , i j k = g ( 1 − g ) w_{\eta,ijk}=g(1-g) w η , ij k = g ( 1 − g ) obeys, on the working region,
∣ ∂ Δ w η , i j k ∣ = ∣ w η ( 1 − 2 g ) f ′ ( Δ i j ) ∣ ≤ 1 4 F 1 , ∥ ∇ γ w η , i j k ∥ 2 = ∣ w η ( 1 − 2 g ) ∣ ∥ Y i j k ∥ 2 ≤ 1 4 κ , |\partial_\Delta w_{\eta,ijk}|=|w_\eta(1-2g)f'(\Delta_{ij})|\leq\frac{1}{4}F_1,\qquad\|\nabla_\gamma w_{\eta,ijk}\|_2=|w_\eta(1-2g)|\|Y_{ijk}\|_2\leq\frac{1}{4}\kappa, ∣ ∂ Δ w η , ij k ∣ = ∣ w η ( 1 − 2 g ) f ′ ( Δ ij ) ∣ ≤ 4 1 F 1 , ∥ ∇ γ w η , ij k ∥ 2 = ∣ w η ( 1 − 2 g ) ∣∥ Y ij k ∥ 2 ≤ 4 1 κ ,
and ∇ β w η , i j k = 0 \nabla_\beta w_{\eta,ijk}=0 ∇ β w η , ij k = 0 ; with ∣ Δ i j ( s ) − Δ i j ( s ′ ) ∣ ≤ 2 ∥ s − s ′ ∥ ∞ |\Delta_{ij}(s)-\Delta_{ij}(s')|\leq2\|s-s'\|_\infty ∣ Δ ij ( s ) − Δ ij ( s ′ ) ∣ ≤ 2∥ s − s ′ ∥ ∞ and the mean value theorem along the segment (contained in the working region by convexity),
∣ w η , i j k ( s , γ ) − w η , i j k ( s ′ , γ ′ ) ∣ ≤ 1 2 F 1 ∥ s − s ′ ∥ ∞ + 1 4 κ ∥ γ − γ ′ ∥ 2 . |w_{\eta,ijk}(s,\gamma)-w_{\eta,ijk}(s',\gamma')|\leq\frac{1}{2}F_1\|s-s'\|_\infty+\frac{1}{4}\kappa\|\gamma-\gamma'\|_2. ∣ w η , ij k ( s , γ ) − w η , ij k ( s ′ , γ ′ ) ∣ ≤ 2 1 F 1 ∥ s − s ′ ∥ ∞ + 4 1 κ ∥ γ − γ ′ ∥ 2 .
Since ∥ Y i j k Y i j k T ∥ 2 ≤ κ 2 \|Y_{ijk}Y_{ijk}^{\mathsf T}\|_2\leq\kappa^2 ∥ Y ij k Y ij k T ∥ 2 ≤ κ 2 and there are ≤ ( n + 1 2 ) m ∗ ≤ n 2 m ∗ \leq\binom{n+1}{2}m^*\leq n^2m^* ≤ ( 2 n + 1 ) m ∗ ≤ n 2 m ∗ summands, the triangle inequality gives
∥ A γ γ ( s , γ ) − A γ γ ( s ′ , γ ′ ) ∥ 2 ≤ κ 2 m ∗ n 2 ( 1 2 F 1 ∥ s − s ′ ∥ ∞ + 1 4 κ ∥ γ − γ ′ ∥ 2 ) . \|A_{\gamma\gamma}(s,\gamma)-A_{\gamma\gamma}(s',\gamma')\|_2\leq\kappa^2m^*n^2\left(\frac{1}{2}F_1\|s-s'\|_\infty+\frac{1}{4}\kappa\|\gamma-\gamma'\|_2\right). ∥ A γ γ ( s , γ ) − A γ γ ( s ′ , γ ′ ) ∥ 2 ≤ κ 2 m ∗ n 2 ( 2 1 F 1 ∥ s − s ′ ∥ ∞ + 4 1 κ ∥ γ − γ ′ ∥ 2 ) .
For the direction block, at fixed data c i j k ∈ { 0 , 1 } c_{ijk}\in\{0,1\} c ij k ∈ { 0 , 1 } , ∣ ∂ Δ ( c w θ ) ∣ ≤ 1 4 |\partial_\Delta(cw_\theta)|\leq\frac{1}{4} ∣ ∂ Δ ( c w θ ) ∣ ≤ 4 1 , ∥ ∇ β ( c w θ ) ∥ 2 ≤ 1 4 κ \|\nabla_\beta(cw_\theta)\|_2\leq\frac{1}{4}\kappa ∥ ∇ β ( c w θ ) ∥ 2 ≤ 4 1 κ , ∇ γ ( c w θ ) = 0 \nabla_\gamma(cw_\theta)=0 ∇ γ ( c w θ ) = 0 , and the same assembly applies with Z i j k Z_{ijk} Z ij k in place of Y i j k Y_{ijk} Y ij k . Summing the two blocks yields the claim with L A : = κ 2 m ∗ max { F 1 + 1 , κ } L_A:=\kappa^2m^*\max\{F_1+1,\kappa\} L A := κ 2 m ∗ max { F 1 + 1 , κ } .
Lemma 11 (Upper curvature bound). Assume A1–A3. With W max : = 1 4 F 1 2 + 1 4 + F 2 W_{\max}:=\frac{1}{4}F_1^2+\frac{1}{4}+F_2 W m a x := 4 1 F 1 2 + 4 1 + F 2 and C max : = 2 m ∗ W max C_{\max}:=2m^*W_{\max} C m a x := 2 m ∗ W m a x , the merit Hessian satisfies, for every s s s with ∥ s ∥ ∞ ≤ C s + 1 2 \|s\|_\infty\leq C_s+\frac{1}{2} ∥ s ∥ ∞ ≤ C s + 2 1 and every fixed ( γ , β ) (\gamma,\beta) ( γ , β ) ,
∥ J ( s , γ , β ) ∥ 2 ≤ C max n . \|J(s,\gamma,\beta)\|_2\leq C_{\max}n. ∥ J ( s , γ , β ) ∥ 2 ≤ C m a x n . The same bound holds for the leave-one-out Hessian J ( m ) J^{(m)} J ( m ) of (B.28).
Proof. J J J is the graph Laplacian with edge weights W i j = ∑ k W i j k W_{ij}=\sum_k W_{ijk} W ij = ∑ k W ij k , where, from (B.15), the first two terms of W i j k W_{ijk} W ij k are nonnegative and the saboteur is bounded by F 2 F_2 F 2 , so
− W max ≤ − ∣ c i j k − g i j k ∣ ∣ f ′ ′ ( Δ i j ) ∣ ≤ W i j k ≤ 1 4 F 1 2 + 1 4 + F 2 = W max , -W_{\max}\leq-|c_{ijk}-g_{ijk}||f''(\Delta_{ij})|\leq W_{ijk}\leq\frac{1}{4}F_1^2+\frac{1}{4}+F_2=W_{\max}, − W m a x ≤ − ∣ c ij k − g ij k ∣∣ f ′′ ( Δ ij ) ∣ ≤ W ij k ≤ 4 1 F 1 2 + 4 1 + F 2 = W m a x ,
i.e. ∣ W i j k ∣ ≤ W max |W_{ijk}|\leq W_{\max} ∣ W ij k ∣ ≤ W m a x .
For a graph Laplacian J = ∑ i < j W i j ω i j ω i j T J = \sum_{i<j} W_{ij}\omega_{ij}\omega_{ij}^{\mathsf{T}} J = ∑ i < j W ij ω ij ω ij T with weights W i j W_{ij} W ij , row a a a has diagonal ∑ b ≠ a W a b \sum_{b\ne a} W_{ab} ∑ b = a W ab and off-diagonals − W a b -W_{ab} − W ab , so by Gershgorin (Fact 5) every eigenvalue z z z lies in a disc of centre ∑ b ≠ a W a b \sum_{b\ne a} W_{ab} ∑ b = a W ab and radius ∑ b ≠ a ∣ W a b ∣ \sum_{b\ne a} |W_{ab}| ∑ b = a ∣ W ab ∣ for some a a a ; since J J J is symmetric, its eigenvalues are real and
∥ J ∥ 2 = max ∣ z ∣ ≤ max a ( ∣ ∑ b ≠ a W a b ∣ + ∑ b ≠ a ∣ W a b ∣ ) ≤ 2 max a ∑ b ≠ a ∣ W a b ∣ . \lVert J\rVert_2 = \max|z| \le\max_a\left(\left|\sum_{b\ne a} W_{ab}\right|+\sum_{b\ne a}|W_{ab}|\right)\le2\max_a\sum_{b\ne a}|W_{ab}|. ∥ J ∥ 2 = max ∣ z ∣ ≤ a max b = a ∑ W ab + b = a ∑ ∣ W ab ∣ ≤ 2 a max b = a ∑ ∣ W ab ∣.
Each W a b = ∑ k W a b k W_{ab}=\sum_k W_{abk} W ab = ∑ k W abk is a sum of m a b ≤ m ∗ m_{ab}\le m^* m ab ≤ m ∗ comparisons, each ∣ W a b k ∣ ≤ W max |W_{abk}|\le W_{\max} ∣ W abk ∣ ≤ W m a x , with at most n n n neighbours b b b , so ∑ b ≠ a ∣ W a b ∣ ≤ m ∗ n W max \sum_{b\ne a}|W_{ab}|\le m^*nW_{\max} ∑ b = a ∣ W ab ∣ ≤ m ∗ n W m a x . Hence ∥ J ∥ 2 ≤ 2 m ∗ W max n = C max n \lVert J\rVert_2\le2m^*W_{\max}n=C_{\max}n ∥ J ∥ 2 ≤ 2 m ∗ W m a x n = C m a x n . For J ( m ) J^{(m)} J ( m ) the m m m -edge weight also satisfies
∣ W i m k ( m ) ∣ = ∣ ω η f 2 ′ + g ∗ w θ − ( g ∗ − g ) f ′ ′ ∣ ≤ 1 4 F 1 2 + 1 4 + F 2 = W max , |W_{imk}^{(m)}|=\left|\omega_{\eta}f'_2+g^*w_{\theta}-\left(g^*-g\right)f''\right|\le\tfrac14F_1^2+\tfrac14+F_2=W_{\max}, ∣ W imk ( m ) ∣ = ∣ ω η f 2 ′ + g ∗ w θ − ( g ∗ − g ) f ′′ ∣ ≤ 4 1 F 1 2 + 4 1 + F 2 = W m a x ,
so the identical bound holds.
Lemma 12 (Local positive-definiteness of the Hessian). Assume A1–A3, with b 0 b_0 b 0 as in Lemma 1. For any sequences ε n , ρ n = o ( 1 ) \varepsilon_n,\rho_n=o(1) ε n , ρ n = o ( 1 ) the following holds: for every c > 0 c>0 c > 0 there exists n 0 n_0 n 0 such that for all n ≥ n 0 n\ge n_0 n ≥ n 0 , on E s a b \mathcal{E}_{sab} E s ab as in Lemma 4, for every ( γ , β ) (\gamma,\beta) ( γ , β ) with ∥ γ − γ ∗ ∥ 2 , ∥ β − β ∗ ∥ 2 ≤ ρ n \lVert\gamma-\gamma^*\rVert_2,\lVert\beta-\beta^*\rVert_2\le\rho_n ∥ γ − γ ∗ ∥ 2 , ∥ β − β ∗ ∥ 2 ≤ ρ n and every s ∈ B ‾ ∞ ( s ∗ , ε n ) s\in\overline{B}_\infty(s^*,\varepsilon_n) s ∈ B ∞ ( s ∗ , ε n ) ,
λ min , ⊥ ( J ( s , γ , β ) ) ≥ 1 2 n b 0 . \lambda_{\min,\perp}\bigl(J(s,\gamma,\beta)\bigr)\ge\frac{1}{2}nb_0. λ m i n , ⊥ ( J ( s , γ , β ) ) ≥ 2 1 n b 0 .
Proof. We establish the bound at s ∗ s^* s ∗ , where the only randomness is used, then extend to the ball deterministically.
Step 1: positive definiteness at s ∗ s^* s ∗ . With J ∗ = V + E J^*=V+E J ∗ = V + E as in (B.24), Lemma 2 gives λ min , ⊥ ( V ) ≥ n b 0 \lambda_{\min,\perp}(V)\ge nb_0 λ m i n , ⊥ ( V ) ≥ n b 0 and Lemma 4 gives ∥ E ∥ 2 ≤ C E n log n \lVert E\rVert_2\le C_E\sqrt{n\log n} ∥ E ∥ 2 ≤ C E n log n on an event of probability at least 1 − O ( n − c ) 1-O(n^{-c}) 1 − O ( n − c ) . Weyl’s inequality on the centered subspace yields
λ min , ⊥ ( J ∗ ) = λ min , ⊥ ( V + E ) ≥ λ min , ⊥ ( V ) − ∥ E ∥ 2 ≥ n b 0 − C E n log n . \lambda_{\min,\perp}(J^*)=\lambda_{\min,\perp}(V+E)\ge\lambda_{\min,\perp}(V)-\lVert E\rVert_2\ge nb_0-C_E\sqrt{n\log n}. λ m i n , ⊥ ( J ∗ ) = λ m i n , ⊥ ( V + E ) ≥ λ m i n , ⊥ ( V ) − ∥ E ∥ 2 ≥ n b 0 − C E n log n .
Step 2: extension to the ball. Fix ( γ , β ) (\gamma,\beta) ( γ , β ) in the stated balls and s ∈ B ‾ ∞ ( s ∗ , ε n ) s\in\overline{B}_\infty(s^*,\varepsilon_n) s ∈ B ∞ ( s ∗ , ε n ) . For any centered unit vector z z z (1 T z = 0 \mathbf{1}^{\mathsf{T}}z=0 1 T z = 0 , ∥ z ∥ 2 = 1 \lVert z\rVert_2=1 ∥ z ∥ 2 = 1 ),
z T J ( s , γ , β ) z ≥ λ min , ⊥ ( J ∗ ) − ∥ J ( s , γ , β ) − J ∗ ∥ 2 . z^{\mathsf{T}}J(s,\gamma,\beta)z\ge\lambda_{\min,\perp}(J^*)-\lVert J(s,\gamma,\beta)-J^*\rVert_2. z T J ( s , γ , β ) z ≥ λ m i n , ⊥ ( J ∗ ) − ∥ J ( s , γ , β ) − J ∗ ∥ 2 .
Using ∥ ⋅ ∥ 2 ≤ ∥ ⋅ ∥ ∞ \lVert\cdot\rVert_2\le\lVert\cdot\rVert_\infty ∥ ⋅ ∥ 2 ≤ ∥ ⋅ ∥ ∞ for symmetric matrices, the triangle inequality, Lemma 6 (Lipschitz in s s s ) and Lemma 7 (Lipschitz in ( γ , β ) (\gamma,\beta) ( γ , β ) ),
∥ J ( s , γ , β ) − J ∗ ∥ 2 ≤ ∥ J ( s , γ , β ) − J ( s ∗ , γ , β ) ∥ ∞ + ∥ J ( s ∗ , γ , β ) − J ( s ∗ , γ ∗ , β ∗ ) ∥ ∞ ≤ 4 L f m ∗ n ∥ s − s ∗ ∥ ∞ + L ζ n ( ∥ γ − γ ∗ ∥ 2 + ∥ β − β ∗ ∥ 2 ) ≤ 4 L f m ∗ n ε n + 2 L ζ n ρ n = o ( n ) , \begin{aligned}
\lVert J(s,\gamma,\beta)-J^*\rVert_2
&\le\lVert J(s,\gamma,\beta)-J(s^*,\gamma,\beta)\rVert_\infty+\lVert J(s^*,\gamma,\beta)-J(s^*,\gamma^*,\beta^*)\rVert_\infty\\
&\le4L_fm^*n\lVert s-s^*\rVert_\infty+L_\zeta n\left(\lVert\gamma-\gamma^*\rVert_2+\lVert\beta-\beta^*\rVert_2\right)\\
&\le4L_fm^*n\varepsilon_n+2L_\zeta n\rho_n=o(n),
\end{aligned} ∥ J ( s , γ , β ) − J ∗ ∥ 2 ≤ ∥ J ( s , γ , β ) − J ( s ∗ , γ , β ) ∥ ∞ + ∥ J ( s ∗ , γ , β ) − J ( s ∗ , γ ∗ , β ∗ ) ∥ ∞ ≤ 4 L f m ∗ n ∥ s − s ∗ ∥ ∞ + L ζ n ( ∥ γ − γ ∗ ∥ 2 + ∥ β − β ∗ ∥ 2 ) ≤ 4 L f m ∗ n ε n + 2 L ζ n ρ n = o ( n ) ,
since ε n , ρ n = o ( 1 ) \varepsilon_n,\rho_n=o(1) ε n , ρ n = o ( 1 ) (the segment [ s ∗ , s ] [s^*,s] [ s ∗ , s ] lies in the working region for n n n large, so Lemma 6 applies).
Conclusion. On the event above,
λ min , ⊥ ( J ( s , γ , β ) ) = inf z : 1 T z = 0 ∥ z ∥ 2 = 1 z T J ( s , γ , β ) z ≥ n b 0 − C E n log n − o ( n ) ≥ 1 2 n b 0 \lambda_{\min,\perp}\bigl(J(s,\gamma,\beta)\bigr)=\inf_{\substack{z:\mathbf{1}^{\mathsf{T}}z=0\\\lVert z\rVert_2=1}}z^{\mathsf{T}}J(s,\gamma,\beta)z\ge nb_0-C_E\sqrt{n\log n}-o(n)\ge\frac12nb_0 λ m i n , ⊥ ( J ( s , γ , β ) ) = z : 1 T z = 0 ∥ z ∥ 2 = 1 inf z T J ( s , γ , β ) z ≥ n b 0 − C E n log n − o ( n ) ≥ 2 1 n b 0
for n ≥ n 0 n\ge n_0 n ≥ n 0 , since n log n = o ( n ) \sqrt{n\log n}=o(n) n log n = o ( n ) .
Stage 1: existence of a local consistent estimator and l-infinity rates
Recall the merit score map G γ , β = ∇ s ( − ℓ ) G_{\gamma,\beta}=\nabla_s(-\ell) G γ , β = ∇ s ( − ℓ ) from (B.14), valued in 1 ⊥ \mathbf{1}^{\perp} 1 ⊥ . For fixed ( γ , β ) (\gamma,\beta) ( γ , β ) we write s ^ γ , β \widehat{s}_{\gamma,\beta} s γ , β for a zero of G γ , β G_{\gamma,\beta} G γ , β in 1 ⊥ \mathbf{1}^{\perp} 1 ⊥ ; its existence and uniqueness near s ∗ s^* s ∗ are the content of Lemma 13.
Lemma 13 (Stage-1 ℓ ∞ \ell_\infty ℓ ∞ rate). Assume A1–A3. Fix any constant C γ β ≥ 0 C_{\gamma\beta}\ge0 C γ β ≥ 0 and put
ρ n : = C γ β log n n , ε n : = C 5 log n n , C 5 : = C 4 + 1 , \rho_n:=C_{\gamma\beta}\sqrt{\frac{\log n}{n}},\qquad\varepsilon_n:=C_5\sqrt{\frac{\log n}{n}},\qquad C_5:=C_4+1, ρ n := C γ β n log n , ε n := C 5 n log n , C 5 := C 4 + 1 ,
with C 4 = C 4 ( C γ β ) C_4=C_4(C_{\gamma\beta}) C 4 = C 4 ( C γ β ) the constant of Lemma 14 below. Then for every c > 0 c>0 c > 0 there is an integer n 0 = n 0 ( c ) n_0=n_0(c) n 0 = n 0 ( c ) and an event of probability at least 1 − O ( n − c ) 1-O(n^{-c}) 1 − O ( n − c ) on which, for all n ≥ n 0 n\ge n_0 n ≥ n 0 and every ( γ , β ) (\gamma,\beta) ( γ , β ) with max { ∥ γ − γ ∗ ∥ 2 , ∥ β − β ∗ ∥ 2 } ≤ ρ n \max\{\|\gamma-\gamma^*\|_2,\|\beta-\beta^*\|_2\}\le\rho_n max { ∥ γ − γ ∗ ∥ 2 , ∥ β − β ∗ ∥ 2 } ≤ ρ n , the merit equation G γ , β ( s ) = 0 G_{\gamma,\beta}(s)=0 G γ , β ( s ) = 0 has a solution s ^ γ , β \widehat{s}_{\gamma,\beta} s γ , β , unique in B ∞ ( s ∗ , ϵ n ) ∩ ( s ∗ + 1 ⊥ ) \mathcal{B}_\infty(s^*,\epsilon_n)\cap(s^*+1^\perp) B ∞ ( s ∗ , ϵ n ) ∩ ( s ∗ + 1 ⊥ ) , and
∥ s ^ γ , β − s ∗ ∥ ∞ ≤ C 5 log n n . \|\widehat{s}_{\gamma,\beta}-s^*\|_\infty\le C_5\sqrt{\frac{\log n}{n}}. ∥ s γ , β − s ∗ ∥ ∞ ≤ C 5 n log n .
The proof rests on the leave-one-out technique. Fix ( γ , β ) (\gamma,\beta) ( γ , β ) near the truth and run gradient descent on − ℓ ( ⋅ , γ , β ) -\ell(\cdot,\gamma,\beta) − ℓ ( ⋅ , γ , β ) , initialized at s ∗ s^* s ∗ (Algorithm 1); for each item m m m , run the same scheme on the LOO objective ℓ ( m ) \ell^{(m)} ℓ ( m ) of (B.26) (Algorithm 2).
Algorithm 1 Gradient descent on the merit block (fixed γ , β \gamma,\beta γ , β )
Require: step size η = ( C max n ) − 1 \eta=(C_{\max}n)^{-1} η = ( C m a x n ) − 1 , fixed ( γ , β ) (\gamma,\beta) ( γ , β )
s 0 ← s ∗ s^0\leftarrow s^* s 0 ← s ∗
for t = 0 , 1 , … , T − 1 t=0,1,\ldots,T-1 t = 0 , 1 , … , T − 1 do
s t + 1 ← s t − η G γ , β ( s t ) s^{t+1}\leftarrow s^t-\eta G_{\gamma,\beta}(s^t) s t + 1 ← s t − η G γ , β ( s t ) ▷ G γ , β = ∇ s ( − ℓ ) G_{\gamma,\beta}=\nabla_s(-\ell) G γ , β = ∇ s ( − ℓ ) is the merit score (B.14)
end for
return s T s^T s T
Algorithm 2 LOO gradient descent on the merit block (fixed γ , β \gamma,\beta γ , β , item m m m )
Require: step size η = ( C max n ) − 1 \eta=(C_{\max}n)^{-1} η = ( C m a x n ) − 1 , fixed ( γ , β ) (\gamma,\beta) ( γ , β )
s 0 , ( m ) ← s ∗ s^{0,(m)}\leftarrow s^* s 0 , ( m ) ← s ∗
for t = 0 , 1 , … , T − 1 t=0,1,\ldots,T-1 t = 0 , 1 , … , T − 1 do
s t + 1 , ( m ) ← s t , ( m ) − η G γ , β ( m ) ( s t , ( m ) ) s^{t+1,(m)}\leftarrow s^{t,(m)}-\eta G_{\gamma,\beta}^{(m)}(s^{t,(m)}) s t + 1 , ( m ) ← s t , ( m ) − η G γ , β ( m ) ( s t , ( m ) ) ▷ G γ , β ( m ) G_{\gamma,\beta}^{(m)} G γ , β ( m ) as in (B.27)
end for
return s T , ( m ) s^{T,(m)} s T , ( m )
The gauge is preserved automatically. Because G γ , β G_{\gamma,\beta} G γ , β and G γ , β ( m ) G_{\gamma,\beta}^{(m)} G γ , β ( m ) are valued in 1 ⊥ 1^\perp 1 ⊥ (every ω i j \omega_{ij} ω ij is centered) and both trajectories start at s 0 = s 0 , ( m ) = s ∗ ∈ 1 ⊥ s^0=s^{0,(m)}=s^*\in1^\perp s 0 = s 0 , ( m ) = s ∗ ∈ 1 ⊥ , every iterate s t , s t , ( m ) s^t,s^{t,(m)} s t , s t , ( m ) remains in 1 ⊥ 1^\perp 1 ⊥ ; no explicit projection is required. In particular s t − s ∗ ∈ 1 ⊥ s^t-s^*\in1^\perp s t − s ∗ ∈ 1 ⊥ and s t − s t , ( m ) ∈ 1 ⊥ s^t-s^{t,(m)}\in1^\perp s t − s t , ( m ) ∈ 1 ⊥ for all t t t .
Two facts drive the argument. First, because the realized merit Hessian is positive definite on a neighborhood of s ∗ s^* s ∗ (Lemma 12), the iterates contract toward the local maximizer s ^ γ , β \widehat{s}_{\gamma,\beta} s γ , β and, after Θ ( log n ) \Theta(\log n) Θ ( log n ) steps, lie within negligible distance of it; it therefore suffices to control the iterates themselves. Second, the iterates remain close to their starting point s ∗ s^* s ∗ , which yields consistency.
The difficulty is that this confinement is naturally controlled in ℓ 2 \ell_2 ℓ 2 , whereas the target is the sharper, per-coordinate ℓ ∞ \ell_\infty ℓ ∞ rate: an ℓ 2 \ell_2 ℓ 2 bound permits the entire error to concentrate on a single coordinate. To rule this out we use, for each coordinate m m m , the leave-one-out proxy of Appendix B.8: by (B.30) the proxy is F − m \mathcal{F}_{-m} F − m -measurable, which restores the concentration needed to bound the m m m -th coordinate individually. The genuine iterate and its proxy are then shown to remain close, and combining the two gives the uniform ℓ ∞ \ell_\infty ℓ ∞ bound.
Concretely, the argument is a single induction carrying four coupled estimates simultaneously along the trajectory: the ℓ 2 \ell_2 ℓ 2 confinement, the per-coordinate accuracy of the proxy, the closeness of iterate and proxy, and the ℓ ∞ \ell_\infty ℓ ∞ bound, which is the invariant that keeps the trajectory inside the region where positive definiteness holds (Lemma 12) and where the LOO score bound of Lemma 5 is active (Remark 5).
Lemma 14 (Coupled trajectory induction). Assume A1–A3. Fix a horizon T = T ( n ) ≥ 1 T=T(n)\ge1 T = T ( n ) ≥ 1 , a constant C γ β ≥ 0 C_{\gamma\beta}\ge0 C γ β ≥ 0 , and a pair ( γ , β ) (\gamma,\beta) ( γ , β ) with
∥ γ − γ ∗ ∥ 2 ≤ C γ β log n n , ∥ β − β ∗ ∥ 2 ≤ C γ β log n n . \|\gamma-\gamma^*\|_2\le C_{\gamma\beta}\sqrt{\frac{\log n}{n}},\qquad
\|\beta-\beta^*\|_2\le C_{\gamma\beta}\sqrt{\frac{\log n}{n}}. ∥ γ − γ ∗ ∥ 2 ≤ C γ β n log n , ∥ β − β ∗ ∥ 2 ≤ C γ β n log n .
Run Algorithm 1 from s 0 = s ∗ s^0=s^* s 0 = s ∗ , and for each m ∈ { 0 , 1 , … , n } m\in\{0,1,\ldots,n\} m ∈ { 0 , 1 , … , n } run Algorithm 2 from s 0 , ( m ) = s ∗ s^{0,(m)}=s^* s 0 , ( m ) = s ∗ , both with step size η = ( C max n ) − 1 \eta=(C_{\max}n)^{-1} η = ( C m a x n ) − 1 . Define the events
E s c o r e : = { max 0 ≤ a ≤ n ∣ G a ( s ∗ , γ ∗ , β ∗ ) ∣ ≤ C ψ n log n } , \mathcal{E}_{\mathrm{score}}:=\left\{\max_{0\leq a\leq n}\left|G_a(s^*,\gamma^*,\beta^*)\right|\leq C_{\psi}\sqrt{n\log n}\right\}, E score := { 0 ≤ a ≤ n max ∣ G a ( s ∗ , γ ∗ , β ∗ ) ∣ ≤ C ψ n log n } ,
E s a b : = { ∥ E ∥ 2 ≤ C E n log n } , \mathcal{E}_{\mathrm{sab}}:=\left\{\left\lVert E\right\rVert_2\leq C_E\sqrt{n\log n}\right\}, E sab := { ∥ E ∥ 2 ≤ C E n log n } ,
E l o o ( γ , β ) : = ⋂ m = 0 n ⋂ t = 0 T ( { ∥ D γ , β ( m ) ( s t , ( m ) ) ∥ 2 ≤ 2 K n log n } ∪ { ∥ s t , ( m ) − s ∗ ∥ ∞ > 1 2 } ) , \mathcal{E}_{\mathrm{loo}}(\gamma,\beta):=\bigcap_{m=0}^{n}\bigcap_{t=0}^{T}\left(\left\{\left\lVert D_{\gamma,\beta}^{(m)}(s^{t,(m)})\right\rVert_2\leq2K\sqrt{n\log n}\right\}\cup\left\{\left\lVert s^{t,(m)}-s^*\right\rVert_{\infty}>\frac{1}{2}\right\}\right), E loo ( γ , β ) := m = 0 ⋂ n t = 0 ⋂ T ( { D γ , β ( m ) ( s t , ( m ) ) 2 ≤ 2 K n log n } ∪ { s t , ( m ) − s ∗ ∞ > 2 1 } ) ,
with D γ , β ( m ) D_{\gamma,\beta}^{(m)} D γ , β ( m ) as in (B.30) and K K K as in Lemma 5, and set E : = E s c o r e ∩ E s a b ∩ E l o o ( γ , β ) \mathcal{E}:=\mathcal{E}_{\mathrm{score}}\cap\mathcal{E}_{\mathrm{sab}}\cap\mathcal{E}_{\mathrm{loo}}(\gamma,\beta) E := E score ∩ E sab ∩ E loo ( γ , β ) . Then for every c > 0 c>0 c > 0 there exist constants C 1 , C 2 , C 3 , C 4 > 0 C_1,C_2,C_3,C_4>0 C 1 , C 2 , C 3 , C 4 > 0 , depending only on c c c , on the constants of A1–A3, and on C γ β C_{\gamma\beta} C γ β (not on n n n or t t t ), and an integer n 0 = n 0 ( c ) n_0=n_0(c) n 0 = n 0 ( c ) , such that for all n ≥ n 0 n\geq n_0 n ≥ n 0 ,
P ( E ) ≥ 1 − O ( ( T + 1 ) n 1 − c ) . \mathbb{P}(\mathcal{E})\geq1-O\left((T+1)n^{1-c}\right). P ( E ) ≥ 1 − O ( ( T + 1 ) n 1 − c ) .
and on E \mathcal{E} E the following hold simultaneously for all 0 ≤ t ≤ T 0\leq t\leq T 0 ≤ t ≤ T :
∥ s t − s ∗ ∥ 2 ≤ C 1 log n , (42) \left\lVert s^t-s^*\right\rVert_2\leq C_1\sqrt{\log n}, \tag*{(42)} s t − s ∗ 2 ≤ C 1 log n , (42)
max 0 ≤ m ≤ n ∣ s m t , ( m ) − s m ∗ ∣ ≤ C 2 log n n , (43) \max_{0\leq m\leq n}\left|s_m^{t,(m)}-s_m^*\right|\leq C_2\sqrt{\frac{\log n}{n}}, \tag*{(43)} 0 ≤ m ≤ n max s m t , ( m ) − s m ∗ ≤ C 2 n log n , (43)
max 0 ≤ m ≤ n ∥ s t − s t , ( m ) ∥ 2 ≤ C 3 log n n , (44) \max_{0\leq m\leq n}\left\lVert s^t-s^{t,(m)}\right\rVert_2\leq C_3\sqrt{\frac{\log n}{n}}, \tag*{(44)} 0 ≤ m ≤ n max s t − s t , ( m ) 2 ≤ C 3 n log n , (44)
∥ s t − s ∗ ∥ ∞ ≤ C 4 log n n . (45) \left\lVert s^t-s^*\right\rVert_{\infty}\leq C_4\sqrt{\frac{\log n}{n}}. \tag*{(45)} s t − s ∗ ∞ ≤ C 4 n log n . (45)
Proof. We proceed by induction on t t t . The constants are fixed at the outset, in the acyclic order C 1 → C 2 → C 3 → C 4 C_1\to C_2\to C_3\to C_4 C 1 → C 2 → C 3 → C 4 recorded in the closing paragraph; they depend only on the stated quantities and not on n 0 n_0 n 0 , so n 0 n_0 n 0 may then be chosen large enough for the finitely many o ( 1 ) o(1) o ( 1 ) absorptions below (in particular, for every invocation of Lemma 12 at the radii C 4 log n / n C_4\sqrt{\log n/n} C 4 log n / n , ( C 3 + C 4 ) log n / n (C_3+C_4)\sqrt{\log n/n} ( C 3 + C 4 ) log n / n and ρ n \rho_n ρ n , all o ( 1 ) o(1) o ( 1 ) , whose defining event is E s a b ⊇ E \mathcal{E}_{\mathrm{sab}}\supseteq\mathcal{E} E sab ⊇ E ). Let the induction hypothesis H ( t ) \mathcal{H}(t) H ( t ) assert that (B.33a)–(B.33d) hold for all 0 ≤ τ ≤ t 0\leq\tau\leq t 0 ≤ τ ≤ t . Since both trajectories start at s ∗ s^* s ∗ , H ( 0 ) \mathcal{H}(0) H ( 0 ) holds trivially. Assume H ( t ) \mathcal{H}(t) H ( t ) for some 0 ≤ t < T 0\leq t<T 0 ≤ t < T . Throughout, s t , s t , ( m ) ∈ 1 ⊥ s^t,s^{t,(m)}\in1^\perp s t , s t , ( m ) ∈ 1 ⊥ and the differences s t − s ∗ , s t − s t , ( m ) ∈ 1 ⊥ s^t-s^*,s^t-s^{t,(m)}\in1^\perp s t − s ∗ , s t − s t , ( m ) ∈ 1 ⊥ (gauge preservation). Note also that H ( t ) \mathcal{H}(t) H ( t ) activates the LOO score bound: by (B.33c) and (B.33d), ∥ s τ , ( m ) − s ∗ ∥ ∞ ≤ ( C 3 + C 4 ) log n / n ≤ 1 2 \left\lVert s^{\tau,(m)}-s^*\right\rVert_{\infty}\leq(C_3+C_4)\sqrt{\log n/n}\leq\frac12 s τ , ( m ) − s ∗ ∞ ≤ ( C 3 + C 4 ) log n / n ≤ 2 1 for all τ ≤ t \tau\leq t τ ≤ t and n ≥ n 0 n\geq n_0 n ≥ n 0 , so on E l o o \mathcal{E}_{\mathrm{loo}} E loo the first alternative of the dichotomy is in force and
∥ D γ , β ( m ) ( s τ , ( m ) ) ∥ 2 ≤ 2 K n log n , 0 ≤ m ≤ n , 0 ≤ τ ≤ t . (46) \left\lVert D_{\gamma,\beta}^{(m)}(s^{\tau,(m)})\right\rVert_2\leq2K\sqrt{n\log n},\qquad0\leq m\leq n,\ 0\leq\tau\leq t. \tag*{(46)} D γ , β ( m ) ( s τ , ( m ) ) 2 ≤ 2 K n log n , 0 ≤ m ≤ n , 0 ≤ τ ≤ t . (46)
Step 1: ∥ s t + 1 − s ∗ ∥ 2 ≤ C 1 log n \left\lVert s^{t+1}-s^*\right\rVert_2\leq C_1\sqrt{\log n} s t + 1 − s ∗ 2 ≤ C 1 log n . By Algorithm 1,
s t + 1 − s ∗ = s t − s ∗ − η G γ , β ( s ∗ ) − η ( G γ , β ( s t ) − G γ , β ( s ∗ ) ) . s^{t+1}-s^*=s^t-s^*-\eta G_{\gamma,\beta}(s^*)-\eta\left(G_{\gamma,\beta}(s^t)-G_{\gamma,\beta}(s^*)\right). s t + 1 − s ∗ = s t − s ∗ − η G γ , β ( s ∗ ) − η ( G γ , β ( s t ) − G γ , β ( s ∗ ) ) .
Writing J ‾ : = ∫ 0 1 J ( s ∗ + τ ( s t − s ∗ ) ) d τ \overline{J}:=\int_0^1J(s^*+\tau(s^t-s^*))\,\mathrm{d}\tau J := ∫ 0 1 J ( s ∗ + τ ( s t − s ∗ )) d τ , the fundamental theorem of calculus gives G γ , β ( s t ) − G γ , β ( s ∗ ) = J ‾ ( s t − s ∗ ) G_{\gamma,\beta}(s^t)-G_{\gamma,\beta}(s^*)=\overline{J}(s^t-s^*) G γ , β ( s t ) − G γ , β ( s ∗ ) = J ( s t − s ∗ ) , so
s t + 1 − s ∗ = ( I − η J ‾ ) ( s t − s ∗ ) − η G γ , β ( s ∗ ) . s^{t+1}-s^*=(I-\eta\overline{J})(s^t-s^*)-\eta G_{\gamma,\beta}(s^*). s t + 1 − s ∗ = ( I − η J ) ( s t − s ∗ ) − η G γ , β ( s ∗ ) .
By (B.33d) the segment [ s ∗ , s t ] [s^*,s^t] [ s ∗ , s t ] lies in B ∞ ( s ∗ , C 4 log n / n ) B_{\infty}(s^*,C_4\sqrt{\log n/n}) B ∞ ( s ∗ , C 4 log n / n ) , hence in the working region for n ≥ n 0 n\geq n_0 n ≥ n 0 ; Lemma 11 gives η ≤ 1 / λ max ( J ‾ ) \eta\leq1/\lambda_{\max}(\overline{J}) η ≤ 1/ λ m a x ( J ) and Lemma 12 gives λ min , ⊥ ( J ‾ ) ≥ 1 2 n b 0 \lambda_{\min,\perp}(\overline{J})\geq\frac12nb_0 λ m i n , ⊥ ( J ) ≥ 2 1 n b 0 . Since s t − s ∗ ∈ 1 ⊥ s^t-s^*\in1^\perp s t − s ∗ ∈ 1 ⊥ , (B.22) gives
∥ I − η J ‾ ∥ 2 , ⊥ = 1 − η λ min , ⊥ ( J ‾ ) ≤ 1 − 1 2 η n b 0 . \left\lVert I-\eta\overline{J}\right\rVert_{2,\perp}=1-\eta\lambda_{\min,\perp}(\overline{J})\leq1-\frac12\eta nb_0. I − η J 2 , ⊥ = 1 − η λ m i n , ⊥ ( J ) ≤ 1 − 2 1 η n b 0 .
Hence, with (B.33a),
∥ s t + 1 − s ∗ ∥ 2 ≤ ( 1 − 1 2 η n b 0 ) C 1 log n + η ∥ G γ , β ( s ∗ ) ∥ 2 . \left\lVert s^{t+1}-s^*\right\rVert_2\leq\left(1-\frac12\eta nb_0\right)C_1\sqrt{\log n}+\eta\left\lVert G_{\gamma,\beta}(s^*)\right\rVert_2. s t + 1 − s ∗ 2 ≤ ( 1 − 2 1 η n b 0 ) C 1 log n + η ∥ G γ , β ( s ∗ ) ∥ 2 .
For the residual, by the fundamental theorem of calculus in ( γ , β ) (\gamma,\beta) ( γ , β ) ,
G γ , β ( s ∗ ) = G γ ∗ , β ∗ ( s ∗ ) + ( ∫ 0 1 ∂ G ( s ∗ , ⋅ ) ∂ ( γ , β ) ( ( γ ∗ , β ∗ ) + τ ( ( γ , β ) − ( γ ∗ , β ∗ ) ) ) d τ ) ( ( γ , β ) − ( γ ∗ , β ∗ ) ) , G_{\gamma,\beta}(s^*) = G_{\gamma^*,\beta^*}(s^*)+\left(\int_0^1 \frac{\partial G(s^*,\cdot)}{\partial(\gamma,\beta)}\bigl((\gamma^*,\beta^*)+\tau((\gamma,\beta)-(\gamma^*,\beta^*))\bigr)\,\mathrm{d}\tau\right)\bigl((\gamma,\beta)-(\gamma^*,\beta^*)\bigr), G γ , β ( s ∗ ) = G γ ∗ , β ∗ ( s ∗ ) + ( ∫ 0 1 ∂ ( γ , β ) ∂ G ( s ∗ , ⋅ ) ( ( γ ∗ , β ∗ ) + τ (( γ , β ) − ( γ ∗ , β ∗ )) ) d τ ) ( ( γ , β ) − ( γ ∗ , β ∗ ) ) ,
so
∥ G γ , β ( s ∗ ) ∥ 2 ≤ ∥ G γ ∗ , β ∗ ( s ∗ ) ∥ 2 + sup ( γ ′ , β ′ ) ∥ ∂ G ( s ∗ , ⋅ ) ∂ ( γ , β ) ( γ ′ , β ′ ) ∥ 2 ∥ ( γ , β ) − ( γ ∗ , β ∗ ) ∥ 2 . \lVert G_{\gamma,\beta}(s^*)\rVert_2 \le\lVert G_{\gamma^*,\beta^*}(s^*)\rVert_2+\sup_{(\gamma',\beta')}\left\lVert\frac{\partial G(s^*,\cdot)}{\partial(\gamma,\beta)}(\gamma',\beta')\right\rVert_2\lVert(\gamma,\beta)-(\gamma^*,\beta^*)\rVert_2. ∥ G γ , β ( s ∗ ) ∥ 2 ≤ ∥ G γ ∗ , β ∗ ( s ∗ ) ∥ 2 + ( γ ′ , β ′ ) sup ∂ ( γ , β ) ∂ G ( s ∗ , ⋅ ) ( γ ′ , β ′ ) 2 ∥( γ , β ) − ( γ ∗ , β ∗ ) ∥ 2 .
On E s c o r e \mathcal{E}_{\mathrm{score}} E score , ∥ G γ ∗ , β ∗ ( s ∗ ) ∥ ∞ ≤ C ψ n log n \lVert G_{\gamma^*,\beta^*}(s^*)\rVert_\infty\le C_\psi\sqrt{n}\log n ∥ G γ ∗ , β ∗ ( s ∗ ) ∥ ∞ ≤ C ψ n log n , hence ∥ G γ ∗ , β ∗ ( s ∗ ) ∥ 2 ≤ n + 1 C ψ n log n ≤ 2 C ψ n log n \lVert G_{\gamma^*,\beta^*}(s^*)\rVert_2\le\sqrt{n+1}C_\psi\sqrt{n}\log n\le\sqrt{2}C_\psi n\sqrt{\log n} ∥ G γ ∗ , β ∗ ( s ∗ ) ∥ 2 ≤ n + 1 C ψ n log n ≤ 2 C ψ n log n . By Lemma 8 (applied at s ∗ s^* s ∗ , which lies in the working region, uniformly over all ( γ ′ , β ′ ) (\gamma',\beta') ( γ ′ , β ′ ) ) the supremum is ≤ C M n 3 / 2 \le C_Mn^{3/2} ≤ C M n 3/2 , and ∥ ( γ , β ) − ( γ ∗ , β ∗ ) ∥ 2 ≤ ∥ γ − γ ∗ ∥ 2 + ∥ β − β ∗ ∥ 2 ≤ 2 C γ β log n / n \lVert(\gamma,\beta)-(\gamma^*,\beta^*)\rVert_2\le\lVert\gamma-\gamma^*\rVert_2+\lVert\beta-\beta^*\rVert_2\le2C_{\gamma\beta}\sqrt{\log n/n} ∥( γ , β ) − ( γ ∗ , β ∗ ) ∥ 2 ≤ ∥ γ − γ ∗ ∥ 2 + ∥ β − β ∗ ∥ 2 ≤ 2 C γ β log n / n . Therefore
∥ G γ , β ( s ∗ ) ∥ 2 ≤ C 6 n log n , C 6 : = 2 C ψ + 2 C M C γ β , (47) \lVert G_{\gamma,\beta}(s^*)\rVert_2\le C_6n\sqrt{\log n},\qquad C_6:=\sqrt{2}C_\psi+2C_MC_{\gamma\beta}, \tag*{(47)} ∥ G γ , β ( s ∗ ) ∥ 2 ≤ C 6 n log n , C 6 := 2 C ψ + 2 C M C γ β , (47)
and, with η = ( C max n ) − 1 \eta=(C_{\max}n)^{-1} η = ( C m a x n ) − 1 ,
∥ s t + 1 − s ∗ ∥ 2 ≤ ( 1 − 1 2 η n b 0 ) C 1 log n + η C 6 n log n . \lVert s^{t+1}-s^*\rVert_2\le\left(1-\frac{1}{2}\eta nb_0\right)C_1\sqrt{\log n}+\eta C_6n\sqrt{\log n}. ∥ s t + 1 − s ∗ ∥ 2 ≤ ( 1 − 2 1 η n b 0 ) C 1 log n + η C 6 n log n .
This is ≤ C 1 log n \le C_1\sqrt{\log n} ≤ C 1 log n provided
C 1 ≥ 2 C 6 b 0 . C_1\ge\frac{2C_6}{b_0}. C 1 ≥ b 0 2 C 6 .
Step 2: max 0 ≤ m ≤ n ∣ s m t + 1 , ( m ) − s m ∗ ∣ ≤ C 2 log n / n \max_{0\le m\le n}\lvert s_m^{t+1,(m)}-s_m^*\rvert\le C_2\sqrt{\log n/n} max 0 ≤ m ≤ n ∣ s m t + 1 , ( m ) − s m ∗ ∣ ≤ C 2 log n / n . Fix m ∈ { 0 , … , n } m\in\{0,\ldots,n\} m ∈ { 0 , … , n } . By Algorithm 2, the m m m -th coordinate updates as s m t + 1 , ( m ) = s m t , ( m ) − η [ G γ , β ( m ) ( s t , ( m ) ) ] m s_m^{t+1,(m)}=s_m^{t,(m)}-\eta[G_{\gamma,\beta}^{(m)}(s^{t,(m)})]_m s m t + 1 , ( m ) = s m t , ( m ) − η [ G γ , β ( m ) ( s t , ( m ) ) ] m , where the only terms of the LOO score involving coordinate m m m are
[ G γ , β ( m ) ( s ) ] m = − ∑ i ≠ m ∑ k [ ( g m i k ∗ − g m i k ) f ′ ( Δ m i ) + g m i k ∗ ( d m i k ∗ − d m i k ) ] . [G_{\gamma,\beta}^{(m)}(s)]_m=-\sum_{i\ne m}\sum_k\left[(g_{mik}^*-g_{mik})f'(\Delta_{mi})+g_{mik}^*(d_{mik}^*-d_{mik})\right]. [ G γ , β ( m ) ( s ) ] m = − i = m ∑ k ∑ [ ( g mik ∗ − g mik ) f ′ ( Δ mi ) + g mik ∗ ( d mik ∗ − d mik ) ] .
Since [ G γ ∗ , β ∗ ( m ) ( s ∗ ) ] m = 0 [G_{\gamma^*,\beta^*}^{(m)}(s^*)]_m=0 [ G γ ∗ , β ∗ ( m ) ( s ∗ ) ] m = 0 ,
s m t + 1 , ( m ) − s m ∗ = ( s m t , ( m ) − s m ∗ ) − η [ G γ , β ( m ) ( s t , ( m ) ) − G γ , β ( m ) ( s ∗ ) ] m ⏟ = : S − η [ G γ , β ( m ) ( s ∗ ) − G γ ∗ , β ∗ ( m ) ( s ∗ ) ] m ⏟ = : Z . s_m^{t+1,(m)}-s_m^*=(s_m^{t,(m)}-s_m^*)-\eta\underbrace{\left[G_{\gamma,\beta}^{(m)}(s^{t,(m)})-G_{\gamma,\beta}^{(m)}(s^*)\right]_m}_{=:S}-\eta\underbrace{\left[G_{\gamma,\beta}^{(m)}(s^*)-G_{\gamma^*,\beta^*}^{(m)}(s^*)\right]_m}_{=:Z}. s m t + 1 , ( m ) − s m ∗ = ( s m t , ( m ) − s m ∗ ) − η =: S [ G γ , β ( m ) ( s t , ( m ) ) − G γ , β ( m ) ( s ∗ ) ] m − η =: Z [ G γ , β ( m ) ( s ∗ ) − G γ ∗ , β ∗ ( m ) ( s ∗ ) ] m .
Control of S S S . By the mean value theorem applied to the scalar map [ G γ , β ( m ) ] m [G_{\gamma,\beta}^{(m)}]_m [ G γ , β ( m ) ] m , there is s ˉ \bar{s} s ˉ on [ s ∗ , s t , ( m ) ] [s^*,s^{t,(m)}] [ s ∗ , s t , ( m ) ] with S = ( ∇ s [ G γ , β ( m ) ] m ( s ˉ ) ) T ( s t , ( m ) − s ∗ ) S=(\nabla_s[G_{\gamma,\beta}^{(m)}]_m(\bar{s}))^\mathsf{T}(s^{t,(m)}-s^*) S = ( ∇ s [ G γ , β ( m ) ] m ( s ˉ ) ) T ( s t , ( m ) − s ∗ ) . Since G γ , β ( m ) = ∇ s ( − ℓ ( m ) ) G_{\gamma,\beta}^{(m)}=\nabla_s(-\ell^{(m)}) G γ , β ( m ) = ∇ s ( − ℓ ( m ) ) , one has ∇ s [ G γ , β ( m ) ] m = ( J ( m ) ) m ⋅ \nabla_s[G_{\gamma,\beta}^{(m)}]_m=(J^{(m)})_{m\cdot} ∇ s [ G γ , β ( m ) ] m = ( J ( m ) ) m ⋅ , the m m m -th row of (B.28). Hence
S = ( J ( m ) ) m m ( s ˉ ) ( s m t , ( m ) − s m ∗ ) + ∑ i ≠ m ( J ( m ) ) m i ( s ˉ ) ( s i t , ( m ) − s i ∗ ) . S=(J^{(m)})_{mm}(\bar{s})(s_m^{t,(m)}-s_m^*)+\sum_{i\ne m}(J^{(m)})_{mi}(\bar{s})(s_i^{t,(m)}-s_i^*). S = ( J ( m ) ) mm ( s ˉ ) ( s m t , ( m ) − s m ∗ ) + i = m ∑ ( J ( m ) ) mi ( s ˉ ) ( s i t , ( m ) − s i ∗ ) .
At the true parameters the saboteur in (B.29) vanishes, hence W i m k ( m ) ( s ∗ , γ ∗ , β ∗ ) = w ˉ i m k ≥ b 0 W_{imk}^{(m)}(s^*,\gamma^*,\beta^*)=\bar{w}_{imk}\ge b_0 W imk ( m ) ( s ∗ , γ ∗ , β ∗ ) = w ˉ imk ≥ b 0 (Lemma 1 and Remark 4) and, summing over the ≥ n \ge n ≥ n comparisons meeting m m m ,
( J ( m ) ) m m ( s ∗ , γ ∗ , β ∗ ) ≥ n b 0 . (J^{(m)})_{mm}(s^*,\gamma^*,\beta^*)\ge nb_0. ( J ( m ) ) mm ( s ∗ , γ ∗ , β ∗ ) ≥ n b 0 .
At the working pair ( γ , β ) ≠ ( γ ∗ , β ∗ ) (\gamma,\beta)\ne(\gamma^*,\beta^*) ( γ , β ) = ( γ ∗ , β ∗ ) the saboteur does not vanish at s ∗ s^* s ∗ , so we instead transfer the floor by perturbation. By (B.33c) and (B.33d) the segment point s ˉ \bar{s} s ˉ lies in B ∞ ( s ∗ , ( C 3 + C 4 ) log n / n ) \mathcal{B}_\infty(s^*,(C_3+C_4)\sqrt{\log n/n}) B ∞ ( s ∗ , ( C 3 + C 4 ) log n / n ) (the ball being convex), hence in the working region for n ≥ n 0 n\ge n_0 n ≥ n 0 . Since any diagonal entry of a symmetric matrix A A A satisfies ∣ A m m ∣ ≤ ∥ A ∥ ∞ \lvert A_{mm}\rvert\le\lVert A\rVert_\infty ∣ A mm ∣ ≤ ∥ A ∥ ∞ , the leave-one-out variants of Lemma 6 and Lemma 7, applied at s ˉ \bar{s} s ˉ with ∥ γ − γ ∗ ∥ 2 , ∥ β − β ∗ ∥ 2 ≤ C γ β log n / n \lVert\gamma-\gamma^*\rVert_2,\lVert\beta-\beta^*\rVert_2\le C_{\gamma\beta}\sqrt{\log n/n} ∥ γ − γ ∗ ∥ 2 , ∥ β − β ∗ ∥ 2 ≤ C γ β log n / n , give
∣ ( J ( m ) ) m m ( s ˉ , γ , β ) − ( J ( m ) ) m m ( s ∗ , γ ∗ , β ∗ ) ∣ \left\lvert(J^{(m)})_{mm}(\bar{s},\gamma,\beta)-(J^{(m)})_{mm}(s^*,\gamma^*,\beta^*)\right\rvert ( J ( m ) ) mm ( s ˉ , γ , β ) − ( J ( m ) ) mm ( s ∗ , γ ∗ , β ∗ )
≤ 4 L f m ∗ n ∥ s ˉ − s ∗ ∥ ∞ + L ζ n ( ∥ γ − γ ∗ ∥ 2 + ∥ β − β ∗ ∥ 2 ) ≤ [ 4 L f m ∗ ( C 3 + C 4 ) + 2 L ζ C γ β ] n log n = o ( n ) , \begin{aligned}
\le4L_fm^*n\lVert\bar{s}-s^*\rVert_\infty+L_\zeta n(\lVert\gamma-\gamma^*\rVert_2+\lVert\beta-\beta^*\rVert_2) \\
&\le[4L_fm^*(C_3+C_4)+2L_\zeta C_{\gamma\beta}]\sqrt{n\log n}=o(n),
\end{aligned} ≤ 4 L f m ∗ n ∥ s ˉ − s ∗ ∥ ∞ + L ζ n (∥ γ − γ ∗ ∥ 2 + ∥ β − β ∗ ∥ 2 ) ≤ [ 4 L f m ∗ ( C 3 + C 4 ) + 2 L ζ C γ β ] n log n = o ( n ) ,
so that
( J ( m ) ) m m ( s ˉ , γ , β ) ≥ n b 0 − o ( n ) ≥ 1 2 n b 0 for n ≥ n 0 . (J^{(m)})_{mm}(\bar{s},\gamma,\beta)\ge nb_0-o(n)\ge\frac{1}{2}nb_0\qquad\text{for }n\ge n_0. ( J ( m ) ) mm ( s ˉ , γ , β ) ≥ n b 0 − o ( n ) ≥ 2 1 n b 0 for n ≥ n 0 .
For the off-diagonal part, ∣ ( J ( m ) ) m i ( s ˉ ) ∣ = ∣ W m i ( m ) ( s ˉ ) ∣ ≤ m ∗ W max \lvert(J^{(m)})_{mi}(\bar{s})\rvert=\lvert W_{mi}^{(m)}(\bar{s})\rvert\le m^*W_{\max} ∣( J ( m ) ) mi ( s ˉ )∣ = ∣ W mi ( m ) ( s ˉ )∣ ≤ m ∗ W m a x for each i ≠ m i\ne m i = m , so
∥ ( J ( m ) ) m , − m ( s ˉ ) ∥ 2 : = ( ∑ i ≠ m ∣ ( J ( m ) ) m i ( s ˉ ) ∣ 2 ) 1 / 2 ≤ n m ∗ W max = C max 2 n . \lVert(J^{(m)})_{m,-m}(\bar{s})\rVert_2:=\left(\sum_{i\ne m}\lvert(J^{(m)})_{mi}(\bar{s})\rvert^2\right)^{1/2}\le\sqrt{n}m^*W_{\max}=\frac{C_{\max}}{2}\sqrt{n}. ∥( J ( m ) ) m , − m ( s ˉ ) ∥ 2 := i = m ∑ ∣( J ( m ) ) mi ( s ˉ ) ∣ 2 1/2 ≤ n m ∗ W m a x = 2 C m a x n .
By the triangle inequality with (B.33a) and (B.33c),
∥ s t , ( m ) − s ∗ ∥ 2 ≤ ∥ s t , ( m ) − s t ∥ 2 + ∥ s t − s ∗ ∥ 2 ≤ C 3 log n n + C 1 log n = ( C 1 + o ( 1 ) ) log n . \lVert s^{t,(m)}-s^*\rVert_2\le\lVert s^{t,(m)}-s^t\rVert_2+\lVert s^t-s^*\rVert_2\le C_3\sqrt{\frac{\log n}{n}}+C_1\sqrt{\log n}=(C_1+o(1))\sqrt{\log n}. ∥ s t , ( m ) − s ∗ ∥ 2 ≤ ∥ s t , ( m ) − s t ∥ 2 + ∥ s t − s ∗ ∥ 2 ≤ C 3 n log n + C 1 log n = ( C 1 + o ( 1 )) log n .
Hence, by Cauchy–Schwarz,
∣ ∑ i ≠ m ( J ( m ) ) m i ( s ˉ ) ( s i t , ( m ) − s i ∗ ) ∣ ≤ ∥ ( J ( m ) ) m , − m ( s ˉ ) ∥ 2 ∥ s t , ( m ) − s ∗ ∥ 2 ≤ C max 2 ( C 1 + o ( 1 ) ) n log n . \left\lvert\sum_{i\ne m}(J^{(m)})_{mi}(\bar{s})(s_i^{t,(m)}-s_i^*)\right\rvert\le\lVert(J^{(m)})_{m,-m}(\bar{s})\rVert_2\lVert s^{t,(m)}-s^*\rVert_2\le\frac{C_{\max}}{2}(C_1+o(1))\sqrt{n\log n}. i = m ∑ ( J ( m ) ) mi ( s ˉ ) ( s i t , ( m ) − s i ∗ ) ≤ ∥( J ( m ) ) m , − m ( s ˉ ) ∥ 2 ∥ s t , ( m ) − s ∗ ∥ 2 ≤ 2 C m a x ( C 1 + o ( 1 )) n log n .
Control of Z Z Z . By the mean value theorem in ( γ , β ) (\gamma,\beta) ( γ , β ) there is ( γ ′ , β ′ ) (\gamma',\beta') ( γ ′ , β ′ ) on [ ( γ ∗ , β ∗ ) , ( γ , β ) ] [(\gamma^*,\beta^*),(\gamma,\beta)] [( γ ∗ , β ∗ ) , ( γ , β )] with ∣ Z ∣ ≤ ∥ ∂ ( γ , β ) [ G ( m ) ] m ( γ ′ , β ′ ) ∥ 2 ∥ ( γ , β ) − ( γ ∗ , β ∗ ) ∥ 2 \lvert Z\rvert\le\lVert\partial_{(\gamma,\beta)}[G^{(m)}]_m(\gamma',\beta')\rVert_2\lVert(\gamma,\beta)-(\gamma^*,\beta^*)\rVert_2 ∣ Z ∣ ≤ ∥ ∂ ( γ , β ) [ G ( m ) ] m ( γ ′ , β ′ ) ∥ 2 ∥( γ , β ) − ( γ ∗ , β ∗ ) ∥ 2 . The per-coordinate Jacobian rows (evaluated at s ∗ s^* s ∗ , in the working region) are
∂ γ [ G ( m ) ] m = − ∑ i ≠ m , k w η , m i k f ′ ( Δ m i ∗ ) Y m i k T , ∂ β [ G ( m ) ] m = − ∑ i ≠ m , k g m i k ∗ w θ , m i k Z m i k T , \partial_\gamma[G^{(m)}]_m=-\sum_{i\ne m,k}w_{\eta,mik}f'(\Delta^*_{mi})Y_{mik}^{\mathsf T},\qquad\partial_\beta[G^{(m)}]_m=-\sum_{i\ne m,k}g^*_{mik}w_{\theta,mik}Z_{mik}^{\mathsf T}, ∂ γ [ G ( m ) ] m = − i = m , k ∑ w η , mik f ′ ( Δ mi ∗ ) Y mik T , ∂ β [ G ( m ) ] m = − i = m , k ∑ g mik ∗ w θ , mik Z mik T ,
each of the ≤ m ∗ n \le m^*n ≤ m ∗ n summands bounded in ℓ 2 \ell_2 ℓ 2 norm by 1 4 F 1 κ \frac14F_1\kappa 4 1 F 1 κ (resp. 1 4 κ \frac14\kappa 4 1 κ ), so ∥ ∂ ( γ , β ) [ G ( m ) ] m ∥ 2 ≤ 1 4 ( F 1 + 1 ) κ m ∗ n = : C ′ n \lVert\partial_{(\gamma,\beta)}[G^{(m)}]_m\rVert_2\le\frac14(F_1+1)\kappa m^*n=:C'n ∥ ∂ ( γ , β ) [ G ( m ) ] m ∥ 2 ≤ 4 1 ( F 1 + 1 ) κ m ∗ n =: C ′ n . With ∥ ( γ , β ) − ( γ ∗ , β ∗ ) ∥ 2 ≤ 2 C γ β log n / n \lVert(\gamma,\beta)-(\gamma^*,\beta^*)\rVert_2\le2C_{\gamma\beta}\sqrt{\log n/n} ∥( γ , β ) − ( γ ∗ , β ∗ ) ∥ 2 ≤ 2 C γ β log n / n ,
∣ Z ∣ ≤ 2 C ′ C γ β n log n n = 2 C ′ C γ β n log n . \lvert Z\rvert\le2C'C_{\gamma\beta}n\sqrt{\frac{\log n}{n}}=2C'C_{\gamma\beta}\sqrt{n\log n}. ∣ Z ∣ ≤ 2 C ′ C γ β n n log n = 2 C ′ C γ β n log n .
Note also the ceiling: ( J ( m ) ) m m ( s ˉ ) = ∑ i ≠ m , k W i m k ( m ) ( s ˉ ) ≤ m ∗ n W max = 1 2 C max n (J^{(m)})_{mm}(\bar{s})=\sum_{i\ne m,k}W_{imk}^{(m)}(\bar{s})\le m^*nW_{\max}=\frac12C_{\max}n ( J ( m ) ) mm ( s ˉ ) = ∑ i = m , k W imk ( m ) ( s ˉ ) ≤ m ∗ n W m a x = 2 1 C m a x n , so η ( J ( m ) ) m m ( s ˉ ) ≤ 1 2 < 1 \eta(J^{(m)})_{mm}(\bar{s})\le\frac12<1 η ( J ( m ) ) mm ( s ˉ ) ≤ 2 1 < 1 ; combined with the floor ( J ( m ) ) m m ( s ˉ ) ≥ 1 2 n b 0 (J^{(m)})_{mm}(\bar{s})\ge\frac12nb_0 ( J ( m ) ) mm ( s ˉ ) ≥ 2 1 n b 0 , the coefficient of the leading term below satisfies 0 < 1 − η ( J ( m ) ) m m ( s ˉ ) ≤ 1 − 1 2 η n b 0 0<1-\eta(J^{(m)})_{mm}(\bar{s})\le1-\frac12\eta nb_0 0 < 1 − η ( J ( m ) ) mm ( s ˉ ) ≤ 1 − 2 1 η n b 0 . Collecting the contributions,
∣ s m t + 1 , ( m ) − s m ∗ ∣ ≤ ( 1 − 1 2 η n b 0 ) ∣ s m t , ( m ) − s m ∗ ∣ + η C max 2 ( C 1 + o ( 1 ) ) n log n + η 2 C ′ C γ β n log n . \lvert s_m^{t+1,(m)}-s_m^*\rvert\le\left(1-\frac12\eta nb_0\right)\lvert s_m^{t,(m)}-s_m^*\rvert+\frac{\eta C_{\max}}{2}(C_1+o(1))\sqrt{n\log n}+\eta\,2C'C_{\gamma\beta}\sqrt{n\log n}. ∣ s m t + 1 , ( m ) − s m ∗ ∣ ≤ ( 1 − 2 1 η n b 0 ) ∣ s m t , ( m ) − s m ∗ ∣ + 2 η C m a x ( C 1 + o ( 1 )) n log n + η 2 C ′ C γ β n log n .
By (B.33b) and η = ( C max n ) − 1 \eta=(C_{\max}n)^{-1} η = ( C m a x n ) − 1 , and using C 1 + o ( 1 ) ≤ C 1 + 1 C_1+o(1)\le C_1+1 C 1 + o ( 1 ) ≤ C 1 + 1 for n ≥ n 0 n\ge n_0 n ≥ n 0 , the right-hand side is ≤ C 2 log n / n \le C_2\sqrt{\log n/n} ≤ C 2 log n / n provided
C 2 ≥ C max ( C 1 + 1 ) + 4 C ′ C γ β b 0 . C_2\ge\frac{C_{\max}(C_1+1)+4C'C_{\gamma\beta}}{b_0}. C 2 ≥ b 0 C m a x ( C 1 + 1 ) + 4 C ′ C γ β .
Step 3: max 0 ≤ m ≤ n ∥ s t + 1 − s t + 1 , ( m ) ∥ 2 ≤ C 3 log n / n \max_{0\le m\le n}\lVert s^{t+1}-s^{t+1,(m)}\rVert_2\le C_3\sqrt{\log n/n} max 0 ≤ m ≤ n ∥ s t + 1 − s t + 1 , ( m ) ∥ 2 ≤ C 3 log n / n . Writing the two updates and inserting ± G γ , β ( s t , ( m ) ) \pm G_{\gamma,\beta}(s^{t,(m)}) ± G γ , β ( s t , ( m ) ) ,
s t + 1 − s t + 1 , ( m ) = ( I − η J ˉ ) ( s t − s t , ( m ) ) − η ( G γ , β ( s t , ( m ) ) − G γ , β ( m ) ( s t , ( m ) ) ) , s^{t+1}-s^{t+1,(m)}=(I-\eta\bar{J})(s^t-s^{t,(m)})-\eta\left(G_{\gamma,\beta}(s^{t,(m)})-G_{\gamma,\beta}^{(m)}(s^{t,(m)})\right), s t + 1 − s t + 1 , ( m ) = ( I − η J ˉ ) ( s t − s t , ( m ) ) − η ( G γ , β ( s t , ( m ) ) − G γ , β ( m ) ( s t , ( m ) ) ) ,
where now J ˉ : = ∫ 0 1 J ( s t , ( m ) + τ ( s t − s t , ( m ) ) ) d τ \bar{J}:=\int_0^1J(s^{t,(m)}+\tau(s^t-s^{t,(m)}))\,d\tau J ˉ := ∫ 0 1 J ( s t , ( m ) + τ ( s t − s t , ( m ) )) d τ . By (B.33c) and (B.33d) the whole segment [ s t , ( m ) , s t ] [s^{t,(m)},s^t] [ s t , ( m ) , s t ] lies in B ∞ ( s ∗ , ( C 3 + C 4 ) log n / n ) B_\infty(s^*,(C_3+C_4)\sqrt{\log n/n}) B ∞ ( s ∗ , ( C 3 + C 4 ) log n / n ) , so Lemmas 11 and 12 apply to J ˉ \bar{J} J ˉ and, since s t − s t , ( m ) ∈ 1 ⊥ s^t-s^{t,(m)}\in1^\perp s t − s t , ( m ) ∈ 1 ⊥ , (B.22) gives
∥ I − η J ˉ ∥ 2 , ⊥ = 1 − η λ min , ⊥ ( J ˉ ) ≤ 1 − 1 2 η n b 0 . \lVert I-\eta\bar{J}\rVert_{2,\perp}=1-\eta\lambda_{\min,\perp}(\bar{J})\le1-\frac12\eta nb_0. ∥ I − η J ˉ ∥ 2 , ⊥ = 1 − η λ m i n , ⊥ ( J ˉ ) ≤ 1 − 2 1 η n b 0 .
For the second term, the identity (B.30) and the activated LOO bound (B.34) give
∥ G γ , β ( s t , ( m ) ) − G γ , β ( m ) ( s t , ( m ) ) ∥ 2 = ∥ D γ , β ( m ) ( s t , ( m ) ) ∥ 2 ≤ 2 K n log n on E l o o . \left\lVert G_{\gamma,\beta}(s^{t,(m)})-G_{\gamma,\beta}^{(m)}(s^{t,(m)})\right\rVert_2=\lVert D_{\gamma,\beta}^{(m)}(s^{t,(m)})\rVert_2\le2K\sqrt{n\log n}\quad\text{on }\mathcal{E}_{\mathrm{loo}}. G γ , β ( s t , ( m ) ) − G γ , β ( m ) ( s t , ( m ) ) 2 = ∥ D γ , β ( m ) ( s t , ( m ) ) ∥ 2 ≤ 2 K n log n on E loo .
Hence
∥ s t + 1 − s t + 1 , ( m ) ∥ 2 ≤ ( 1 − 1 2 η n b 0 ) ∥ s t − s t , ( m ) ∥ 2 + 2 η K n log n . \lVert s^{t+1}-s^{t+1,(m)}\rVert_2\le\left(1-\frac12\eta nb_0\right)\lVert s^t-s^{t,(m)}\rVert_2+2\eta K\sqrt{n\log n}. ∥ s t + 1 − s t + 1 , ( m ) ∥ 2 ≤ ( 1 − 2 1 η n b 0 ) ∥ s t − s t , ( m ) ∥ 2 + 2 η K n log n .
With η = ( C max n ) − 1 \eta=(C_{\max}n)^{-1} η = ( C m a x n ) − 1 , 1 2 η n b 0 = b 0 2 C max \frac{1}{2}\eta nb_0=\frac{b_0}{2C_{\max}} 2 1 η n b 0 = 2 C m a x b 0 and 2 η K n log n = 2 K C max log n / n 2\eta K\sqrt{n\log n}=\frac{2K}{C_{\max}}\sqrt{\log n/n} 2 η K n log n = C m a x 2 K log n / n ; using (B.33c),
∥ s t + 1 − s t + 1 , ( m ) ∥ 2 ≤ ( ( 1 − b 0 2 C max ) C 3 + 2 K C max ) log n n . \left\|s^{t+1}-s^{t+1,(m)}\right\|_2\leq\left(\left(1-\frac{b_0}{2C_{\max}}\right)C_3+\frac{2K}{C_{\max}}\right)\sqrt{\frac{\log n}{n}}. s t + 1 − s t + 1 , ( m ) 2 ≤ ( ( 1 − 2 C m a x b 0 ) C 3 + C m a x 2 K ) n log n .
The bracket is ≤ C 3 \leq C_3 ≤ C 3 iff 2 K C max ≤ b 0 2 C max C 3 \frac{2K}{C_{\max}}\leq\frac{b_0}{2C_{\max}}C_3 C m a x 2 K ≤ 2 C m a x b 0 C 3 , i.e. C 3 ≥ 4 K b 0 C_3\geq\frac{4K}{b_0} C 3 ≥ b 0 4 K .
Step 4: ∥ s t + 1 − s ∗ ∥ ∞ ≤ C 4 log n / n \left\|s^{t+1}-s^*\right\|_\infty\leq C_4\sqrt{\log n/n} s t + 1 − s ∗ ∞ ≤ C 4 log n / n . For each m m m ,
∣ s m t + 1 − s m ∗ ∣ ≤ ∣ s m t + 1 − s m t + 1 , ( m ) ∣ + ∣ s m t + 1 , ( m ) − s m ∗ ∣ ≤ max 0 ≤ m ≤ n ∥ s t + 1 − s t + 1 , ( m ) ∥ 2 + max 0 ≤ m ≤ n ∣ s m t + 1 , ( m ) − s m ∗ ∣ , \begin{aligned}
\left|s_m^{t+1}-s_m^*\right|\leq\left|s_m^{t+1}-s_m^{t+1,(m)}\right|+\left|s_m^{t+1,(m)}-s_m^*\right| \\
&\leq\max_{0\leq m\leq n}\left\|s^{t+1}-s^{t+1,(m)}\right\|_2+\max_{0\leq m\leq n}\left|s_m^{t+1,(m)}-s_m^*\right|,
\end{aligned} s m t + 1 − s m ∗ ≤ s m t + 1 − s m t + 1 , ( m ) + s m t + 1 , ( m ) − s m ∗ ≤ 0 ≤ m ≤ n max s t + 1 − s t + 1 , ( m ) 2 + 0 ≤ m ≤ n max s m t + 1 , ( m ) − s m ∗ ,
so by Steps 2–3, ∥ s t + 1 − s ∗ ∥ ∞ ≤ ( C 2 + C 3 ) log n / n ≤ C 4 log n / n \left\|s^{t+1}-s^*\right\|_\infty\leq(C_2+C_3)\sqrt{\log n/n}\leq C_4\sqrt{\log n/n} s t + 1 − s ∗ ∞ ≤ ( C 2 + C 3 ) log n / n ≤ C 4 log n / n provided C 4 ≥ C 2 + C 3 C_4\geq C_2+C_3 C 4 ≥ C 2 + C 3 .
Closing the constants and the probability. Choose, in order, C 1 ≥ 2 C 6 / b 0 C_1\geq2C_6/b_0 C 1 ≥ 2 C 6 / b 0 with C 6 = 2 C ψ + 2 C M C γ β C_6=\sqrt{2}C_\psi+2C_M C_\gamma\beta C 6 = 2 C ψ + 2 C M C γ β , then C 2 ≥ ( C max ( C 1 + 1 ) + 4 C ′ C γ β ) / b 0 C_2\geq(C_{\max}(C_1+1)+4C'C_\gamma\beta)/b_0 C 2 ≥ ( C m a x ( C 1 + 1 ) + 4 C ′ C γ β ) / b 0 , then C 3 ≥ 4 K / b 0 C_3\geq4K/b_0 C 3 ≥ 4 K / b 0 , then C 4 ≥ C 2 + C 3 C_4\geq C_2+C_3 C 4 ≥ C 2 + C 3 ; this is the acyclic order C 1 → C 2 → C 3 → C 4 C_1\to C_2\to C_3\to C_4 C 1 → C 2 → C 3 → C 4 , so the choices are consistent and depend only on the stated quantities. Each step preserved (B.33a)–(B.33d), completing the induction. Finally, by Lemma 3, P ( E s c o r e ) ≥ 1 − O ( n − c ) \mathbb{P}(\mathcal{E}_{\mathrm{score}})\geq1-O(n^{-c}) P ( E score ) ≥ 1 − O ( n − c ) ; by Lemma 4, P ( E s a b ) ≥ 1 − O ( n − c ) \mathbb{P}(\mathcal{E}_{\mathrm{sab}})\geq1-O(n^{-c}) P ( E sab ) ≥ 1 − O ( n − c ) ; and for E l o o ( γ , β ) \mathcal{E}_{\mathrm{loo}}(\gamma,\beta) E loo ( γ , β ) , each iterate s t , ( m ) s^{t,(m)} s t , ( m ) is F − m \mathcal{F}_{-m} F − m -measurable (gauge plus Appendix B.8), and ∥ s t , ( m ) − s ∗ ∥ ∞ ≤ 1 2 \left\|s^{t,(m)}-s^*\right\|_\infty\leq\frac{1}{2} s t , ( m ) − s ∗ ∞ ≤ 2 1 implies ∥ s t , ( m ) ∥ ∞ ≤ C s + 1 2 \left\|s^{t,(m)}\right\|_\infty\leq C_s+\frac{1}{2} s t , ( m ) ∞ ≤ C s + 2 1 (A2), so Lemma 5 bounds each term of the complement,
P ( { ∥ D γ , β ( m ) ( s t , ( m ) ) ∥ 2 > 2 K n log n } ∩ { ∥ s t , ( m ) − s ∗ ∥ ∞ ≤ 1 2 } ) = O ( n − c ) , \mathbb{P}\left(\left\{\left\|D_{\gamma,\beta}^{(m)}(s^{t,(m)})\right\|_2>2K\sqrt{n\log n}\right\}\cap\left\{\left\|s^{t,(m)}-s^*\right\|_\infty\leq\frac{1}{2}\right\}\right)=O(n^{-c}), P ( { D γ , β ( m ) ( s t , ( m ) ) 2 > 2 K n log n } ∩ { s t , ( m ) − s ∗ ∞ ≤ 2 1 } ) = O ( n − c ) ,
a fortiori at threshold 2 K ≥ K 2K\geq K 2 K ≥ K ; a union bound over the ( n + 1 ) ( T + 1 ) (n+1)(T+1) ( n + 1 ) ( T + 1 ) pairs ( m , t ) (m,t) ( m , t ) gives P ( E l o o ( γ , β ) ) ≥ 1 − O ( ( T + 1 ) n 1 − c ) \mathbb{P}(\mathcal{E}_{\mathrm{loo}}(\gamma,\beta))\geq1-O((T+1)n^{1-c}) P ( E loo ( γ , β )) ≥ 1 − O (( T + 1 ) n 1 − c ) .
Lemma 14 holds at a fixed ( γ , β ) (\gamma,\beta) ( γ , β ) . To obtain Lemma 13 we run Algorithm 1 for T = Θ ( log n ) T=\Theta(\log n) T = Θ ( log n ) steps and feed the coupled bounds into a degree-theoretic existence argument. This requires the LOO event to hold uniformly over all ( γ , β ) (\gamma,\beta) ( γ , β ) in the structural ball, which is the next lemma.
Lemma 15 (Uniformity of E l o o \mathcal{E}_{\mathrm{loo}} E loo over the structural ball). Assume A1–A3, and let T = T ( n ) T=T(n) T = T ( n ) be any horizon with T ≤ C T log n T\leq C_T\log n T ≤ C T log n for a constant C T C_T C T . For ζ = ( γ , β ) \zeta=(\gamma,\beta) ζ = ( γ , β ) in the balls of Lemma 14, let { s t , ( m ) ( ζ ) } t ≤ T \{s^{t,(m)}(\zeta)\}_{t\leq T} { s t , ( m ) ( ζ ) } t ≤ T be the LOO trajectories of Algorithm 2, and write C 4 ′ : = C 3 + C 4 C'_4:=C_3+C_4 C 4 ′ := C 3 + C 4 with C 3 , C 4 C_3,C_4 C 3 , C 4 the constants of Lemma 14. Then for every c > 0 c>0 c > 0 there is n 0 = n 0 ( c ) n_0=n_0(c) n 0 = n 0 ( c ) such that for all n ≥ n 0 n\geq n_0 n ≥ n 0 , on a single event E ~ l o o \widetilde{\mathcal{E}}_{\mathrm{loo}} E loo with P ( E ~ l o o ) ≥ 1 − O ( n − c ) \mathbb{P}(\widetilde{\mathcal{E}}_{\mathrm{loo}})\geq1-O(n^{-c}) P ( E loo ) ≥ 1 − O ( n − c ) , the following hold simultaneously, with Θ : = B ( γ ∗ , ρ n ) × B ( β ∗ , ρ n ) \Theta:=B(\gamma^*,\rho_n)\times B(\beta^*,\rho_n) Θ := B ( γ ∗ , ρ n ) × B ( β ∗ , ρ n ) :
(a) (uniform LOO bound) sup ζ ∈ Θ max 0 ≤ m ≤ n max 0 ≤ t ≤ T ∥ D ζ ( m ) ( s t , ( m ) ( ζ ) ) ∥ 2 ≤ 2 K n log n ; \sup_{\zeta\in\Theta}\max_{0\leq m\leq n}\max_{0\leq t\leq T}\left\|D_\zeta^{(m)}(s^{t,(m)}(\zeta))\right\|_2\leq2K\sqrt{n\log n}; sup ζ ∈ Θ max 0 ≤ m ≤ n max 0 ≤ t ≤ T D ζ ( m ) ( s t , ( m ) ( ζ )) 2 ≤ 2 K n log n ;
(b) (uniform confinement) sup ζ ∈ Θ max 0 ≤ m ≤ n max 0 ≤ t ≤ T ∥ s t , ( m ) ( ζ ) − s ∗ ∥ ∞ ≤ ( C 4 ′ + 1 ) log n / n . \sup_{\zeta\in\Theta}\max_{0\leq m\leq n}\max_{0\leq t\leq T}\left\|s^{t,(m)}(\zeta)-s^*\right\|_\infty\leq(C'_4+1)\sqrt{\log n/n}. sup ζ ∈ Θ max 0 ≤ m ≤ n max 0 ≤ t ≤ T s t , ( m ) ( ζ ) − s ∗ ∞ ≤ ( C 4 ′ + 1 ) log n / n .
Moreover E ~ l o o ⊆ E s c o r e ∩ E s a b \widetilde{\mathcal{E}}_{\mathrm{loo}}\subseteq\mathcal{E}_{\mathrm{score}}\cap\mathcal{E}_{\mathrm{sab}} E loo ⊆ E score ∩ E sab , and on E ~ l o o \widetilde{\mathcal{E}}_{\mathrm{loo}} E loo the event E \mathcal{E} E of Lemma 14 holds at every ζ \zeta ζ in the stated balls.
Proof. Write p : = p γ + p β p:=p_\gamma+p_\beta p := p γ + p β (fixed). Throughout, “the working region” means { s : ∥ s ∥ ∞ ≤ C s + 1 2 } \{s:\|s\|_\infty\leq C_s+\frac{1}{2}\} { s : ∥ s ∥ ∞ ≤ C s + 2 1 } , on which the bounds F 1 , F 2 , F 3 F_1,F_2,F_3 F 1 , F 2 , F 3 of A3 apply to every gap (Remark 1).
Step A: sensitivity bounds on the working region. Fix m m m and ζ , ζ ′ ∈ Θ \zeta,\zeta'\in\Theta ζ , ζ ′ ∈ Θ .
(i) Point sensitivity of D ( m ) D^{(m)} D ( m ) . For s , s ′ s,s' s , s ′ in the working region, each summand χ i m k \chi_{imk} χ imk of (B.30) is, at fixed ζ \zeta ζ , a function of the gap Δ i m \Delta_{im} Δ im alone, with
∣ ∂ χ i m k ∂ Δ i m ∣ = ∣ ( c i m k − g i m k ∗ ) ( f ′ ′ ( Δ i m ) − w θ , i m k ) ∣ ≤ F 2 + 1 4 = : L χ \left|\frac{\partial\chi_{imk}}{\partial\Delta_{im}}\right|=\left|(c_{imk}-g^*_{imk})(f''(\Delta_{im})-w_{\theta,imk})\right|\leq F_2+\frac{1}{4}=:L_\chi ∂ Δ im ∂ χ imk = ∣ ( c imk − g imk ∗ ) ( f ′′ ( Δ im ) − w θ , imk ) ∣ ≤ F 2 + 4 1 =: L χ
on the region; hence ∣ χ i m k ( s ) − χ i m k ( s ′ ) ∣ ≤ L χ ∣ Δ i m ( s ) − Δ i m ( s ′ ) ∣ ≤ 2 L χ ∥ s − s ′ ∥ ∞ \lvert\chi_{imk}(s)-\chi_{imk}(s')\rvert\le L_{\chi}\lvert\Delta_{im}(s)-\Delta_{im}(s')\rvert\le2L_{\chi}\lVert s-s'\rVert_{\infty} ∣ χ imk ( s ) − χ imk ( s ′ )∣ ≤ L χ ∣ Δ im ( s ) − Δ im ( s ′ )∣ ≤ 2 L χ ∥ s − s ′ ∥ ∞ . The difference D ζ ( m ) ( s ) − D ζ ( m ) ( s ′ ) D_{\zeta}^{(m)}(s)-D_{\zeta}^{(m)}(s') D ζ ( m ) ( s ) − D ζ ( m ) ( s ′ ) has m m m -th coordinate a sum of ≤ m ∗ n \le m^{*}n ≤ m ∗ n increments and, for each i ≠ m i\ne m i = m , an i i i -th coordinate consisting of one pair’s ≤ m ∗ \le m^{*} ≤ m ∗ increments; summing in ℓ 2 \ell_{2} ℓ 2 and using ∥ ⋅ ∥ ∞ ≤ ∥ ⋅ ∥ 2 \lVert\cdot\rVert_{\infty}\le\lVert\cdot\rVert_{2} ∥ ⋅ ∥ ∞ ≤ ∥ ⋅ ∥ 2 ,
∥ D ζ ( m ) ( s ) − D ζ ( m ) ( s ′ ) ∥ 2 ≤ 2 L χ m ∗ n 2 + n ∥ s − s ′ ∥ ∞ ≤ C D n ∥ s − s ′ ∥ 2 \left\lVert D_{\zeta}^{(m)}(s)-D_{\zeta}^{(m)}(s')\right\rVert_{2}\le2L_{\chi}m^{*}\sqrt{n^{2}+n}\,\lVert s-s'\rVert_{\infty}\le C_{D}n\lVert s-s'\rVert_{2} D ζ ( m ) ( s ) − D ζ ( m ) ( s ′ ) 2 ≤ 2 L χ m ∗ n 2 + n ∥ s − s ′ ∥ ∞ ≤ C D n ∥ s − s ′ ∥ 2
with C D : = 4 L χ m ∗ C_{D}:=4L_{\chi}m^{*} C D := 4 L χ m ∗ .
(ii) Coefficient sensitivity of D ( m ) D^{(m)} D ( m ) . At fixed s s s in the working region, D ζ ( m ) D_{\zeta}^{(m)} D ζ ( m ) depends on ζ \zeta ζ only through d i m k = σ ( Δ i m + Z i m k T β ) d_{imk}=\sigma(\Delta_{im}+Z_{imk}^{\mathsf{T}}\beta) d imk = σ ( Δ im + Z imk T β ) (no γ \gamma γ -dependence, by (B.30)); as σ \sigma σ is 1 4 \frac{1}{4} 4 1 -Lipschitz and ∥ Z i m k ∥ 2 ≤ κ \lVert Z_{imk}\rVert_{2}\le\kappa ∥ Z imk ∥ 2 ≤ κ , each summand moves by at most ∣ c i m k − g i m k ∗ ∣ ⋅ 1 4 κ ∥ β − β ′ ∥ 2 ≤ 1 4 κ ∥ ζ − ζ ′ ∥ 2 \lvert c_{imk}-g_{imk}^{*}\rvert\cdot\frac{1}{4}\kappa\lVert\beta-\beta'\rVert_{2}\le\frac{1}{4}\kappa\lVert\zeta-\zeta'\rVert_{2} ∣ c imk − g imk ∗ ∣ ⋅ 4 1 κ ∥ β − β ′ ∥ 2 ≤ 4 1 κ ∥ ζ − ζ ′ ∥ 2 , and the same ℓ 2 \ell_{2} ℓ 2 assembly as in (i) gives
∥ D ζ ( m ) ( s ) − D ζ ′ ( m ) ( s ) ∥ 2 ≤ C c o e f n ∥ ζ − ζ ′ ∥ 2 , C c o e f : = κ m ∗ . \left\lVert D_{\zeta}^{(m)}(s)-D_{\zeta'}^{(m)}(s)\right\rVert_{2}\le C_{\mathrm{coef}}n\lVert\zeta-\zeta'\rVert_{2},\qquad C_{\mathrm{coef}}:=\kappa m^{*}. D ζ ( m ) ( s ) − D ζ ′ ( m ) ( s ) 2 ≤ C coef n ∥ ζ − ζ ′ ∥ 2 , C coef := κ m ∗ .
(iii) One-step map. Writing one LOO step as s t + 1 , ( m ) = Φ ζ ( m ) ( s t , ( m ) ) s^{t+1,(m)}=\Phi_{\zeta}^{(m)}(s^{t,(m)}) s t + 1 , ( m ) = Φ ζ ( m ) ( s t , ( m ) ) with Φ ζ ( m ) ( s ) = s − η G ζ ( m ) ( s ) \Phi_{\zeta}^{(m)}(s)=s-\eta G_{\zeta}^{(m)}(s) Φ ζ ( m ) ( s ) = s − η G ζ ( m ) ( s ) : on the working region, ∂ s Φ ζ ( m ) = I − η J ( m ) \partial_{s}\Phi_{\zeta}^{(m)}=I-\eta J^{(m)} ∂ s Φ ζ ( m ) = I − η J ( m ) has ∥ ⋅ ∥ 2 ≤ 2 \lVert\cdot\rVert_{2}\le2 ∥ ⋅ ∥ 2 ≤ 2 (by Lemma 11 and η = ( C max n ) − 1 \eta=(C_{\max}n)^{-1} η = ( C m a x n ) − 1 , the spectrum of the symmetric matrix η J ( m ) \eta J^{(m)} η J ( m ) lies in [ − 1 , 1 ] [-1,1] [ − 1 , 1 ] , hence that of I − η J ( m ) I-\eta J^{(m)} I − η J ( m ) in [ 0 , 2 ] [0,2] [ 0 , 2 ] ), and
∥ ∂ ζ Φ ζ ( m ) ∥ 2 = η ∥ ∂ ζ G ζ ( m ) ∥ 2 ≤ η C M n 3 / 2 = C M C max − 1 n \left\lVert\partial_{\zeta}\Phi_{\zeta}^{(m)}\right\rVert_{2}=\eta\left\lVert\partial_{\zeta}G_{\zeta}^{(m)}\right\rVert_{2}\le\eta C_{M}n^{3/2}=C_{M}C_{\max}^{-1}\sqrt{n} ∂ ζ Φ ζ ( m ) 2 = η ∂ ζ G ζ ( m ) 2 ≤ η C M n 3/2 = C M C m a x − 1 n
(Lemma 8, leave-one-out variant). All three bounds require the argument(s) to lie in the working region; establishing that they do, along the trajectories at arbitrary ζ ∈ Θ \zeta\in\Theta ζ ∈ Θ , is the purpose of Step C.
Step B: net, good event, and behavior at net points. Since T ≤ C T log n T\le C_{T}\log n T ≤ C T log n , we have 2 T + 1 ≤ n κ T 2^{T+1}\le n^{\kappa_{T}} 2 T + 1 ≤ n κ T with κ T : = 2 C T \kappa_{T}:=2C_{T} κ T := 2 C T for n ≥ n 0 n\ge n_{0} n ≥ n 0 . Let N ⊂ Θ \mathcal{N}\subset\Theta N ⊂ Θ be a δ \delta δ -net in ℓ 2 \ell_{2} ℓ 2 with
δ : = n − ( κ T + 2 ) , so ∣ N ∣ ≤ ( 3 ρ n δ ) p = n O ( 1 ) , log ∣ N ∣ = O ( log n ) . \delta:=n^{-(\kappa_{T}+2)},\qquad\text{so}\qquad|\mathcal{N}|\le\left(\frac{3\rho n}{\delta}\right)^{p}=n^{O(1)},\qquad\log|\mathcal{N}|=O(\log n). δ := n − ( κ T + 2 ) , so ∣ N ∣ ≤ ( δ 3 ρ n ) p = n O ( 1 ) , log ∣ N ∣ = O ( log n ) .
Define
E ~ l o o : = E s c o r e ∩ E s a b ∩ ⋂ ζ j ∈ N ⋂ m = 0 n ⋂ t = 0 T ( { ∥ D ζ j ( m ) ( s t , ( m ) ( ζ j ) ) ∥ 2 ≤ K n log n } ∪ { ∥ s t , ( m ) ( ζ j ) − s ∗ ∥ ∞ > 1 2 } ) . \widetilde{\mathcal{E}}_{\mathrm{loo}}:=\mathcal{E}_{\mathrm{score}}\cap\mathcal{E}_{\mathrm{sab}}\cap\bigcap_{\zeta_{j}\in\mathcal{N}}\bigcap_{m=0}^{n}\bigcap_{t=0}^{T}\left(\left\{\left\lVert D_{\zeta_{j}}^{(m)}(s^{t,(m)}(\zeta_{j}))\right\rVert_{2}\le K\sqrt{n\log n}\right\}\cup\left\{\left\lVert s^{t,(m)}(\zeta_{j})-s^{*}\right\rVert_{\infty}>\frac{1}{2}\right\}\right). E loo := E score ∩ E sab ∩ ζ j ∈ N ⋂ m = 0 ⋂ n t = 0 ⋂ T ( { D ζ j ( m ) ( s t , ( m ) ( ζ j )) 2 ≤ K n log n } ∪ { s t , ( m ) ( ζ j ) − s ∗ ∞ > 2 1 } ) .
Each ζ j ∈ N \zeta_{j}\in\mathcal{N} ζ j ∈ N is deterministic, so s t , ( m ) ( ζ j ) s^{t,(m)}(\zeta_{j}) s t , ( m ) ( ζ j ) is F − m \mathcal{F}_{-m} F − m -measurable (Appendix B.8), and Lemma 5 bounds the probability of each complementary event { ∥ D ζ j ( m ) ∥ 2 > K n log n } ∩ { ∥ s t , ( m ) ( ζ j ) − s ∗ ∥ ∞ ≤ 1 2 } \{\lVert D_{\zeta_{j}}^{(m)}\rVert_{2}>K\sqrt{n\log n}\}\cap\{\lVert s^{t,(m)}(\zeta_{j})-s^{*}\rVert_{\infty}\le\frac{1}{2}\} {∥ D ζ j ( m ) ∥ 2 > K n log n } ∩ {∥ s t , ( m ) ( ζ j ) − s ∗ ∥ ∞ ≤ 2 1 } by O ( n − c ′ ) O(n^{-c'}) O ( n − c ′ ) : running the lemma at grade c ′ c' c ′ and union-bounding over N × { 0 , … , n } × { 0 , … , T } \mathcal{N}\times\{0,\ldots,n\}\times\{0,\ldots,T\} N × { 0 , … , n } × { 0 , … , T } , a set of cardinality n O ( 1 ) n^{O(1)} n O ( 1 ) (with exponent depending only on p , κ T p,\kappa_{T} p , κ T ), gives P ( E ~ l o o ) ≥ 1 − O ( n − c ) \mathbb{P}(\widetilde{\mathcal{E}}_{\mathrm{loo}})\ge1-O(n^{-c}) P ( E loo ) ≥ 1 − O ( n − c ) for c ′ c' c ′ large enough (depending on c , p , C T c,p,C_{T} c , p , C T ), after intersecting with the events of Lemmas 3 and 4.
On E ~ l o o \widetilde{\mathcal{E}}_{\mathrm{loo}} E loo , for every net point ζ j \zeta_{j} ζ j the event E = E s c o r e ∩ E s a b ∩ E l o o ( ζ j ) \mathcal{E}=\mathcal{E}_{\mathrm{score}}\cap\mathcal{E}_{\mathrm{sab}}\cap\mathcal{E}_{\mathrm{loo}}(\zeta_{j}) E = E score ∩ E sab ∩ E loo ( ζ j ) of Lemma 14 holds: the net dichotomy at threshold K K K implies the defining dichotomy at threshold $2K$ a fortiori (same escape set, weaker bound). Hence (B.33a)–(B.33d) are in force at every ζ j \zeta_{j} ζ j for all t ≤ T t\le T t ≤ T ; combining (B.33c) (with $\lVert\cdot\rVert_{\infty}\le\lVert\cdot\rVert_{2}) and (B.33d) yields the net-point confinement of the LOO trajectories ,
∥ s t , ( m ) ( ζ j ) − s ∗ ∥ ∞ ≤ C 4 ′ log n n , ζ j ∈ N , 0 ≤ m ≤ n , 0 ≤ t ≤ T , (48) \left\lVert s^{t,(m)}(\zeta_{j})-s^{*}\right\rVert_{\infty}\le C_{4}'\sqrt{\frac{\log n}{n}},\qquad\zeta_{j}\in\mathcal{N},\ 0\le m\le n,\ 0\le t\le T, \tag*{(48)} s t , ( m ) ( ζ j ) − s ∗ ∞ ≤ C 4 ′ n log n , ζ j ∈ N , 0 ≤ m ≤ n , 0 ≤ t ≤ T , (48)
with C 4 ′ = C 3 + C 4 C_{4}'=C_{3}+C_{4} C 4 ′ = C 3 + C 4 as in the statement.
Step C: confinement induction off the net. For ζ ∈ Θ \zeta\in\Theta ζ ∈ Θ fix a nearest net point ζ j = ζ j ( ζ ) \zeta_{j}=\zeta_{j}(\zeta) ζ j = ζ j ( ζ ) , ∥ ζ − ζ j ∥ 2 ≤ δ \lVert\zeta-\zeta_{j}\rVert_{2}\le\delta ∥ ζ − ζ j ∥ 2 ≤ δ . Set Λ t : = ( 2 t − 1 ) C M C max − 1 n \Lambda_{t}:=(2^{t}-1)C_{M}C_{\max}^{-1}\sqrt{n} Λ t := ( 2 t − 1 ) C M C m a x − 1 n , and note
Λ T δ ≤ 2 T C M C max − 1 n n − ( κ T + 2 ) ≤ C M C max − 1 n − 3 / 2 ≤ log n n ( n ≥ n 0 ) . (49) \Lambda_{T}\delta\le2^{T}C_{M}C_{\max}^{-1}\sqrt{n}\,n^{-(\kappa_{T}+2)}\le C_{M}C_{\max}^{-1}n^{-3/2}\le\sqrt{\frac{\log n}{n}}\qquad(n\ge n_{0}). \tag*{(49)} Λ T δ ≤ 2 T C M C m a x − 1 n n − ( κ T + 2 ) ≤ C M C m a x − 1 n − 3/2 ≤ n log n ( n ≥ n 0 ) . (49)
We prove by induction on t t t the hypothesis
H ( t ) : ∥ s τ , ( m ) ( ζ ) − s τ , ( m ) ( ζ j ) ∥ 2 ≤ Λ τ δ for all ζ ∈ Θ , 0 ≤ m ≤ n , 0 ≤ τ ≤ t . H(t):\qquad\left\|s^{\tau,(m)}(\zeta)-s^{\tau,(m)}(\zeta_j)\right\|_2\leq\Lambda_\tau\delta\quad\text{for all }\zeta\in\Theta,\ 0\leq m\leq n,\ 0\leq\tau\leq t. H ( t ) : s τ , ( m ) ( ζ ) − s τ , ( m ) ( ζ j ) 2 ≤ Λ τ δ for all ζ ∈ Θ , 0 ≤ m ≤ n , 0 ≤ τ ≤ t .
H ( t ) H(t) H ( t ) has the following consequence, by (B.36), (B.37) and ∥ ⋅ ∥ ∞ ≤ ∥ ⋅ ∥ 2 \|\cdot\|_\infty\leq\|\cdot\|_2 ∥ ⋅ ∥ ∞ ≤ ∥ ⋅ ∥ 2 : for all τ ≤ t \tau\leq t τ ≤ t ,
∥ s τ , ( m ) ( ζ ) − s ∗ ∥ ∞ ≤ C 4 ′ log n / n + Λ τ δ ≤ ( C 4 ′ + 1 ) log n / n , (50) \left\|s^{\tau,(m)}(\zeta)-s^*\right\|_\infty\leq C'_4\sqrt{\log n/n}+\Lambda_\tau\delta\leq(C'_4+1)\sqrt{\log n/n}, \tag*{(50)} s τ , ( m ) ( ζ ) − s ∗ ∞ ≤ C 4 ′ log n / n + Λ τ δ ≤ ( C 4 ′ + 1 ) log n / n , (50)
so that every point s τ , ( m ) ( ζ ) s^{\tau,(m)}(\zeta) s τ , ( m ) ( ζ ) , every net-point iterate, and (by convexity of the ℓ ∞ \ell_\infty ℓ ∞ ball) every segment between them lies in the working region for n ≥ n 0 n\geq n_0 n ≥ n 0 .
H ( 0 ) H(0) H ( 0 ) holds trivially, both trajectories starting at s ∗ s^* s ∗ . Assume H ( t ) H(t) H ( t ) , t < T t<T t < T . Decompose one step as
s t + 1 , ( m ) ( ζ ) − s t + 1 , ( m ) ( ζ j ) = Φ ζ ( m ) ( s t , ( m ) ( ζ ) ) − Φ ζ ( m ) ( s t , ( m ) ( ζ j ) ) ⏟ = : P + Φ ζ ( m ) ( s t , ( m ) ( ζ j ) ) − Φ ζ j ( m ) ( s t , ( m ) ( ζ j ) ) ⏟ = : Q . s^{t+1,(m)}(\zeta)-s^{t+1,(m)}(\zeta_j)=\underbrace{\Phi^{(m)}_\zeta\left(s^{t,(m)}(\zeta)\right)-\Phi^{(m)}_\zeta\left(s^{t,(m)}(\zeta_j)\right)}_{=:P}
+\underbrace{\Phi^{(m)}_\zeta\left(s^{t,(m)}(\zeta_j)\right)-\Phi^{(m)}_{\zeta_j}\left(s^{t,(m)}(\zeta_j)\right)}_{=:Q}. s t + 1 , ( m ) ( ζ ) − s t + 1 , ( m ) ( ζ j ) = =: P Φ ζ ( m ) ( s t , ( m ) ( ζ ) ) − Φ ζ ( m ) ( s t , ( m ) ( ζ j ) ) + =: Q Φ ζ ( m ) ( s t , ( m ) ( ζ j ) ) − Φ ζ j ( m ) ( s t , ( m ) ( ζ j ) ) .
For P P P : by (B.38) the segment between the two arguments lies in the working region, so Step A(iii) and the mean value theorem give ∥ P ∥ 2 ≤ 2 ∥ s t , ( m ) ( ζ ) − s t , ( m ) ( ζ j ) ∥ 2 ≤ 2 Λ t δ \|P\|_2\leq2\left\|s^{t,(m)}(\zeta)-s^{t,(m)}(\zeta_j)\right\|_2\leq2\Lambda_t\delta ∥ P ∥ 2 ≤ 2 s t , ( m ) ( ζ ) − s t , ( m ) ( ζ j ) 2 ≤ 2 Λ t δ . For Q Q Q : the argument s t , ( m ) ( ζ j ) s^{t,(m)}(\zeta_j) s t , ( m ) ( ζ j ) lies in the working region by (B.36), and the segment [ ζ j , ζ ] [\zeta_j,\zeta] [ ζ j , ζ ] lies in the convex set Θ \Theta Θ , so Step A(iii) gives ∥ Q ∥ 2 ≤ C M C max − 1 n δ \|Q\|_2\leq C_M C_{\max}^{-1}\sqrt n\delta ∥ Q ∥ 2 ≤ C M C m a x − 1 n δ . Hence
∥ s t + 1 , ( m ) ( ζ ) − s t + 1 , ( m ) ( ζ j ) ∥ 2 ≤ ( 2 Λ t + C M C max − 1 n ) δ = Λ t + 1 δ , \left\|s^{t+1,(m)}(\zeta)-s^{t+1,(m)}(\zeta_j)\right\|_2\leq\left(2\Lambda_t+C_M C_{\max}^{-1}\sqrt n\right)\delta=\Lambda_{t+1}\delta, s t + 1 , ( m ) ( ζ ) − s t + 1 , ( m ) ( ζ j ) 2 ≤ ( 2 Λ t + C M C m a x − 1 n ) δ = Λ t + 1 δ ,
by the recursion Λ t + 1 = 2 Λ t + C M C max − 1 n \Lambda_{t+1}=2\Lambda_t+C_M C_{\max}^{-1}\sqrt n Λ t + 1 = 2 Λ t + C M C m a x − 1 n . This proves H ( t + 1 ) H(t+1) H ( t + 1 ) , and the induction delivers H ( T ) H(T) H ( T ) ; consequence (B.38) at τ ≤ T \tau\leq T τ ≤ T is claim (b).
Step D: off-net extension of the LOO bound. Fix ζ ∈ Θ \zeta\in\Theta ζ ∈ Θ , m m m , t t t , and ζ j = ζ j ( ζ ) \zeta_j=\zeta_j(\zeta) ζ j = ζ j ( ζ ) . By (B.36) the net-point iterate is confined, ∥ s t , ( m ) ( ζ j ) − s ∗ ∥ ∞ ≤ C 4 ′ log n / n ≤ 1 2 \left\|s^{t,(m)}(\zeta_j)-s^*\right\|_\infty\leq C'_4\sqrt{\log n/n}\leq\frac12 s t , ( m ) ( ζ j ) − s ∗ ∞ ≤ C 4 ′ log n / n ≤ 2 1 for n ≥ n 0 n\geq n_0 n ≥ n 0 , so on E ~ l o o \widetilde{\mathcal E}_{\mathrm{loo}} E loo the first alternative of the net dichotomy is active: ∥ D ζ j ( m ) ( s t , ( m ) ( ζ j ) ) ∥ 2 ≤ K n log n \left\|D_{\zeta_j}^{(m)}\left(s^{t,(m)}(\zeta_j)\right)\right\|_2\leq K\sqrt{n\log n} D ζ j ( m ) ( s t , ( m ) ( ζ j ) ) 2 ≤ K n log n . All remaining points lie in the working region by (B.36) and (B.38), so Steps A(i)--(ii) apply:
∥ D ζ ( m ) ( s t , ( m ) ( ζ ) ) ∥ 2 ≤ ∥ D ζ j ( m ) ( s t , ( m ) ( ζ j ) ) ∥ 2 + C D n ∥ s t , ( m ) ( ζ ) − s t , ( m ) ( ζ j ) ∥ 2 + C c o e f n ∥ ζ − ζ j ∥ 2 . \left\|D_\zeta^{(m)}\left(s^{t,(m)}(\zeta)\right)\right\|_2
\leq\left\|D_{\zeta_j}^{(m)}\left(s^{t,(m)}(\zeta_j)\right)\right\|_2
+C_Dn\left\|s^{t,(m)}(\zeta)-s^{t,(m)}(\zeta_j)\right\|_2
+C_{\mathrm{coef}}n\left\|\zeta-\zeta_j\right\|_2. D ζ ( m ) ( s t , ( m ) ( ζ ) ) 2 ≤ D ζ j ( m ) ( s t , ( m ) ( ζ j ) ) 2 + C D n s t , ( m ) ( ζ ) − s t , ( m ) ( ζ j ) 2 + C coef n ∥ ζ − ζ j ∥ 2 .
The first term is ≤ K n log n \leq K\sqrt{n\log n} ≤ K n log n ; by H ( T ) H(T) H ( T ) and (B.37) the second is ≤ C D n ⋅ C M C max − 1 n − 3 / 2 = O ( n − 1 / 2 ) \leq C_Dn\cdot C_M C_{\max}^{-1}n^{-3/2}=O(n^{-1/2}) ≤ C D n ⋅ C M C m a x − 1 n − 3/2 = O ( n − 1/2 ) ; the third is ≤ C c o e f n 1 − ( κ T + 2 ) = O ( n − 1 ) \leq C_{\mathrm{coef}}n^{1-(\kappa T+2)}=O(n^{-1}) ≤ C coef n 1 − ( κ T + 2 ) = O ( n − 1 ) . For n ≥ n 0 n\geq n_0 n ≥ n 0 the last two are together ≤ K n log n \leq K\sqrt{n\log n} ≤ K n log n , whence the total is ≤ 2 K n log n \leq2K\sqrt{n\log n} ≤ 2 K n log n . Taking suprema over ζ ∈ Θ \zeta\in\Theta ζ ∈ Θ and maxima over m , t m,t m , t gives claim (a). The final sentence of the statement restates that E \mathcal E E of Lemma 14 holds at every ζ \zeta ζ : its first two constituents hold by construction, and E l o o ( ζ ) \mathcal E_{\mathrm{loo}}(\zeta) E loo ( ζ ) holds because the actual bound of claim (a) implies the dichotomy at threshold 2 K 2K 2 K a fortiori. □
Proof of Lemma 13. Let D ≥ 1 D\geq1 D ≥ 1 be a constant and set
ϱ : = 1 − b 0 2 C max ∈ ( 0 , 1 ) , T : = ⌈ ( D + 2 ) log n log ( 1 / ϱ ) ⌉ = Θ ( log n ) , \varrho:=1-\frac{b_0}{2C_{\max}}\in(0,1),\qquad T:=\left\lceil\frac{(D+2)\log n}{\log(1/\varrho)}\right\rceil=\Theta(\log n), ϱ := 1 − 2 C m a x b 0 ∈ ( 0 , 1 ) , T := ⌈ log ( 1/ ϱ ) ( D + 2 ) log n ⌉ = Θ ( log n ) ,
so that T + 1 ≤ C T log n T+1\leq C_T\log n T + 1 ≤ C T log n with C T : = ( D + 3 ) / log ( 1 / ϱ ) C_T:=(D+3)/\log(1/\varrho) C T := ( D + 3 ) / log ( 1/ ϱ ) for n ≥ n 0 n\geq n_0 n ≥ n 0 . Set E ~ : = E s c o r e ∩ E s a b ∩ E ~ l o o \widetilde{\mathcal E}:=\mathcal E_{\mathrm{score}}\cap\mathcal E_{\mathrm{sab}}\cap\widetilde{\mathcal E}_{\mathrm{loo}} E := E score ∩ E sab ∩ E loo , with E ~ l o o \widetilde{\mathcal E}_{\mathrm{loo}} E loo from Lemma 15 at horizon T + 1 T+1 T + 1 . As E s c o r e , E s a b \mathcal E_{\mathrm{score}},\mathcal E_{\mathrm{sab}} E score , E sab are evaluated at ( γ ∗ , β ∗ ) (\gamma^*,\beta^*) ( γ ∗ , β ∗ ) , the union bound gives P ( E ~ ) ≥ 1 − O ( n − c ) \mathbb P(\widetilde{\mathcal E})\geq1-O(n^{-c}) P ( E ) ≥ 1 − O ( n − c ) .
Fix any ( γ , β ) (\gamma,\beta) ( γ , β ) with max { ∥ γ − γ ∗ ∥ 2 , ∥ β − β ∗ ∥ 2 } ≤ ρ n \max\{\|\gamma-\gamma^*\|_2,\|\beta-\beta^*\|_2\}\leq\rho_n max { ∥ γ − γ ∗ ∥ 2 , ∥ β − β ∗ ∥ 2 } ≤ ρ n . By the final assertion of Lemma 15, E ~ \widetilde{\mathcal E} E implies the event E \mathcal E E of Lemma 14 for this ( γ , β ) (\gamma,\beta) ( γ , β ) ; run Algorithm 1 for T + 1 T+1 T + 1 steps, applying Lemma 14 with horizon T + 1 T+1 T + 1 . All statements below hold on E \mathcal E E ; the four bounds (B.33a)--(B.33d) are in force for $0\leq t\leq T+1. Step 1 (near-stationarity). Since s t + 1 − s t = − η G γ , β ( s t ) s^{t+1}-s^t=-\eta G_{\gamma,\beta}(s^t) s t + 1 − s t = − η G γ , β ( s t ) , consecutive differences satisfy s t + 1 − s t = ( I − η J ˉ t ) ( s t − s t − 1 ) s^{t+1}-s^t=(I-\eta\bar{J}_t)(s^t-s^{t-1}) s t + 1 − s t = ( I − η J ˉ t ) ( s t − s t − 1 ) with J ˉ t : = ∫ 0 1 J ( s t − 1 + τ ( s t − s t − 1 ) ) d τ \bar{J}_t:=\int_0^1 J(s^{t-1}+\tau(s^t-s^{t-1}))\,\mathrm{d}\tau J ˉ t := ∫ 0 1 J ( s t − 1 + τ ( s t − s t − 1 )) d τ . By (B.33d) the segment [ s t − 1 , s t ] ⊂ B ‾ ∞ ( s ∗ , C 4 log n / n ) [s^{t-1},s^t]\subset\overline{B}_{\infty}(s^*,C_4\sqrt{\log n/n}) [ s t − 1 , s t ] ⊂ B ∞ ( s ∗ , C 4 log n / n ) , so Lemma 11 gives η ≤ 1 / λ max ( J ˉ t ) \eta\leq1/\lambda_{\max}(\bar{J}_t) η ≤ 1/ λ m a x ( J ˉ t ) and Lemma 12 gives λ min , ⊥ ( J ˉ t ) ≥ 1 2 n b 0 \lambda_{\min,\perp}(\bar{J}_t)\geq\frac{1}{2}nb_0 λ m i n , ⊥ ( J ˉ t ) ≥ 2 1 n b 0 ; with s t − s t − 1 ∈ 1 ⊥ s^t-s^{t-1}\in1^\perp s t − s t − 1 ∈ 1 ⊥ , (B.22) gives ∥ I − η J ˉ t ∥ 2 , ⊥ ≤ ϱ \lVert I-\eta\bar{J}_t\rVert_{2,\perp}\leq\varrho ∥ I − η J ˉ t ∥ 2 , ⊥ ≤ ϱ for 1 ≤ t ≤ T + 1 1\leq t\leq T+1 1 ≤ t ≤ T + 1 . Iterating from s 0 = s ∗ s^0=s^* s 0 = s ∗ and using ∥ G γ , β ( s ∗ ) ∥ 2 ≤ C 6 n log n \lVert G_{\gamma,\beta}(s^*)\rVert_2\leq C_6n\sqrt{\log n} ∥ G γ , β ( s ∗ ) ∥ 2 ≤ C 6 n log n on E E E (display (B.35) in the proof of Lemma 14, with C 6 = 2 C ψ + 2 C M C γ β C_6=\sqrt{2}C_\psi+2C_MC_{\gamma\beta} C 6 = 2 C ψ + 2 C M C γ β ),
∥ s 1 − s 0 ∥ 2 = η ∥ G γ , β ( s ∗ ) ∥ 2 ≤ C 6 C max log n , ∥ s t + 1 − s t ∥ 2 ≤ ϱ t C 6 C max log n . \lVert s^1-s^0\rVert_2=\eta\lVert G_{\gamma,\beta}(s^*)\rVert_2\leq\frac{C_6}{C_{\max}}\sqrt{\log n},\qquad\lVert s^{t+1}-s^t\rVert_2\leq\varrho^t\frac{C_6}{C_{\max}}\sqrt{\log n}. ∥ s 1 − s 0 ∥ 2 = η ∥ G γ , β ( s ∗ ) ∥ 2 ≤ C m a x C 6 log n , ∥ s t + 1 − s t ∥ 2 ≤ ϱ t C m a x C 6 log n .
Therefore, using s T + 1 − s T = − η G γ , β ( s T ) s^{T+1}-s^T=-\eta G_{\gamma,\beta}(s^T) s T + 1 − s T = − η G γ , β ( s T ) and η = ( C max n ) − 1 \eta=(C_{\max}n)^{-1} η = ( C m a x n ) − 1 ,
∥ G γ , β ( s T ) ∥ 2 = 1 η ∥ s T + 1 − s T ∥ 2 ≤ C 6 n log n ϱ T ≤ n − D , \lVert G_{\gamma,\beta}(s^T)\rVert_2=\frac{1}{\eta}\lVert s^{T+1}-s^T\rVert_2\leq C_6n\sqrt{\log n}\,\varrho^T\leq n^{-D}, ∥ G γ , β ( s T ) ∥ 2 = η 1 ∥ s T + 1 − s T ∥ 2 ≤ C 6 n log n ϱ T ≤ n − D ,
the last step by the choice of T T T (for n n n large).
Step 2 (existence, via degree theory on a small ball). If G γ , β ( s T ) = 0 G_{\gamma,\beta}(s^T)=0 G γ , β ( s T ) = 0 , set s ^ γ , β : = s T \widehat{s}_{\gamma,\beta}:=s^T s γ , β := s T and pass directly to Step 3. Otherwise, put
r : = 4 ∥ G γ , β ( s T ) ∥ 2 n b 0 ∈ ( 0 , 4 n − D b 0 ] , Ω : = B 2 ( s T , r ) ∩ ( s ∗ + 1 ⊥ ) . r:=\frac{4\lVert G_{\gamma,\beta}(s^T)\rVert_2}{nb_0}\in\left(0,\frac{4n^{-D}}{b_0}\right],\qquad\Omega:=B_2(s^T,r)\cap(s^*+1^\perp). r := n b 0 4 ∥ G γ , β ( s T ) ∥ 2 ∈ ( 0 , b 0 4 n − D ] , Ω := B 2 ( s T , r ) ∩ ( s ∗ + 1 ⊥ ) .
Because ∥ s T − s ∗ ∥ ∞ ≤ C 4 log n / n \lVert s^T-s^*\rVert_\infty\leq C_4\sqrt{\log n/n} ∥ s T − s ∗ ∥ ∞ ≤ C 4 log n / n (by (B.33d)) and r = o ( log n / n ) r=o(\sqrt{\log n/n}) r = o ( log n / n ) , the whole ball satisfies Ω ‾ ⊂ B ∞ ( s ∗ , C 5 log n / n ) \overline{\Omega}\subset B_\infty(s^*,C_5\sqrt{\log n/n}) Ω ⊂ B ∞ ( s ∗ , C 5 log n / n ) for n ≥ n 0 n\geq n_0 n ≥ n 0 , since C 5 = C 4 + 1 C_5=C_4+1 C 5 = C 4 + 1 . For x ∈ ∂ Ω x\in\partial\Omega x ∈ ∂ Ω write v : = x − s T ∈ 1 ⊥ v:=x-s^T\in1^\perp v := x − s T ∈ 1 ⊥ (as s T ∈ 1 ⊥ s^T\in1^\perp s T ∈ 1 ⊥ ), ∥ v ∥ 2 = r \lVert v\rVert_2=r ∥ v ∥ 2 = r , and J ˉ : = ∫ 0 1 J ( s T + τ v ) d τ \bar{J}:=\int_0^1J(s^T+\tau v)\,\mathrm{d}\tau J ˉ := ∫ 0 1 J ( s T + τ v ) d τ . Lemma 12 applies to J ˉ \bar{J} J ˉ (segment inside B ‾ ∞ ( s ∗ , ε n ) \overline{B}_\infty(s^*,\varepsilon_n) B ∞ ( s ∗ , ε n ) ), so
v ⊤ G γ , β ( x ) = v ⊤ G γ , β ( s T ) + v ⊤ J ˉ v ≥ − r ∥ G γ , β ( s T ) ∥ 2 + 1 2 n b 0 r 2 = 1 2 n b 0 r ( r − 2 ∥ G γ , β ( s T ) ∥ 2 n b 0 ) > 0. \begin{aligned}
v^\top G_{\gamma,\beta}(x)=v^\top G_{\gamma,\beta}(s^T)+v^\top\bar{J}v\geq-r\lVert G_{\gamma,\beta}(s^T)\rVert_2+\frac{1}{2}nb_0r^2 \\
&=\frac{1}{2}nb_0r\left(r-\frac{2\lVert G_{\gamma,\beta}(s^T)\rVert_2}{nb_0}\right)>0.
\end{aligned} v ⊤ G γ , β ( x ) = v ⊤ G γ , β ( s T ) + v ⊤ J ˉ v ≥ − r ∥ G γ , β ( s T ) ∥ 2 + 2 1 n b 0 r 2 = 2 1 n b 0 r ( r − n b 0 2 ∥ G γ , β ( s T ) ∥ 2 ) > 0.
By the degree-theoretic existence criterion (Fact 4, with anchor s T s^T s T and F = G γ , β F=G_{\gamma,\beta} F = G γ , β restricted to the affine slice s ∗ + 1 ⊥ s^*+1^\perp s ∗ + 1 ⊥ , identified with R n \mathbb{R}^n R n and on which G γ , β G_{\gamma,\beta} G γ , β takes values in 1 ⊥ 1^\perp 1 ⊥ ), G γ , β G_{\gamma,\beta} G γ , β has a zero s ^ γ , β ∈ Ω ‾ \widehat{s}_{\gamma,\beta}\in\overline{\Omega} s γ , β ∈ Ω .
Step 3 (rate). Since s ^ γ , β ∈ B ‾ 2 ( s T , r ) \widehat{s}_{\gamma,\beta}\in\overline{B}_2(s^T,r) s γ , β ∈ B 2 ( s T , r ) ,
∥ s ^ γ , β − s ∗ ∥ ∞ ≤ ∥ s T − s ∗ ∥ ∞ + ∥ s ^ γ , β − s T ∥ ∞ ≤ C 4 log n n + r ≤ C 5 log n n \lVert\widehat{s}_{\gamma,\beta}-s^*\rVert_\infty\leq\lVert s^T-s^*\rVert_\infty+\lVert\widehat{s}_{\gamma,\beta}-s^T\rVert_\infty\leq C_4\sqrt{\frac{\log n}{n}}+r\leq C_5\sqrt{\frac{\log n}{n}} ∥ s γ , β − s ∗ ∥ ∞ ≤ ∥ s T − s ∗ ∥ ∞ + ∥ s γ , β − s T ∥ ∞ ≤ C 4 n log n + r ≤ C 5 n log n
for n ≥ n 0 n\geq n_0 n ≥ n 0 , as r = O ( n − D ) = o ( log n / n ) r=O(n^{-D})=o(\sqrt{\log n/n}) r = O ( n − D ) = o ( log n / n ) for D ≥ 1 D\geq1 D ≥ 1 and C 5 = C 4 + 1 C_5=C_4+1 C 5 = C 4 + 1 ; in particular s ^ γ , β ∈ B ∞ ( s ∗ , ε n ) \widehat{s}_{\gamma,\beta}\in B_\infty(s^*,\varepsilon_n) s γ , β ∈ B ∞ ( s ∗ , ε n ) .
Step 4 (uniqueness). The set B ∞ ( s ∗ , ε n ) ∩ ( s ∗ + 1 ⊥ ) B_\infty(s^*,\varepsilon_n)\cap(s^*+1^\perp) B ∞ ( s ∗ , ε n ) ∩ ( s ∗ + 1 ⊥ ) is convex, and for distinct x , y x,y x , y in it, with J ˉ : = ∫ 0 1 J ( y + τ ( x − y ) ) d τ \bar{J}:=\int_0^1J(y+\tau(x-y))\,\mathrm{d}\tau J ˉ := ∫ 0 1 J ( y + τ ( x − y )) d τ and x − y ∈ 1 ⊥ x-y\in1^\perp x − y ∈ 1 ⊥ , Lemma 12 gives
( G γ , β ( x ) − G γ , β ( y ) ) ⊤ ( x − y ) = ( x − y ) ⊤ J ˉ ( x − y ) ≥ 1 2 n b 0 ∥ x − y ∥ 2 2 > 0. \left(G_{\gamma,\beta}(x)-G_{\gamma,\beta}(y)\right)^\top(x-y)=(x-y)^\top\bar{J}(x-y)\geq\frac{1}{2}nb_0\lVert x-y\rVert_2^2>0. ( G γ , β ( x ) − G γ , β ( y ) ) ⊤ ( x − y ) = ( x − y ) ⊤ J ˉ ( x − y ) ≥ 2 1 n b 0 ∥ x − y ∥ 2 2 > 0.
Hence G γ , β G_{\gamma,\beta} G γ , β is strictly monotone there and has at most one zero; with Step 2, s ^ γ , β \widehat{s}_{\gamma,\beta} s γ , β is the unique zero in B ∞ ( s ∗ , ε n ) ∩ ( s ∗ + 1 ⊥ ) B_\infty(s^*,\varepsilon_n)\cap(s^*+1^\perp) B ∞ ( s ∗ , ε n ) ∩ ( s ∗ + 1 ⊥ ) .
Stage 2: the profiled likelihood and joint consistency
The population information. At the truth, the Fisher information of the model is the expected Hessian I n : = E [ ∇ 2 ( − ℓ ) ( s ∗ , γ ∗ , β ∗ ) ] \mathcal{I}_n:=\mathbb{E}\left[\nabla^2(-\ell)(s^*,\gamma^*,\beta^*)\right] I n := E [ ∇ 2 ( − ℓ ) ( s ∗ , γ ∗ , β ∗ ) ] , whose blocks are obtained from (B.17)–(B.20) by replacing c i j k c_{ijk} c ij k with g i j k ∗ g_{ijk}^{*} g ij k ∗ (the remaining quantities being deterministic at the truth) and whose merit block is V V V of (B.24). Explicitly, with
A p o p : = [ A γ γ ( s ∗ , γ ∗ ) 0 0 ∑ i < j ∑ k g i j k ∗ w θ , i j k ∗ Z i j k Z i j k T ] , A^{\mathrm{pop}} := \left[ \begin{array}{cc} A_{\gamma\gamma}(s^{*},\gamma^{*}) & 0 \\ 0 & \sum_{i<j}\sum_k g_{ijk}^{*}w_{\theta,ijk}^{*}Z_{ijk}Z_{ijk}^{\mathsf T} \end{array} \right], A pop := [ A γ γ ( s ∗ , γ ∗ ) 0 0 ∑ i < j ∑ k g ij k ∗ w θ , ij k ∗ Z ij k Z ij k T ] ,
B p o p : = [ B γ ( s ∗ , γ ∗ ) | ∑ i < j ∑ k g i j k ∗ w θ , i j k ∗ ω i j Z i j k T ] . B^{\mathrm{pop}} := \left[ B_{\gamma}(s^{*},\gamma^{*}) \,\middle|\, \sum_{i<j}\sum_k g_{ijk}^{*}w_{\theta,ijk}^{*}\omega_{ij}Z_{ijk}^{\mathsf T} \right]. B pop := [ B γ ( s ∗ , γ ∗ ) i < j ∑ k ∑ g ij k ∗ w θ , ij k ∗ ω ij Z ij k T ] .
(note A γ γ A_{\gamma\gamma} A γ γ and B γ B_{\gamma} B γ carry no data, hence equal their population versions at the truth), define the efficient information for the structural block as the Schur complement of the merit block in I n \mathcal{I}_n I n ,
Σ p o p : = A p o p − ( B p o p ) T V − 1 B p o p , (51) \Sigma^{\mathrm{pop}} := A^{\mathrm{pop}} - \left(B^{\mathrm{pop}}\right)^{\mathsf T}V^{-1}B^{\mathrm{pop}}, \tag*{(51)} Σ pop := A pop − ( B pop ) T V − 1 B pop , (51)
with V − 1 V^{-1} V − 1 taken on 1 ⊥ 1^{\perp} 1 ⊥ (legitimate by Lemma 2).
Assumption 6 (Efficient information floor). There is a constant c Σ > 0 c_{\Sigma} > 0 c Σ > 0 such that λ min ( Σ p o p ) ≥ c Σ n 2 \lambda_{\min}\left(\Sigma^{\mathrm{pop}}\right) \ge c_{\Sigma} n^{2} λ m i n ( Σ pop ) ≥ c Σ n 2 for all n n n large enough.
Remark 6. Since ( B p o p ) T V − 1 B p o p ⪰ 0 \left(B^{\mathrm{pop}}\right)^{\mathsf T}V^{-1}B^{\mathrm{pop}} \succeq0 ( B pop ) T V − 1 B pop ⪰ 0 (the columns of B p o p B^{\mathrm{pop}} B pop lie in 1 ⊥ 1^{\perp} 1 ⊥ , where V ≻ 0 V \succ0 V ≻ 0 ), A6 forces λ min ( A p o p ) ≥ c Σ n 2 \lambda_{\min}\left(A^{\mathrm{pop}}\right) \ge c_{\Sigma}n^{2} λ m i n ( A pop ) ≥ c Σ n 2 ; A p o p A^{\mathrm{pop}} A pop is block-diagonal with blocks the matrices of A4 and A5, Assumption A6 implies both with c 0 = c Σ c_0 = c_{\Sigma} c 0 = c Σ . We nevertheless retain A4–A5 as standing conditions for their interpretability. For asymptotic normality, which is beyond the scope of this appendix, A6 would be strengthened to the existence of the limit Σ ‾ : = lim n N − 1 Σ p o p ≻ 0 \overline{\Sigma} := \lim_n N^{-1}\Sigma^{\mathrm{pop}} \succ0 Σ := lim n N − 1 Σ pop ≻ 0 , N = ( n + 1 2 ) N = \binom{n+1}{2} N = ( 2 n + 1 ) ; only the floor is needed for consistency.
Lemma 16 (Information concentration at the truth). Assume A1–A3. For every c > 0 c > 0 c > 0 there is an absolute constant C Σ C_{\Sigma} C Σ and an event E Σ \mathcal{E}_{\Sigma} E Σ with P ( E Σ ) ≥ 1 − O ( n − c ) \mathbb{P}(\mathcal{E}_{\Sigma}) \ge1 - O(n^{-c}) P ( E Σ ) ≥ 1 − O ( n − c ) on which
∥ A β β ( s ∗ , β ∗ ) − A β β p o p ∥ 2 ≤ C Σ n log n , ∥ B β ( s ∗ , γ ∗ , β ∗ ) − B β p o p ∥ 2 ≤ C Σ n log n , \left\|A_{\beta\beta}(s^{*},\beta^{*}) - A_{\beta\beta}^{\mathrm{pop}}\right\|_{2} \le C_{\Sigma} n\sqrt{\log n}, \qquad\left\|B_{\beta}(s^{*},\gamma^{*},\beta^{*}) - B_{\beta}^{\mathrm{pop}}\right\|_{2} \le C_{\Sigma} n\sqrt{\log n}, A β β ( s ∗ , β ∗ ) − A β β pop 2 ≤ C Σ n log n , B β ( s ∗ , γ ∗ , β ∗ ) − B β pop 2 ≤ C Σ n log n ,
where A β β p o p A_{\beta\beta}^{\mathrm{pop}} A β β pop and B β p o p B_{\beta}^{\mathrm{pop}} B β pop denote the direction blocks of A p o p A^{\mathrm{pop}} A pop and B p o p B^{\mathrm{pop}} B pop .
Proof. Direction block of A. Write A β β ( s ∗ , β ∗ ) − A β β p o p = ∑ i < j , k X i j k A_{\beta\beta}(s^{*},\beta^{*}) - A_{\beta\beta}^{\mathrm{pop}} = \sum_{i<j,k}X_{ijk} A β β ( s ∗ , β ∗ ) − A β β pop = ∑ i < j , k X ij k with X_{ijk} := (c_{ijk} - g_{ijk}^{})w_{\theta,ijk}^{ }Z_{ijk}Z_{ijk}^{\mathsf T}, independent, symmetric, mean-zero, of dimension p β p_{\beta} p β (fixed). Then ∥ X i j k ∥ 2 ≤ 1 4 κ 2 = : R \|X_{ijk}\|_{2} \le\frac{1}{4}\kappa^{2} =: R ∥ X ij k ∥ 2 ≤ 4 1 κ 2 =: R and ∥ ∑ E X i j 2 ∥ 2 ≤ ∑ ∥ E X i j 2 ∥ 2 ≤ 1 16 κ 4 m ∗ n 2 = : ν \left\|\sum\mathbb{E}X_{ij}^{2}\right\|_{2} \le\sum\left\|\mathbb{E}X_{ij}^{2}\right\|_{2} \le\frac{1}{16}\kappa^{4}m^{*}n^{2} =: \nu ∑ E X ij 2 2 ≤ ∑ E X ij 2 2 ≤ 16 1 κ 4 m ∗ n 2 =: ν , so matrix Bernstein (Fact 2) with t = C Σ n log n t = C_{\Sigma}n\sqrt{\log n} t = C Σ n log n gives exponent ≲ − C 2 log n / ( 2 ⋅ 1 16 κ 4 m ∗ ) \lesssim-C^{2}\log n/(2\cdot\frac{1}{16}\kappa^{4}m^{*}) ≲ − C 2 log n / ( 2 ⋅ 16 1 κ 4 m ∗ ) for n n n large, hence probability O ( n − ( c + 1 ) ) O(n^{-(c+1)}) O ( n − ( c + 1 ) ) for C C C large enough.
Direction block of B. Fix a column index l ≤ p β l \le p_{\beta} l ≤ p β . The l l l -th column of the difference is ∑ i < j , k ς i j k ω i j ∈ 1 ⊥ \sum_{i<j,k}\varsigma_{ijk}\omega_{ij} \in1^{\perp} ∑ i < j , k ς ij k ω ij ∈ 1 ⊥ with ς i j k : = ( c i j k − g i j k ∗ ) w θ , i j k ∗ Z i j k , l \varsigma_{ijk} := (c_{ijk} - g_{ijk}^{*})w_{\theta,ijk}^{*}Z_{ijk,l} ς ij k := ( c ij k − g ij k ∗ ) w θ , ij k ∗ Z ij k , l , independent and mean-zero across ( i , j , k ) (i,j,k) ( i , j , k ) ; ∥ ς i j k ω i j ∥ 2 ≤ 2 4 κ = : R \|\varsigma_{ijk}\omega_{ij}\|_{2} \le\frac{\sqrt{2}}{4}\kappa=: R ∥ ς ij k ω ij ∥ 2 ≤ 4 2 κ =: R and ∑ E ∥ ς i j k ω i j ∥ 2 2 ≤ 1 8 κ 2 m ∗ n 2 = : ν \sum\mathbb{E}\|\varsigma_{ijk}\omega_{ij}\|_{2}^{2} \le\frac{1}{8}\kappa^{2}m^{*}n^{2} =: \nu ∑ E ∥ ς ij k ω ij ∥ 2 2 ≤ 8 1 κ 2 m ∗ n 2 =: ν . Vector Bernstein (Fact 1) in dimension n + 1 n+1 n + 1 with t = C ′ n log n t = C'n\sqrt{\log n} t = C ′ n log n gives, as in Lemma 5, probability O ( n − ( c + 1 ) ) O(n^{-(c+1)}) O ( n − ( c + 1 ) ) for C ′ C' C ′ large. A union bound over the p β p_{\beta} p β columns and ∥ M ∥ 2 ≤ p β max l ∥ M ⋅ l ∥ 2 \|M\|_{2} \le\sqrt{p_{\beta}}\max_l\|M_{\cdot l}\|_{2} ∥ M ∥ 2 ≤ p β max l ∥ M ⋅ l ∥ 2 complete the proof, with C Σ : = max { C , p β C ′ } C_{\Sigma} := \max\{C,\sqrt{p_{\beta}}C'\} C Σ := max { C , p β C ′ } . □ \square □
Lemma 17 (Uniform information floor on the joint neighborhood). Assume A1–A3 and A6, and let K : = B ∞ ( s ∗ , ε n ) × B ( γ ∗ , ρ n ) × B ( β ∗ , ρ n ) \mathcal{K} := B_{\infty}(s^{*},\varepsilon_n) \times B(\gamma^{*},\rho_n) \times B(\beta^{*},\rho_n) K := B ∞ ( s ∗ , ε n ) × B ( γ ∗ , ρ n ) × B ( β ∗ , ρ n ) with ε n , ρ n = O ( log n / n ) \varepsilon_n,\rho_n = O(\sqrt{\log n/n}) ε n , ρ n = O ( log n / n ) . Write, for ( s , ζ ) ∈ K (s,\zeta) \in\mathcal{K} ( s , ζ ) ∈ K ,
Σ ( s , ζ ) : = A ( s , ζ ) − B ( s , ζ ) J ( s , ζ ) − 1 B ( s , ζ ) , \Sigma(s,\zeta) := A(s,\zeta) - B(s,\zeta)J(s,\zeta)^{-1}B(s,\zeta), Σ ( s , ζ ) := A ( s , ζ ) − B ( s , ζ ) J ( s , ζ ) − 1 B ( s , ζ ) , A : = ( A γ γ 0 0 A β β ) , B : = [ B γ ∣ B β ] , A := \begin{pmatrix} A_{\gamma\gamma} & 0 \\ 0 & A_{\beta\beta} \end{pmatrix}, \qquad B := [B_{\gamma} \mid B_{\beta}], A := ( A γ γ 0 0 A β β ) , B := [ B γ ∣ B β ] , with J − 1 J^{-1} J − 1 taken on 1 ⊥ 1^{\perp} 1 ⊥ . For every c > 0 c > 0 c > 0 there is n 0 n_0 n 0 such that for all n ≥ n 0 n \ge n_0 n ≥ n 0 , on E Σ ∩ E s a b \mathcal{E}_{\Sigma} \cap\mathcal{E}_{\mathrm{sab}} E Σ ∩ E sab ,
Σ ( s , ζ ) ≥ 1 2 c Σ n 2 I p for every ( s , ζ ) ∈ K , \Sigma(s,\zeta) \ge\frac{1}{2}c_{\Sigma}n^{2}I_p \qquad\text{for every }(s,\zeta) \in\mathcal{K}, Σ ( s , ζ ) ≥ 2 1 c Σ n 2 I p for every ( s , ζ ) ∈ K , where p = p γ + p β p = p_{\gamma} + p_{\beta} p = p γ + p β .
Proof. Throughout, J − 1 J^{-1} J − 1 is well defined on 1 ⊥ 1^\perp 1 ⊥ with ∥ J ( s , ζ ) − 1 ∥ 2 , 1 ≤ 2 / ( n b 0 ) \lVert J(s,\zeta)^{-1}\rVert_{2,1}\le2/(nb_0) ∥ J ( s , ζ ) − 1 ∥ 2 , 1 ≤ 2/ ( n b 0 ) for all ( s , ζ ) ∈ K (s,\zeta)\in\mathcal{K} ( s , ζ ) ∈ K , by Lemma 12 (whose event is E s a b E_{\mathrm{sab}} E sab ); likewise ∥ V − 1 ∥ 2 , 1 ≤ 1 / ( n b 0 ) \lVert V^{-1}\rVert_{2,1}\le1/(nb_0) ∥ V − 1 ∥ 2 , 1 ≤ 1/ ( n b 0 ) by Lemma 2. Also ∥ B ( s , ζ ) ∥ 2 ≤ C M n 3 / 2 \lVert B(s,\zeta)\rVert_2\le C_Mn^{3/2} ∥ B ( s , ζ ) ∥ 2 ≤ C M n 3/2 on K \mathcal{K} K by Lemma 8, and ∥ B p o p ∥ 2 ≤ C M n 3 / 2 \lVert B^{\mathrm{pop}}\rVert_2\le C_Mn^{3/2} ∥ B pop ∥ 2 ≤ C M n 3/2 by the identical Cauchy–Schwarz argument (with g i j k ∗ w θ , i j k ∗ ≤ 1 4 g_{ijk}^{*}w_{\theta,ijk}^{*}\le\frac{1}{4} g ij k ∗ w θ , ij k ∗ ≤ 4 1 ).
Step 1: from Σ p o p \Sigma^{\mathrm{pop}} Σ pop to Σ ( s ∗ , ζ ∗ ) \Sigma(s^{*},\zeta^{*}) Σ ( s ∗ , ζ ∗ ) . Set δ A : = A ( s ∗ , ζ ∗ ) − A p o p \delta A:=A(s^{*},\zeta^{*})-A^{\mathrm{pop}} δ A := A ( s ∗ , ζ ∗ ) − A pop and δ B : = B ( s ∗ , ζ ∗ ) − B p o p \delta B:=B(s^{*},\zeta^{*})-B^{\mathrm{pop}} δ B := B ( s ∗ , ζ ∗ ) − B pop ; only their direction blocks are nonzero, and on E Σ E_{\Sigma} E Σ both have spectral norm ≤ C Σ n log n \le C_{\Sigma}n\sqrt{\log n} ≤ C Σ n log n (Lemma 16). Telescoping,
B T J − 1 B − ( B p o p ) T V − 1 B p o p = δ B T J − 1 B + ( B p o p ) T J − 1 δ B + ( B p o p ) T ( J − 1 − V − 1 ) B p o p , B^{\mathsf T}J^{-1}B-(B^{\mathrm{pop}})^{\mathsf T}V^{-1}B^{\mathrm{pop}}=\delta B^{\mathsf T}J^{-1}B+(B^{\mathrm{pop}})^{\mathsf T}J^{-1}\delta B+(B^{\mathrm{pop}})^{\mathsf T}(J^{-1}-V^{-1})B^{\mathrm{pop}}, B T J − 1 B − ( B pop ) T V − 1 B pop = δ B T J − 1 B + ( B pop ) T J − 1 δ B + ( B pop ) T ( J − 1 − V − 1 ) B pop ,
all matrices evaluated at ( s ∗ , ζ ∗ ) (s^{*},\zeta^{*}) ( s ∗ , ζ ∗ ) . On E s a b E_{\mathrm{sab}} E sab , J − 1 − V − 1 = − J − 1 E V − 1 J^{-1}-V^{-1}=-J^{-1}EV^{-1} J − 1 − V − 1 = − J − 1 E V − 1 with ∥ E ∥ 2 ≤ C E n log n \lVert E\rVert_2\le C_E\sqrt{n\log n} ∥ E ∥ 2 ≤ C E n log n (Lemma 4), so the three terms are bounded in spectral norm by
2 ⋅ C M n 3 / 2 ⋅ 2 n b 0 ⋅ C Σ n log n + C M 2 n 3 ⋅ 2 C E n log n n 2 b 0 2 = O ( n 3 / 2 log n ) = o ( n 2 ) , 2\cdot C_Mn^{3/2}\cdot\frac{2}{nb_0}\cdot C_{\Sigma}n\sqrt{\log n}+C_M^2n^3\cdot\frac{2C_E\sqrt{n\log n}}{n^2b_0^2}=O(n^{3/2}\sqrt{\log n})=o(n^2), 2 ⋅ C M n 3/2 ⋅ n b 0 2 ⋅ C Σ n log n + C M 2 n 3 ⋅ n 2 b 0 2 2 C E n log n = O ( n 3/2 log n ) = o ( n 2 ) ,
whence, with ∥ δ A ∥ 2 = O ( n log n ) \lVert\delta A\rVert_2=O(n\sqrt{\log n}) ∥ δ A ∥ 2 = O ( n log n ) and Weyl, Σ ( s ∗ , ζ ∗ ) ⪰ Σ p o p − o ( n 2 ) I p ⪰ 3 4 c Σ n 2 I p \Sigma(s^{*},\zeta^{*})\succeq\Sigma^{\mathrm{pop}}-o(n^2)I_p\succeq\frac{3}{4}c_{\Sigma}n^2I_p Σ ( s ∗ , ζ ∗ ) ⪰ Σ pop − o ( n 2 ) I p ⪰ 4 3 c Σ n 2 I p for n ≥ n 0 n\ge n_0 n ≥ n 0 , by A6.
Step 2: across K \mathcal{K} K . Fix ( s , ζ ) ∈ K (s,\zeta)\in\mathcal{K} ( s , ζ ) ∈ K . For the A A A -block, Lemma 10 gives ∥ A ( s , ζ ) − A ( s ∗ , ζ ∗ ) ∥ 2 ≤ L A n 2 ( ε n + 2 ρ n ) = o ( n 2 ) \lVert A(s,\zeta)-A(s^{*},\zeta^{*})\rVert_2\le L_A n^2(\varepsilon_n+2\rho_n)=o(n^2) ∥ A ( s , ζ ) − A ( s ∗ , ζ ∗ ) ∥ 2 ≤ L A n 2 ( ε n + 2 ρ n ) = o ( n 2 ) . For the cross block, Lemma 9 gives ∥ B ( s , ζ ) − B ( s ∗ , ζ ∗ ) ∥ 2 ≤ L B n 3 / 2 ( ε n + 2 ρ n ) = o ( n 3 / 2 ) \lVert B(s,\zeta)-B(s^{*},\zeta^{*})\rVert_2\le L_B n^{3/2}(\varepsilon_n+2\rho_n)=o(n^{3/2}) ∥ B ( s , ζ ) − B ( s ∗ , ζ ∗ ) ∥ 2 ≤ L B n 3/2 ( ε n + 2 ρ n ) = o ( n 3/2 ) . For the inverse,
∥ J ( s , ζ ) − 1 − J ( s ∗ , ζ ∗ ) − 1 ∥ 2 , ⊥ ≤ 4 n 2 b 0 2 ∥ J ( s , ζ ) − J ( s ∗ , ζ ∗ ) ∥ 2 ≤ 4 n 2 b 0 2 ( 4 L f m ∗ n ε n + 2 L ζ n ρ n ) = o ( n − 1 ) \begin{aligned}
\lVert J(s,\zeta)^{-1}-J(s^{*},\zeta^{*})^{-1}\rVert_{2,\perp}\le\frac{4}{n^2b_0^2}\lVert J(s,\zeta)-J(s^{*},\zeta^{*})\rVert_2 \\
&\le\frac{4}{n^2b_0^2}\left(4L_{fm^{*}}n\varepsilon_n+2L_{\zeta}n\rho_n\right)=o(n^{-1})
\end{aligned} ∥ J ( s , ζ ) − 1 − J ( s ∗ , ζ ∗ ) − 1 ∥ 2 , ⊥ ≤ n 2 b 0 2 4 ∥ J ( s , ζ ) − J ( s ∗ , ζ ∗ ) ∥ 2 ≤ n 2 b 0 2 4 ( 4 L f m ∗ n ε n + 2 L ζ n ρ n ) = o ( n − 1 )
by Lemmas 6 and 7. Telescoping B T J − 1 B B^{\mathsf T}J^{-1}B B T J − 1 B between the two points as in Step 1, each of the three terms is O ( n 3 / 2 ) ⋅ O ( n − 1 ) ⋅ o ( n 3 / 2 ) O(n^{3/2})\cdot O(n^{-1})\cdot o(n^{3/2}) O ( n 3/2 ) ⋅ O ( n − 1 ) ⋅ o ( n 3/2 ) or O ( n 3 ) ⋅ o ( n − 1 ) O(n^3)\cdot o(n^{-1}) O ( n 3 ) ⋅ o ( n − 1 ) , hence o ( n 2 ) o(n^2) o ( n 2 ) ; with the A A A -increment and Weyl, Σ ( s , ζ ) ⪰ Σ ( s ∗ , ζ ∗ ) − o ( n 2 ) I p ⪰ 1 2 c Σ n 2 I p \Sigma(s,\zeta)\succeq\Sigma(s^{*},\zeta^{*})-o(n^2)I_p\succeq\frac{1}{2}c_{\Sigma}n^2I_p Σ ( s , ζ ) ⪰ Σ ( s ∗ , ζ ∗ ) − o ( n 2 ) I p ⪰ 2 1 c Σ n 2 I p for n ≥ n 0 n\ge n_0 n ≥ n 0 , uniformly over K \mathcal{K} K since every o ( n 2 ) o(n^2) o ( n 2 ) bound above depends on ( s , ζ ) (s,\zeta) ( s , ζ ) only through ε n , ρ n \varepsilon_n,\rho_n ε n , ρ n .
Structural scores and Hessian blocks. Write the stacked structural score of − ℓ -\ell − ℓ as
T ( s , ζ ) : = ∇ ζ ( − ℓ ) ( s , ζ ) = ( G γ ( s , γ ) T , G β ( s , β ) T ) T ∈ R p , p = p γ + p β , T(s,\zeta):=\nabla_{\zeta}(-\ell)(s,\zeta)=\left(G_{\gamma}(s,\gamma)^{\mathsf T},\,G_{\beta}(s,\beta)^{\mathsf T}\right)^{\mathsf T}\in\mathbb{R}^{p},\qquad p=p_{\gamma}+p_{\beta}, T ( s , ζ ) := ∇ ζ ( − ℓ ) ( s , ζ ) = ( G γ ( s , γ ) T , G β ( s , β ) T ) T ∈ R p , p = p γ + p β ,
where, from (B.11)–(B.12),
G γ ( s , γ ) : = ∇ γ ( − ℓ ) = ∑ i < j ∑ k ( g i j k − c i j k ) Y i j k , G_{\gamma}(s,\gamma):=\nabla_{\gamma}(-\ell)=\sum_{i<j}\sum_k(g_{ijk}-c_{ijk})Y_{ijk}, G γ ( s , γ ) := ∇ γ ( − ℓ ) = i < j ∑ k ∑ ( g ij k − c ij k ) Y ij k ,
G β ( s , β ) : = ∇ β ( − ℓ ) = ∑ i < j ∑ k ( c i j k d i j k − a i j k 1 ) Z i j k , G_{\beta}(s,\beta):=\nabla_{\beta}(-\ell)=\sum_{i<j}\sum_k(c_{ijk}d_{ijk}-a_{ijk}^{1})Z_{ijk}, G β ( s , β ) := ∇ β ( − ℓ ) = i < j ∑ k ∑ ( c ij k d ij k − a ij k 1 ) Z ij k ,
with the corresponding curvature blocks A γ γ A_{\gamma\gamma} A γ γ and A β β A_{\beta\beta} A β β as in (B.17) and (B.18). Since g i j k = σ ( f ( Δ i j ) + Y i j k T γ ) g_{ijk}=\sigma(f(\Delta_{ij})+Y_{ijk}^{\mathsf T}\gamma) g ij k = σ ( f ( Δ ij ) + Y ij k T γ ) carries no β \beta β and d i j k = σ ( Δ i j + Z i j k T β ) d_{ijk}=\sigma(\Delta_{ij}+Z_{ijk}^{\mathsf T}\beta) d ij k = σ ( Δ ij + Z ij k T β ) carries no γ \gamma γ (the data c , a 1 c,a^1 c , a 1 being fixed), at a fixed merit s s s the map G γ G_{\gamma} G γ depends on γ \gamma γ alone and G β G_{\beta} G β on β \beta β alone—equivalently ∇ γ β 2 ( − ℓ ) = 0 \nabla_{\gamma\beta}^{2}(-\ell)=0 ∇ γ β 2 ( − ℓ ) = 0 by (B.21)—so the structural block A A A of the Hessian is block-diagonal. The two channels are nonetheless not independent estimation problems: both share the merit vector, and profiling out s s s re-couples them through the cross blocks, the profile Hessian being the Schur complement Σ = A − B T J − 1 B \Sigma=A-B^{\mathsf T}J^{-1}B Σ = A − B T J − 1 B of Lemma 18, whose off-diagonal block B γ T J − 1 B β B_{\gamma}^{\mathsf T}J^{-1}B_{\beta} B γ T J − 1 B β is nonzero in general. The joint argument below therefore treats ζ = ( γ , β ) \zeta=(\gamma,\beta) ζ = ( γ , β ) as a single p p p -dimensional parameter.
The profiled likelihood. Let Θ 2 : = B ( γ ∗ , 2 ρ n ) × B ( β ∗ , 2 ρ n ) \Theta_2:=\mathcal{B}(\gamma^{*},2\rho_n)\times\mathcal{B}(\beta^{*},2\rho_n) Θ 2 := B ( γ ∗ , 2 ρ n ) × B ( β ∗ , 2 ρ n ) denote the doubled structural ball. Lemmas 13 and 15 hold verbatim with ρ n \rho_n ρ n replaced by $2\rho_n$ (their hypotheses require only a radius of the form const ⋅ log n / n \cdot\sqrt{\log n/n} ⋅ log n / n , now with constant 2 C γ β 2C_{\gamma\beta} 2 C γ β ); we henceforth work on the corresponding event E ~ ∞ \widetilde{E}_{\infty} E ∞ of Lemma 15 run at radius 2 ρ n 2\rho_n 2 ρ n (and horizon $T+1 as in the proof of Lemma 13), on which, by the final assertion of Lemma 15, the conclusions of Lemma 13 hold simultaneously for every ζ ∈ Θ 2 \zeta\in\Theta_2 ζ ∈ Θ 2 —including the degenerate radius C γ β = 0 C_{\gamma\beta}=0 C γ β = 0 at ζ = ζ ∗ : = ( γ ∗ , β ∗ ) \zeta=\zeta^*:=(\gamma^*,\beta^*) ζ = ζ ∗ := ( γ ∗ , β ∗ ) . For ζ ∈ Θ 2 \zeta\in\Theta_2 ζ ∈ Θ 2 write s ^ ζ \widehat{s}_\zeta s ζ for the unique zero of G ζ G_\zeta G ζ in B ‾ ∞ ( s ∗ , ε n ) ∩ ( s ∗ + 1 ⊥ ) \overline{B}_\infty(s^*,\varepsilon_n)\cap(s^*+1^\perp) B ∞ ( s ∗ , ε n ) ∩ ( s ∗ + 1 ⊥ ) furnished by Lemma 13, with ε n = C 5 log n / n \varepsilon_n=C_5\sqrt{\log n/n} ε n = C 5 log n / n and C 5 C_5 C 5 the Stage-1 constant at the doubled radius. The profiled negative log-likelihood is
φ ( ζ ) : = − ℓ ( s ^ ζ , ζ ) , ζ ∈ Θ 2 . (52) \varphi(\zeta):=-\ell(\widehat{s}_\zeta,\zeta),\qquad\zeta\in\Theta_2. \tag*{(52)} φ ( ζ ) := − ℓ ( s ζ , ζ ) , ζ ∈ Θ 2 . (52)
Since − ℓ ( ⋅ , ζ ) -\ell(\cdot,\zeta) − ℓ ( ⋅ , ζ ) is strictly convex on B ‾ ∞ ( s ∗ , ε n ) ∩ ( s ∗ + 1 ⊥ ) \overline{B}_\infty(s^*,\varepsilon_n)\cap(s^*+1^\perp) B ∞ ( s ∗ , ε n ) ∩ ( s ∗ + 1 ⊥ ) (Lemma 12) with interior critical point s ^ ζ \widehat{s}_\zeta s ζ , one has equivalently φ ( ζ ) = min { − ℓ ( s , ζ ) : s ∈ B ‾ ∞ ( s ∗ , ε n ) ∩ ( s ∗ + 1 ⊥ ) } \varphi(\zeta)=\min\{-\ell(s,\zeta):s\in\overline{B}_\infty(s^*,\varepsilon_n)\cap(s^*+1^\perp)\} φ ( ζ ) = min { − ℓ ( s , ζ ) : s ∈ B ∞ ( s ∗ , ε n ) ∩ ( s ∗ + 1 ⊥ )} , so φ \varphi φ is the merit-profiled objective in the usual sense.
Lemma 18 (Profile regularity and envelope identities). Assume A1–A3. For every c > 0 c>0 c > 0 there is n 0 n_0 n 0 such that for all n ≥ n 0 n\ge n_0 n ≥ n 0 , on E ~ l o o \widetilde{\mathcal{E}}_{\mathrm{loo}} E loo , the following hold.
(a) (Regularity of the profile root.) The map ζ ↦ s ^ ζ \zeta\mapsto\widehat{s}_\zeta ζ ↦ s ζ is C 1 C^1 C 1 on the interior of Θ 2 \Theta_2 Θ 2 , with values in B ‾ ∞ ( s ∗ , ε n ) ∩ ( s ∗ + 1 ⊥ ) \overline{B}_\infty(s^*,\varepsilon_n)\cap(s^*+1^\perp) B ∞ ( s ∗ , ε n ) ∩ ( s ∗ + 1 ⊥ ) and Jacobian
d s ^ ζ d ζ = − J ( s ^ ζ , ζ ) − 1 B ( s ^ ζ , ζ ) ∈ R ( n + 1 ) × p , \frac{d\widehat{s}_\zeta}{d\zeta}=-J(\widehat{s}_\zeta,\zeta)^{-1}B(\widehat{s}_\zeta,\zeta)\in\mathbb{R}^{(n+1)\times p}, d ζ d s ζ = − J ( s ζ , ζ ) − 1 B ( s ζ , ζ ) ∈ R ( n + 1 ) × p , where J − 1 J^{-1} J − 1 is taken on 1 ⊥ 1^\perp 1 ⊥ and B = [ B γ ∣ B β ] B=[B_\gamma\mid B_\beta] B = [ B γ ∣ B β ] as in Lemma 9.
(b) (Envelope identities.) φ \varphi φ is C 2 C^2 C 2 on the interior of Θ 2 \Theta_2 Θ 2 , with
∇ φ ( ζ ) = T ( s ^ ζ , ζ ) , \nabla\varphi(\zeta)=T(\widehat{s}_\zeta,\zeta), ∇ φ ( ζ ) = T ( s ζ , ζ ) , ∇ 2 φ ( ζ ) = Σ ( s ^ ζ , ζ ) = A ( s ^ ζ , ζ ) − B ( s ^ ζ , ζ ) T J ( s ^ ζ , ζ ) − 1 B ( s ^ ζ , ζ ) . \nabla^2\varphi(\zeta)=\Sigma(\widehat{s}_\zeta,\zeta)=A(\widehat{s}_\zeta,\zeta)-B(\widehat{s}_\zeta,\zeta)^\mathsf{T}J(\widehat{s}_\zeta,\zeta)^{-1}B(\widehat{s}_\zeta,\zeta). ∇ 2 φ ( ζ ) = Σ ( s ζ , ζ ) = A ( s ζ , ζ ) − B ( s ζ , ζ ) T J ( s ζ , ζ ) − 1 B ( s ζ , ζ ) . (c) (Profile curvature floor.) If in addition A6 holds then, on E ~ l o o ∩ E Σ \widetilde{\mathcal{E}}_{\mathrm{loo}}\cap\mathcal{E}_{\Sigma} E loo ∩ E Σ ,
∇ 2 φ ( ζ ) ⪰ 1 2 c Σ n 2 I p for every ζ in the interior of Θ 2 . \nabla^2\varphi(\zeta)\succeq\frac{1}{2}c_{\Sigma}n^2I_p\qquad\text{for every }\zeta\text{ in the interior of }\Theta_2. ∇ 2 φ ( ζ ) ⪰ 2 1 c Σ n 2 I p for every ζ in the interior of Θ 2 .
Proof. (a). Parametrize the gauge slice by 1 ⊥ 1^\perp 1 ⊥ : writing s = s ∗ + u s=s^*+u s = s ∗ + u with u ∈ 1 ⊥ u\in1^\perp u ∈ 1 ⊥ , the map F ( u , ζ ) : = G ζ ( s ∗ + u ) ∈ 1 ⊥ F(u,\zeta):=G_\zeta(s^*+u)\in1^\perp F ( u , ζ ) := G ζ ( s ∗ + u ) ∈ 1 ⊥ is C 2 C^2 C 2 on an open neighborhood of its domain (under A3, f ∈ C 3 f\in C^3 f ∈ C 3 makes ℓ \ell ℓ three times continuously differentiable in all arguments, and G ζ = ∇ s ( − ℓ ) G_\zeta=\nabla_s(-\ell) G ζ = ∇ s ( − ℓ ) ). Fix ζ 0 \zeta_0 ζ 0 in the interior of Θ 2 \Theta_2 Θ 2 and set u 0 : = s ^ ζ 0 − s ∗ u_0:=\widehat{s}_{\zeta_0}-s^* u 0 := s ζ 0 − s ∗ , so F ( u 0 , ζ 0 ) = 0 F(u_0,\zeta_0)=0 F ( u 0 , ζ 0 ) = 0 with ∥ u 0 ∥ ∞ < ε n \|u_0\|_\infty<\varepsilon_n ∥ u 0 ∥ ∞ < ε n (the root is interior to the ℓ ∞ \ell_\infty ℓ ∞ ball since, by the proof of Lemma 13, ∥ s ^ ζ − s ∗ ∥ ∞ ≤ C 4 log n / n + o ( log n / n ) \|\widehat{s}_\zeta-s^*\|_\infty\le C_4\sqrt{\log n/n}+o(\sqrt{\log n/n}) ∥ s ζ − s ∗ ∥ ∞ ≤ C 4 log n / n + o ( log n / n ) and C 5 = C 4 + 1 C_5=C_4+1 C 5 = C 4 + 1 ). The partial derivative ∂ u F = J ( s ∗ + u , ζ ) ∣ 1 ⊥ \partial_uF=J(s^*+u,\zeta)|_{1^\perp} ∂ u F = J ( s ∗ + u , ζ ) ∣ 1 ⊥ is invertible at ( u 0 , ζ 0 ) (u_0,\zeta_0) ( u 0 , ζ 0 ) with λ min , ⊥ ≥ 1 2 n b 0 \lambda_{\min,\perp}\ge\frac{1}{2}nb_0 λ m i n , ⊥ ≥ 2 1 n b 0 (Lemma 12, whose event E s a b ⊇ E ~ l o o \mathcal{E}_{\mathrm{sab}}\supseteq\widetilde{\mathcal{E}}_{\mathrm{loo}} E sab ⊇ E loo is in force, its radii ε n , 2 ρ n \varepsilon_n,2\rho_n ε n , 2 ρ n being o ( 1 ) o(1) o ( 1 ) ). By the implicit function theorem on 1 ⊥ × R p 1^\perp\times\mathbb{R}^p 1 ⊥ × R p , there is a neighborhood of ζ 0 \zeta_0 ζ 0 and a C 1 C^1 C 1 map ζ ↦ u ( ζ ) \zeta\mapsto u(\zeta) ζ ↦ u ( ζ ) with F ( u ( ζ ) , ζ ) = 0 F(u(\zeta),\zeta)=0 F ( u ( ζ ) , ζ ) = 0 and u ( ζ 0 ) = u 0 u(\zeta_0)=u_0 u ( ζ 0 ) = u 0 ; shrinking the neighborhood keeps ∥ u ( ζ ) ∥ ∞ < ε n \|u(\zeta)\|_\infty<\varepsilon_n ∥ u ( ζ ) ∥ ∞ < ε n , so by the uniqueness in Lemma 13 the branch coincides with s ^ ζ − s ∗ \widehat{s}_\zeta-s^* s ζ − s ∗ there. Hence ζ ↦ s ^ ζ \zeta\mapsto\widehat{s}_\zeta ζ ↦ s ζ is C 1 C^1 C 1 at every interior ζ 0 \zeta_0 ζ 0 . Differentiating G ζ ( s ^ ζ ) = 0 G_\zeta(\widehat{s}_\zeta)=0 G ζ ( s ζ ) = 0 in ζ \zeta ζ ,
J ( s ^ ζ , ζ ) d s ^ ζ d ζ + ∂ G ζ ∂ ζ ( s ^ ζ ) = 0 , J(\widehat{s}_\zeta,\zeta)\frac{d\widehat{s}_\zeta}{d\zeta}+\frac{\partial G_\zeta}{\partial\zeta}(\widehat{s}_\zeta)=0, J ( s ζ , ζ ) d ζ d s ζ + ∂ ζ ∂ G ζ ( s ζ ) = 0 ,
and ∂ G ζ / ∂ ζ = B \partial G_\zeta/\partial\zeta=B ∂ G ζ / ∂ ζ = B (proof of Lemma 8); solving on 1 ⊥ 1^\perp 1 ⊥ (both the columns of B B B and the range of d s ^ ζ / d ζ d\widehat{s}_\zeta/d\zeta d s ζ / d ζ lie in 1 ⊥ 1^\perp 1 ⊥ , the root staying in the gauge slice) gives the stated Jacobian.
(b). By the chain rule and part (a),
∇ φ ( ζ ) = ( d s ^ ζ d ζ ) T ∇ s ( − ℓ ) ( s ^ ζ , ζ ) ⏟ = G ζ ( s ^ ζ ) = 0 + ∇ ζ ( − ℓ ) ( s ^ ζ , ζ ) = T ( s ^ ζ , ζ ) . \nabla\varphi(\zeta)=\left(\frac{d\widehat{s}_\zeta}{d\zeta}\right)^\mathsf{T}\underbrace{\nabla_s(-\ell)(\widehat{s}_\zeta,\zeta)}_{=G_\zeta(\widehat{s}_\zeta)=0}+\nabla_\zeta(-\ell)(\widehat{s}_\zeta,\zeta)=T(\widehat{s}_\zeta,\zeta). ∇ φ ( ζ ) = ( d ζ d s ζ ) T = G ζ ( s ζ ) = 0 ∇ s ( − ℓ ) ( s ζ , ζ ) + ∇ ζ ( − ℓ ) ( s ζ , ζ ) = T ( s ζ , ζ ) .
the merit direction contributes exactly zero, by the definition of the profile root. For the Hessian, differentiate once more:
∇ 2 φ ( ζ ) = ∂ T ∂ s ( s ^ ζ , ζ ) d s ^ ζ d ζ + ∂ T ∂ ζ ( s ^ ζ , ζ ) = B T ( − J − 1 B ) + A = Σ ( s ^ ζ , ζ ) , (53) \nabla^2\varphi(\zeta)=\frac{\partial T}{\partial s}(\widehat{s}_\zeta,\zeta)\frac{d\widehat{s}_\zeta}{d\zeta}+\frac{\partial T}{\partial\zeta}(\widehat{s}_\zeta,\zeta)=B^\mathsf{T}(-J^{-1}B)+A=\Sigma(\widehat{s}_\zeta,\zeta), \tag*{(53)} ∇ 2 φ ( ζ ) = ∂ s ∂ T ( s ζ , ζ ) d ζ d s ζ + ∂ ζ ∂ T ( s ζ , ζ ) = B T ( − J − 1 B ) + A = Σ ( s ζ , ζ ) , (53)
using ∂ T / ∂ s = ∇ ζ s 2 ( − ℓ ) = B T \partial T/\partial s=\nabla_{\zeta s}^{2}(-\ell)=B^{\mathsf T} ∂ T / ∂ s = ∇ ζ s 2 ( − ℓ ) = B T and ∂ T / ∂ ζ = ∇ ζ ζ 2 ( − ℓ ) = A \partial T/\partial\zeta=\nabla_{\zeta\zeta}^{2}(-\ell)=A ∂ T / ∂ ζ = ∇ ζ ζ 2 ( − ℓ ) = A , the latter block-diagonal by (B.21). In particular ∇ 2 φ \nabla^{2}\varphi ∇ 2 φ is symmetric, as it must be, and φ \varphi φ is C 2 C^{2} C 2 since all blocks are C 1 C^{1} C 1 compositions of C 1 C^{1} C 1 maps.
(c). By part (a), ( s ^ ζ ^ , ζ ) ∈ B ∞ ( s ∗ , ϵ n ) × Θ 2 ⊆ K (\widehat{s}_{\widehat{\zeta}},\zeta)\in B_{\infty}(s^{*},\epsilon_{n})\times\Theta_{2}\subseteq\mathcal{K} ( s ζ , ζ ) ∈ B ∞ ( s ∗ , ϵ n ) × Θ 2 ⊆ K with the radii of Lemma 17 (there ρ n \rho_{n} ρ n is read as 2 ρ n = O ( log n / n ) 2\rho_{n}=O(\sqrt{\log n/n}) 2 ρ n = O ( log n / n ) ), so that lemma applies at ( s ^ ζ ^ , ζ ) (\widehat{s}_{\widehat{\zeta}},\zeta) ( s ζ , ζ ) and gives ∇ 2 φ ( ζ ) = Σ ( s ^ ζ ^ , ζ ) ⪰ 1 2 c Σ n 2 I p \nabla^{2}\varphi(\zeta)=\Sigma(\widehat{s}_{\widehat{\zeta}},\zeta)\succeq\frac{1}{2}c_{\Sigma}n^{2}I_{p} ∇ 2 φ ( ζ ) = Σ ( s ζ , ζ ) ⪰ 2 1 c Σ n 2 I p .
Lemma 19 (Structural score at the profile anchor). *Assume A1–A3 and n ≥ 2 n\ge2 n ≥ 2 . Let C 5 ′ C_{5}^{\prime} C 5 ′ be the Stage-1 constant of Lemma 13 run with C γ β = 0 C_{\gamma\beta}=0 C γ β = 0 , i.e. C 5 ′ = C 4 ( 0 ) + 1 C_{5}^{\prime}=C_{4}(0)+1 C 5 ′ = C 4 ( 0 ) + 1 with C 4 ( 0 ) C_{4}(0) C 4 ( 0 ) from Lemma 14 at C γ β = 0 C_{\gamma\beta}=0 C γ β = 0 ; in particular C 5 ′ C_{5}^{\prime} C 5 ′ does not depend on C γ β C_{\gamma\beta} C γ β , and on E ~ l o o \widetilde{\mathcal{E}}_{\mathrm{loo}} E loo the merit root s ^ ζ ∗ \widehat{s}_{\zeta^{*}} s ζ ∗ satisfies ∥ s ^ ζ ∗ − s ∗ ∥ ∞ ≤ C 5 ′ log n / n \lVert\widehat{s}_{\zeta^{*}}-s^{*}\rVert_{\infty}\le C_{5}^{\prime}\sqrt{\log n/n} ∥ s ζ ∗ − s ∗ ∥ ∞ ≤ C 5 ′ log n / n . Then, on E ~ l o o \widetilde{\mathcal{E}}_{\mathrm{loo}} E loo ,
∥ T ( s ^ ζ ∗ , ζ ∗ ) ∥ 2 ≤ C ⋆ n 3 / 2 log n , C ⋆ : = 2 ( C κ + C M C 5 ′ ) . \lVert T(\widehat{s}_{\zeta^{*}},\zeta^{*})\rVert_{2}\le C_{\star}n^{3/2}\sqrt{\log n},\qquad C_{\star}:=\sqrt{2}(C_{\kappa}+C_{M}C_{5}^{\prime}). ∥ T ( s ζ ∗ , ζ ∗ ) ∥ 2 ≤ C ⋆ n 3/2 log n , C ⋆ := 2 ( C κ + C M C 5 ′ ) . Proof. At the truth, the two blocks of T ( s ∗ , ζ ∗ ) T(s^{*},\zeta^{*}) T ( s ∗ , ζ ∗ ) are G γ ( s ∗ , γ ∗ ) G_{\gamma}(s^{*},\gamma^{*}) G γ ( s ∗ , γ ∗ ) and G β ( s ∗ , β ∗ ) G_{\beta}(s^{*},\beta^{*}) G β ( s ∗ , β ∗ ) , each of ℓ 2 \ell_{2} ℓ 2 norm ≤ C κ n log n \le C_{\kappa}n\sqrt{\log n} ≤ C κ n log n on E s c o r e ⊇ E ~ l o o \mathcal{E}_{\mathrm{score}}\supseteq\widetilde{\mathcal{E}}_{\mathrm{loo}} E score ⊇ E loo (Lemma 3, structural part), so the stacked vector obeys ∥ T ( s ∗ , ζ ∗ ) ∥ 2 ≤ 2 C κ n log n \lVert T(s^{*},\zeta^{*})\rVert_{2}\le\sqrt{2}C_{\kappa}n\sqrt{\log n} ∥ T ( s ∗ , ζ ∗ ) ∥ 2 ≤ 2 C κ n log n . For the increment, ∂ T / ∂ s = B T \partial T/\partial s=B^{\mathsf T} ∂ T / ∂ s = B T (proof of Lemma 18), so the fundamental theorem of calculus along s ∗ + τ ( s ^ ζ ∗ − s ∗ ) s^{*}+\tau(\widehat{s}_{\zeta^{*}}-s^{*}) s ∗ + τ ( s ζ ∗ − s ∗ ) —a segment contained in B ∞ ( s ∗ , C 5 ′ log n / n ) B_{\infty}(s^{*},C_{5}^{\prime}\sqrt{\log n/n}) B ∞ ( s ∗ , C 5 ′ log n / n ) , hence in the working region—and Lemma 8 give
∥ T ( s ^ ζ ∗ , ζ ∗ ) − T ( s ∗ , ζ ∗ ) ∥ 2 ≤ C M n 3 / 2 ∥ s ^ ζ ∗ − s ∗ ∥ 2 \lVert T(\widehat{s}_{\zeta^{*}},\zeta^{*})-T(s^{*},\zeta^{*})\rVert_{2}\le C_{M}n^{3/2}\lVert\widehat{s}_{\zeta^{*}}-s^{*}\rVert_{2} ∥ T ( s ζ ∗ , ζ ∗ ) − T ( s ∗ , ζ ∗ ) ∥ 2 ≤ C M n 3/2 ∥ s ζ ∗ − s ∗ ∥ 2 ≤ C M n 3 / 2 n + 1 C 5 ′ log n n ≤ 2 C M C 5 ′ n 3 / 2 log n . \le C_{M}n^{3/2}\sqrt{n+1}\,C_{5}^{\prime}\sqrt{\frac{\log n}{n}}\le\sqrt{2}C_{M}C_{5}^{\prime}n^{3/2}\sqrt{\log n}. ≤ C M n 3/2 n + 1 C 5 ′ n log n ≤ 2 C M C 5 ′ n 3/2 log n . Adding the two bounds and using n log n ≤ n 3 / 2 log n n\sqrt{\log n}\le n^{3/2}\sqrt{\log n} n log n ≤ n 3/2 log n gives the claim.
Theorem 2 (Joint ℓ ∞ / ℓ 2 \ell_{\infty}/\ell_{2} ℓ ∞ / ℓ 2 consistency). *Assume A1–A3 and A6. For every c > 0 c>0 c > 0 there is n 0 = n 0 ( c ) n_{0}=n_{0}(c) n 0 = n 0 ( c ) and an event of probability at least 1 − O ( n − c ) 1-O(n^{-c}) 1 − O ( n − c ) on which, for all n ≥ n 0 n\ge n_{0} n ≥ n 0 , there is a point ( s ^ , γ ^ , β ^ ) (\widehat{s},\widehat{\gamma},\widehat{\beta}) ( s , γ , β ) with ∇ s ℓ = ∇ γ ℓ = ∇ β ℓ = 0 \nabla_{s}\ell=\nabla_{\gamma}\ell=\nabla_{\beta}\ell=0 ∇ s ℓ = ∇ γ ℓ = ∇ β ℓ = 0 satisfying
∥ s ^ − s ∗ ∥ ∞ = O ( log n n ) , ∥ γ ^ − γ ∗ ∥ 2 = O ( log n n ) , ∥ β ^ − β ∗ ∥ 2 = O ( log n n ) . \lVert\widehat{s}-s^{*}\rVert_{\infty}=O\left(\sqrt{\frac{\log n}{n}}\right),\qquad
\lVert\widehat{\gamma}-\gamma^{*}\rVert_{2}=O\left(\sqrt{\frac{\log n}{n}}\right),\qquad
\lVert\widehat{\beta}-\beta^{*}\rVert_{2}=O\left(\sqrt{\frac{\log n}{n}}\right). ∥ s − s ∗ ∥ ∞ = O ( n log n ) , ∥ γ − γ ∗ ∥ 2 = O ( n log n ) , ∥ β − β ∗ ∥ 2 = O ( n log n ) . Moreover ζ ^ = ( γ ^ , β ^ ) \widehat{\zeta}=(\widehat{\gamma},\widehat{\beta}) ζ = ( γ , β ) is the unique zero of ∇ φ \nabla\varphi ∇ φ in B ( ζ ∗ , ρ n ) \mathcal{B}(\zeta^{*},\rho_{n}) B ( ζ ∗ , ρ n ) .
Proof. Constants, in acyclic order. First run Lemma 14 at the fixed pair ( γ ∗ , β ∗ ) (\gamma^{*},\beta^{*}) ( γ ∗ , β ∗ ) , i.e. with C γ β = 0 C_{\gamma\beta}=0 C γ β = 0 , and let C 5 ′ C_{5}^{\prime} C 5 ′ be the resulting Stage-1 constant of Lemma 19; C 5 ′ C_{5}^{\prime} C 5 ′ depends only on the constants of A1–A3 and on c c c . Next set
C ⋆ : = 2 ( C κ + C M C 5 ′ ) , C γ β : = 3 C ⋆ c Σ , ρ n : = C γ β log n n . C_{\star}:=\sqrt{2}(C_{\kappa}+C_{M}C_{5}^{\prime}),\qquad
C_{\gamma\beta}:=\frac{3C_{\star}}{c_{\Sigma}},\qquad
\rho_{n}:=C_{\gamma\beta}\sqrt{\frac{\log n}{n}}. C ⋆ := 2 ( C κ + C M C 5 ′ ) , C γ β := c Σ 3 C ⋆ , ρ n := C γ β n log n . Finally run Lemmas 14 and 15 at the doubled radius 2 ρ n 2\rho_{n} 2 ρ n (constant 2 C γ β 2C_{\gamma\beta} 2 C γ β ), obtain the Stage-1 constant C 5 C_{5} C 5 (≥ C 5 ′ \ge C_{5}^{\prime} ≥ C 5 ′ without loss of generality), and set ϵ n : = C 5 log n / n \epsilon_{n}:=C_{5}\sqrt{\log n/n} ϵ n := C 5 log n / n . The chain is C 5 ′ → C ⋆ → C γ β → C 5 C_{5}^{\prime}\to C_{\star}\to C_{\gamma\beta}\to C_{5} C 5 ′ → C ⋆ → C γ β → C 5 , and C 5 C_{5} C 5 is consumed below only through the o ( 1 ) o(1) o ( 1 ) radius ϵ n \epsilon_{n} ϵ n (via Lemmas 17 and 18), never through a score bound; there is no cycle.
Event. Work on E : = E ~ l o o ∩ E Σ \mathcal{E}:=\widetilde{\mathcal{E}}_{\mathrm{loo}}\cap\mathcal{E}_{\Sigma} E := E loo ∩ E Σ , with E ~ l o o \widetilde{\mathcal{E}}_{\mathrm{loo}} E loo the event of Lemma 15 at radius 2 ρ n 2\rho_{n} 2 ρ n (it contains E s c o r e \mathcal{E}_{\mathrm{score}} E score and E s a b \mathcal{E}_{\mathrm{sab}} E sab ) and E Σ \mathcal{E}_{\Sigma} E Σ that of Lemma 16; the union bound gives P ( E ) ≥ 1 − O ( n − c ) \mathbb{P}(\mathcal{E})\ge1-O(n^{-c}) P ( E ) ≥ 1 − O ( n − c ) . On E \mathcal{E} E , the profile root s ^ ζ \widehat{s}_{\zeta} s ζ and the profile φ \varphi φ of (B.40) are defined on Θ 2 \Theta_{2} Θ 2 , and Lemma 18 is in force. Note that the root s ^ ζ ∗ \widehat{s}_{\zeta^{*}} s ζ ∗ of Lemma 19 coincides with the profile root at ζ = ζ ∗ \zeta=\zeta^{*} ζ = ζ ∗ : both are zeros of G γ ∗ , β ∗ G_{\gamma^{*},\beta^{*}} G γ ∗ , β ∗ in B ‾ ∞ ( s ∗ , ϵ n ) ∩ ( s ∗ + 1 ⊥ ) \overline{B}_{\infty}(s^{*},\epsilon_{n})\cap(s^{*}+\mathbbm{1}^{\perp}) B ∞ ( s ∗ , ϵ n ) ∩ ( s ∗ + 1 ⊥ ) (using C 5 ′ ≤ C 5 C_{5}^{\prime}\le C_{5} C 5 ′ ≤ C 5 ), where Lemma 13 guarantees uniqueness.
Existence of a structural root. Apply the degree-theoretic criterion (Fact 4) to F = ∇ φ F=\nabla\varphi F = ∇ φ on Ω = B ( ζ ∗ , ρ n ) \Omega=\mathcal{B}(\zeta^{*},\rho_{n}) Ω = B ( ζ ∗ , ρ n ) with anchor ζ ∗ \zeta^{*} ζ ∗ ; F F F is continuous on Ω ‾ \overline{\Omega} Ω since Ω ‾ \overline{\Omega} Ω lies in the interior of Θ 2 \Theta_{2} Θ 2 , where φ \varphi φ is C 2 C^{2} C 2 (Lemma 18(b)). For ζ ∈ ∂ Ω \zeta\in\partial\Omega ζ ∈ ∂ Ω write u : = ζ − ζ ∗ u:=\zeta-\zeta^{*} u := ζ − ζ ∗ , ∥ u ∥ 2 = ρ n \lVert u\rVert_{2}=\rho_{n} ∥ u ∥ 2 = ρ n . By the fundamental theorem of
calculus along [ ζ ∗ , ζ ] ⊂ Ω ‾ [\zeta^*,\zeta]\subset\overline{\Omega} [ ζ ∗ , ζ ] ⊂ Ω , the envelope identity ∇ φ ( ζ ∗ ) = T ( s ^ ζ ∗ , ζ ∗ ) \nabla\varphi(\zeta^*)=T(\widehat{s}_{\zeta^*},\zeta^*) ∇ φ ( ζ ∗ ) = T ( s ζ ∗ , ζ ∗ ) , and the curvature floor of Lemma 18(c),
u ⊤ ∇ φ ( ζ ) = u ⊤ ∇ φ ( ζ ∗ ) + u ⊤ ( ∫ 0 1 ∇ 2 φ ( ζ ∗ + τ u ) d τ ) u ≥ − ρ n C ⋆ n 3 / 2 log n + 1 2 c Σ n 2 ρ n 2 . u^\top\nabla\varphi(\zeta)=u^\top\nabla\varphi(\zeta^*)+u^\top\left(\int_0^1\nabla^2\varphi(\zeta^*+\tau u)\,d\tau\right)u\geq-\rho_n C_\star n^{3/2}\sqrt{\log n}+\frac{1}{2}c_{\Sigma}n^2\rho_n^2. u ⊤ ∇ φ ( ζ ) = u ⊤ ∇ φ ( ζ ∗ ) + u ⊤ ( ∫ 0 1 ∇ 2 φ ( ζ ∗ + τ u ) d τ ) u ≥ − ρ n C ⋆ n 3/2 log n + 2 1 c Σ n 2 ρ n 2 .
using Lemma 19 for the first term. Hence
u ⊤ ∇ φ ( ζ ) ≥ 1 2 c Σ n 2 ρ n ( ρ n − 2 C ⋆ c Σ log n n ) > 0 , u^\top\nabla\varphi(\zeta)\geq\frac{1}{2}c_{\Sigma}n^2\rho_n\left(\rho_n-\frac{2C_\star}{c_{\Sigma}}\sqrt{\frac{\log n}{n}}\right)>0, u ⊤ ∇ φ ( ζ ) ≥ 2 1 c Σ n 2 ρ n ( ρ n − c Σ 2 C ⋆ n log n ) > 0 ,
the bracket being strictly positive since C γ β = 3 C ⋆ / c Σ > 2 C ⋆ / c Σ C_{\gamma\beta}=3C_\star/c_{\Sigma}>2C_\star/c_{\Sigma} C γ β = 3 C ⋆ / c Σ > 2 C ⋆ / c Σ . By Fact 4, ∇ φ \nabla\varphi ∇ φ has a zero ζ ^ ∈ Ω ‾ \widehat{\zeta}\in\overline{\Omega} ζ ∈ Ω ; since u ⊤ ∇ φ > 0 u^\top\nabla\varphi>0 u ⊤ ∇ φ > 0 on ∂ Ω \partial\Omega ∂ Ω , the zero is interior, ζ ^ ∈ B ( ζ ∗ , ρ n ) \widehat{\zeta}\in B(\zeta^*,\rho_n) ζ ∈ B ( ζ ∗ , ρ n ) .
Uniqueness of the structural root. For distinct ζ 1 , ζ 2 ∈ Ω ‾ \zeta_1,\zeta_2\in\overline{\Omega} ζ 1 , ζ 2 ∈ Ω , with Σ ‾ : = ∫ 0 1 ∇ 2 φ ( ζ 2 + τ ( ζ 1 − ζ 2 ) ) d τ ⪰ 1 2 c Σ n 2 I p \overline{\Sigma}:=\int_0^1\nabla^2\varphi(\zeta_2+\tau(\zeta_1-\zeta_2))\,d\tau\succeq\frac{1}{2}c_{\Sigma}n^2I_p Σ := ∫ 0 1 ∇ 2 φ ( ζ 2 + τ ( ζ 1 − ζ 2 )) d τ ⪰ 2 1 c Σ n 2 I p (Lemma 18(c), the segment lying in Ω ‾ \overline{\Omega} Ω ),
( ∇ φ ( ζ 1 ) − ∇ φ ( ζ 2 ) ) ⊤ ( ζ 1 − ζ 2 ) = ( ζ 1 − ζ 2 ) ⊤ Σ ‾ ( ζ 1 − ζ 2 ) ≥ 1 2 c Σ n 2 ∥ ζ 1 − ζ 2 ∥ 2 2 > 0 , (\nabla\varphi(\zeta_1)-\nabla\varphi(\zeta_2))^\top(\zeta_1-\zeta_2)=(\zeta_1-\zeta_2)^\top\overline{\Sigma}(\zeta_1-\zeta_2)\geq\frac{1}{2}c_{\Sigma}n^2\lVert\zeta_1-\zeta_2\rVert_2^2>0, ( ∇ φ ( ζ 1 ) − ∇ φ ( ζ 2 ) ) ⊤ ( ζ 1 − ζ 2 ) = ( ζ 1 − ζ 2 ) ⊤ Σ ( ζ 1 − ζ 2 ) ≥ 2 1 c Σ n 2 ∥ ζ 1 − ζ 2 ∥ 2 2 > 0 ,
so ∇ φ \nabla\varphi ∇ φ is strictly monotone on Ω ‾ \overline{\Omega} Ω and ζ ^ \widehat{\zeta} ζ is its unique zero there.
Conclusion. Set s ^ : = s ^ ζ ^ \widehat{s}:=\widehat{s}_{\widehat{\zeta}} s := s ζ and ( γ ^ , β ^ ) : = ζ ^ (\widehat{\gamma},\widehat{\beta}):=\widehat{\zeta} ( γ , β ) := ζ . Then ∇ s ℓ ( s ^ , ζ ^ ) = 0 \nabla_s\ell(\widehat{s},\widehat{\zeta})=0 ∇ s ℓ ( s , ζ ) = 0 by the definition of the profile root, and ∇ ζ ℓ ( s ^ , ζ ^ ) = − T ( s ^ ζ ^ , ζ ^ ) = − ∇ φ ( ζ ^ ) = 0 \nabla_{\zeta}\ell(\widehat{s},\widehat{\zeta})=-T(\widehat{s}_{\widehat{\zeta}},\widehat{\zeta})=-\nabla\varphi(\widehat{\zeta})=0 ∇ ζ ℓ ( s , ζ ) = − T ( s ζ , ζ ) = − ∇ φ ( ζ ) = 0 by the envelope identity (Lemma 18(b)). The rates hold with the constants built into the radii: ∥ s ^ − s ∗ ∥ ∞ ≤ ϵ n = C 5 log n / n \lVert\widehat{s}-s^*\rVert_{\infty}\leq\epsilon_n=C_5\sqrt{\log n/n} ∥ s − s ∗ ∥ ∞ ≤ ϵ n = C 5 log n / n and max { ∥ γ ^ − γ ∗ ∥ 2 , ∥ β ^ − β ∗ ∥ 2 } ≤ ∥ ζ ^ − ζ ∗ ∥ 2 ≤ ρ n = C γ β log n / n \max\{\lVert\widehat{\gamma}-\gamma^*\rVert_2,\lVert\widehat{\beta}-\beta^*\rVert_2\}\leq\lVert\widehat{\zeta}-\zeta^*\rVert_2\leq\rho_n=C_{\gamma\beta}\sqrt{\log n/n} max {∥ γ − γ ∗ ∥ 2 , ∥ β − β ∗ ∥ 2 } ≤ ∥ ζ − ζ ∗ ∥ 2 ≤ ρ n = C γ β log n / n .
Corollary 1 (Uniqueness and the MLE). On the event and under the assumptions of Theorem 2, the point ( s ^ , γ ^ , β ^ ) (\widehat{s},\widehat{\gamma},\widehat{\beta}) ( s , γ , β ) is the unique stationary point of ℓ \ell ℓ in K = B ∞ ( s ∗ , ϵ n ) × B ( γ ∗ , ρ n ) × B ( β ∗ , ρ n ) \mathcal{K}=B_{\infty}(s^*,\epsilon_n)\times B(\gamma^*,\rho_n)\times B(\beta^*,\rho_n) K = B ∞ ( s ∗ , ϵ n ) × B ( γ ∗ , ρ n ) × B ( β ∗ , ρ n ) intersected with the gauge slice s ∗ + 1 ⊥ s^*+1^\perp s ∗ + 1 ⊥ , and it maximizes ℓ \ell ℓ over that set; it is in this sense the MLE.
Proof. On 1 ⊥ × R p 1^\perp\times\mathbb{R}^p 1 ⊥ × R p the full Hessian of − ℓ -\ell − ℓ is the bordered matrix M = ( J B B ⊤ A ) M=\begin{pmatrix}J&B\\B^\top&A\end{pmatrix} M = ( J B ⊤ B A ) . Throughout K \mathcal{K} K , J ⪰ 1 2 n b 0 J\succeq\frac{1}{2}nb_0 J ⪰ 2 1 n b 0 on 1 ⊥ 1^\perp 1 ⊥ (Lemma 12) and M / J = Σ ⪰ 1 2 c Σ n 2 I p M/J=\Sigma\succeq\frac{1}{2}c_{\Sigma}n^2I_p M / J = Σ ⪰ 2 1 c Σ n 2 I p (Lemma 17), so by the Schur criterion (Fact 6) M ≻ 0 M\succ0 M ≻ 0 throughout K \mathcal{K} K . Hence − ℓ -\ell − ℓ is strictly convex on the convex set K ∩ [ ( s ∗ + 1 ⊥ ) × R p γ × R p β ] \mathcal{K}\cap[(s^*+1^\perp)\times\mathbb{R}^{p_\gamma}\times\mathbb{R}^{p_\beta}] K ∩ [( s ∗ + 1 ⊥ ) × R p γ × R p β ] and admits at most one stationary point there, which is its minimizer if interior. Theorem 2 exhibits an interior stationary point; it is therefore unique and maximizes ℓ \ell ℓ over K ∩ [ ( s ∗ + 1 ⊥ ) × R p γ × R p β ] \mathcal{K}\cap[(s^*+1^\perp)\times\mathbb{R}^{p_\gamma}\times\mathbb{R}^{p_\beta}] K ∩ [( s ∗ + 1 ⊥ ) × R p γ × R p β ] .
† Counts the number of skills mentioned; a composite of the per-skill dials, hence not counted as a separate recovery target.
‡ Deliberately irrelevant to the role; serves as a leakage check. Hence D = 16 D = 16 D = 16 relevant dials.
∗ Exactly 2 of the 4 niche dials are non-absent in any given letter.
Experiment Details on Interpretable Feature Recovery
This appendix collects the full experimental details for the synthetic cover-letter study of § 4 . Appendix C.1 describes the seeded corpus generation: the planted dials, their levels and sampling distributions, and the generation prompts. Appendix C.2 gives the formal definitions of the recovery metrics (Tracked, Stat.R., Sem.R., Attrib., LK), including the matching procedure between discovered features and planted dials. It also specifies the four baseline configurations (Direct LLM, Direct LLM–Multi-score, Local LLMs, Local LLMs–Multi-score), their prompts, and the API budget of each method. Appendix C.3 provides additional results such as the full list of surfaced features with their fitted parameters and diagnostics, and the judge-written rubric. Finally, Appendix C.4 details the paired-bootstrap protocol behind Figure 15 and Table 7 , and gathers additional recovery figures (dial-recovery heatmaps and the feature-merging threshold analysis).
Table 7. Paired comparisons over B = 50 B = 50 B = 50 bootstrap draws (with replacement) of the 200-letter corpus; every arm sees the same draws, and differences are taken within replicate. Each row reports the mean difference, its 95% percentile CI (*: excludes zero), and Ours better : the share of replicates in which our method wins in the metric’s preferred direction (Δ K \Delta K Δ K is descriptive and carries no share). The panels bracket discovery variance from opposite sides and are not comparable to one another. Both panels score our method at a reduced budget (average degree 25.3 against the designed 40); it leads on several metrics regardless. Design rationale, the conditional-fusion alternative, and budget details: Appendix C.4.2.
Dataset
Job ad. The real job ad was archived here. The job ad obtained after trimming boilerplate content is shown in Figure 11 .
Figure 11. The job advertisement used as the fixed evaluation context. Colored tags mark which requirement grounds which dial, using the same visual vocabulary as the letter template (Figure 12 ): the qualifications ground the education and numbered dials, and the preferred qualifications and responsibilities ground the six role-skill dials. Go is a mandatory requirement realized identically in every letter, not a dial. The remaining dials of Table 3 (school and employer prestige, prose fluency, and the four niche dials) have no anchor in the ad and are described in Appendix § 11 .§ 1 .
Example of template. Figure 12 shows one of the ten templates used for the cover letter generation.
Figure 12. Example cover-letter template. Roman text is fixed template prose; angle-bracketed slots are filled from the ground-truth dial vector. Skill coverage sets how many core units are capability sentences; paragraph count is constant. Fluency is imposed afterward by injecting 0/5/12 surface errors.
In what follows, we describe each dial in more detail: the levels it takes and how those levels are realized in the assembled letter. Everything is also gathered in Table 3 .
Table 3. The 18 engineered ground-truth dials of the cover-letter corpus.
Experience. The experience dial takes one of three values (3, 5 or 7 years) and is stated in the background clause as a tenure at the current employer. It is realized as a single digit and is one of the three numbers on the letter’s digit whitelist.
Education. The education dial takes three levels, BS < MS < PhD, rendered in the opening as a “{EDUCATION} in computer science from {SCHOOL}”. The three degrees are written as two-
word phrases: “Bachelor’s degree”, “Master’s degree”, “Doctoral degree”, so that the dial adds no length signal (the bare token “PhD” would make higher education correlate with shorter letters).
School prestige. The school dial fills the {SCHOOL} slot and takes three ascending selectivity bands: less selective (Cal State East Bay, University of Toledo, Kennesaw State University, Central Michigan University), moderately selective (strong general publics: Penn State, Ohio State, Arizona State, Virginia Tech), and highly selective (MIT, Stanford, UC Berkeley, Princeton). All names are real institutions, including the bottom band, whose intended reading is “an ordinary accredited school”. Band membership is anchored to a published selectivity classification.
Employer prestige. The employer dial fills the {EMPLOYER} slot and takes three ascending recognizability bands: an unknown-startup band of fictional, unrecognizable firms (Northlake, Bluegrove, Quarrypoint, Milldam), a medium band of real mid-recognizability companies (Zillow, Etsy, Expedia, Wayfair), and a big-tech band (Meta, Amazon, Microsoft, Apple). Unlike the school dial, the bottom band is fictional on purpose: invented names give a clean “no prestige prior” floor, whereas a real but obscure firm would carry its own idiosyncratic associations.
Programming languages. This numbered dial takes values { 2 , 5 , 9 } \{2,5,9\} { 2 , 5 , 9 } and is stated in the background clause as “writing Go alongside {NUMBER OF LANGUAGES} other languages.” Go is a fixed, mandatory slot (a minimum qualification); only the count varies. The count is realized as a single whitelisted digit in a clause present in every letter, so it grades a numbered attribute without introducing any length variation across its levels.
Production launches. This numbered dial takes values { 2 , 4 , 6 } \{2,4,6\} { 2 , 4 , 6 } and appears in the background clause as “tested, maintained, and launched {NUMBER OF LAUNCHES} production services.” Like the language count it is a self-contained count (never a duration), realized as a whitelisted digit in a fixed clause. It is assigned orthogonally to the experience dial so that it cannot re-enter the model as an experience proxy.
Skill coverage. The skill-coverage dial sets how many of the six role-skills the letter demonstrates, taking values k ∈ { 2 , 4 , 6 } k \in\{2,4,6\} k ∈ { 2 , 4 , 6 } . Its effect is to determine how many of the six core body units are capability sentences: exactly k k k core units render a present skill, and the remaining 6 − k 6-k 6 − k are fillers : role/domain-motivation sentences that never claim a skill, a niche, or communication ability. Because the body always contains six core units laid out in a fixed paragraph plan regardless of k k k , coverage changes the content of the body but not its length or paragraph count, so it cannot ride the verbosity bias.
Skill identity. Which skills are present (as opposed to how many) is a constrained set rather than an ordered dial. The pool has six letter-addressable items derived from the ad’s preferred qualifications and responsibilities: data-structures depth, Python, applied-AI experience, security assessments, vulnerability-triage automation, and agentic-tooling design. Each present skill is rendered as one concrete capability sentence drawn from the current template’s phrasing, with no numbers and no durations.
Niche dials. There are four niche dials: side project, open-source involvement, security certification, and capture-the-flag (CTF). Each is a three-level ordered factor: absent , a middle level, and a strong level (side project: toy < hard; open source: patches < maintainer; certification: entry (Security+) < advanced (OSCP); CTF: participated < strong). They are subject to a presence constraint: exactly two of the four are non-absent in every letter, so “number of extras mentioned” is held constant and cannot become a richness or length axis. The two active dials are realized, at their assigned level, as evidence clauses in a single dedicated niche paragraph placed at a fixed position after the body.
Fluency. The fluency dial controls surface prose quality and takes three levels anchored to an exact injected-error count: clean (0), mid (5), and degraded (12). Fluency is not a slot but a deterministic post-assembly pass that injects the target number of surface spelling errors into the finished letter; the word count is re-validated in band after injection, so the only thing that varies across fluency levels is the density of errors.
Nuisance factors. Two factors are balanced across the corpus but are not part of the ground truth and are not scored. Each letter is assembled from one of ten templates, which vary the greeting, opening, closing, and the exact wording of every skill, niche, and filler unit while holding the underlying dial content fixed. The applicant’s name, drawn from a fixed pool, appears only in the signature and carries no signal.
Dataset creation. The dataset comprises n = 200 n = 200 n = 200 cover letters. Of these, 184 form a balanced pool: every level of every dial appears an equal number of times, and the joint assignment of levels to letters is chosen as the most orthogonal of 256 independently drawn candidate designs (the candidate minimizing the largest non-structural pairwise ∣ r ∣ |r| ∣ r ∣ ). The remaining 16 are deliberate anchors: 8 polished-but-thin and 8 rough-gem profiles that anti-correlate the “prestige” dials with the substance dials, included so the judge sees the full range of the design, and excluded from the balance and orthogonality gates below.
Each letter is then assembled deterministically from one of 10 templates keyed to its levels. For example, experience, education, and school/employer prestige are stated in the opening; the k ∈ { 2 , 4 , 6 } k \in\{2, 4, 6\} k ∈ { 2 , 4 , 6 } present role-skills are rendered as capability sentences drawn from a pool (one pool per template), the identity of the k k k present skills being chosen by a greedy usage-balancing rule so that skill identity is orthogonal to coverage k k k ; production launches is stated in the background clause; the two active niche dials appear as evidence clauses; and prose fluency is imposed by injecting a target number of surface spelling errors (0/5/12).
On the balanced pool the realized design satisfies ∣ ρ ∣ ≤ 0.15 |\rho| \le0.15 ∣ ρ ∣ ≤ 0.15 over all non-structural dial pairs (max 0.146) and ∣ r ∣ < 0.22 |r| < 0.22 ∣ r ∣ < 0.22 between each dial and letter length (max 0.202); over the full corpus, where the anchors enter, these are 0.195 and 0.219. Letter length is not part of the design objective but it is verified after assembly. Generation is seeded and fully reproducible.
Methods Details
Metrics
Each method yields a set of discovered features that we will need to match to the dials.
Matching the discovered features to the dials. For each method, we perform a Hungarian matching assigning each dial to a distinct feature so as to maximize total cosine similarity in TEXT-EMBEDDING-3-LARGE’s embedding space. The pairs it rejects (features and dials sharing domain vocabulary but, in the assignment’s judgment, not describing the same construct) form an empirical background of unrelated pairs, and we define a threshold η \eta η as the 95th percentile of their alignment (η = 0.406 \eta= 0.406 η = 0.406 ): the level that unrelated pairs almost never attain. A matched pair clearing η \eta η is therefore semantically exceptional relative to background similarity, not merely the least-bad assignment; cosine similarities carry no absolute scale, so the reference distribution, rather than a fixed constant, is what gives the threshold meaning.
Having defined this threshold, we map a method’s set of discovered features to the dial set by performing semantic Hungarian matching, but only keeping matches whose cosine similarity is greater than η \eta η .
In what follows, we also say that a feature k k k tracks a dial d d d if the absolute Pearson correlation ∣ r k , d ∣ |r_{k,d}| ∣ r k , d ∣ between the two is greater than a threshold τ = 0.4 \tau= 0.4 τ = 0.4 fixed a priori and well above the permutation floor (≈ 0.15 \approx0.15 ≈ 0.15 under letter-shuffled nulls; results are stable for τ ∈ [ 0.3 , 0.5 ] \tau\in[0.3, 0.5] τ ∈ [ 0.3 , 0.5 ] ). The metrics mentioned in Section 4 are mathematically defined as follows:
Statistical recovery. For each dial, does some feature track it? We report the fraction of dials with max k ∣ r k , d ∣ > τ \max_k |r_{k,d}| > \tau max k ∣ r k , d ∣ > τ (Tracked), and the mean of max k ∣ r k , d ∣ \max_k |r_{k,d}| max k ∣ r k , d ∣ over all D D D dials, misses included (Stat.R.).
Semantic recovery. For each dial, is there a feature that is both about it and tracks it? Let M m M_m M m assign features to dials by Hungarian matching on semantic alignment, and c m , d c_{m,d} c m , d the alignment of dial d d d with its match M m ( d ) M_m(d) M m ( d ) :
Sem.R. ( m ) = 1 D ∑ d = 1 D 1 { c m , d ≥ η } 1 { ∣ r M m ( d ) , d ∣ > τ } . (54) \operatorname{Sem.R.}(m) = \frac{1}{D}\sum_{d=1}^{D} \mathbf{1}\{c_{m,d}\ge\eta\}\mathbf{1}\{|r_{M_m(d),d}|>\tau\}. \tag*{(54)} Sem.R. ( m ) = D 1 d = 1 ∑ D 1 { c m , d ≥ η } 1 { ∣ r M m ( d ) , d ∣ > τ } . (54)
Attributable features. Analogously, what fraction of a method’s K m K_m K m discovered features measure a single dial? A feature is attributable if it tracks some dial and that dial dominates its attribution, i.e. carries a majority of its squared correlation, π k = max d r k , d 2 / ∑ d r k , d 2 > 1 2 \pi_k = \max_d r_{k,d}^2/\sum_d r_{k,d}^2 > \frac{1}{2} π k = max d r k , d 2 / ∑ d r k , d 2 > 2 1 :
Attrib. ( m ) = 1 K m ∑ k = 1 K m 1 { max d ∣ r k , d ∣ > τ } 1 { π k > 1 2 } . (55) \operatorname{Attrib.}(m) = \frac{1}{K_m}\sum_{k=1}^{K_m}\mathbf{1}\{\max_d|r_{k,d}|>\tau\}\mathbf{1}\left\{\pi_k>\frac{1}{2}\right\}. \tag*{(55)} Attrib. ( m ) = K m 1 k = 1 ∑ K m 1 { d max ∣ r k , d ∣ > τ } 1 { π k > 2 1 } . (55)
Leakage. Do scores leak across dials? Since dials are near-orthogonal by construction, off-target correlations of a valid feature set should be small. With A m A_m A m the attributable features and k m ( d ) = arg max k ∣ r k , d ∣ k_m(d)=\operatorname*{arg\,max}_k|r_{k,d}| k m ( d ) = arg max k ∣ r k , d ∣ the tracker of dial d d d ,
LK ( m ) = 1 D ∑ d = 1 D mean k ∈ A m ∖ { k m ( d ) } ∣ r k , d ∣ (56) \operatorname{LK}(m) = \frac{1}{D}\sum_{d=1}^{D}\underset{k\in A_m\setminus\{k_m(d)\}}{\operatorname{mean}} |r_{k,d}| \tag*{(56)} LK ( m ) = D 1 d = 1 ∑ D k ∈ A m ∖ { k m ( d )} mean ∣ r k , d ∣ (56)
(lower is better). Restricting to attributable features prevents noise features from diluting the average.
Methods Description
Our approach. For our method, the comparison graph on the n = 200 n = 200 n = 200 cover letters is a union of d = 40 d = 40 d = 40 balanced subgraphs (called matchings , each a perfect matching in which every item is compared exactly once). The number d = 40 d = 40 d = 40 was chosen to guarantee high precision.
Prompt 4 was used for the pairwise comparisons (GPT-5.4 MINI), and Prompt 5 was used for the feature taxonomy (GPT-5.4 (HIGH REASONING)). The cost was estimated using the API logs.
Other methods. Other methods feed the entirety of the dataset (Direct LLM)/a sample of it (Local LLMs) to the judge (GPT-5.4 (HIGH REASONING)) and have it draw up a rubric tailored to the samples seen (Prompt 6). This rubric is then used to have GPT-5.4 MINI score each cover letter pointwise on each feature (Prompt 8).
For Local LLMs, only 40 letters of the dataset are given to the judge during the feature elicitation step, enabling the LLM to focus its attention on a smaller sample of data. To ensure that each cover letter is seen at least once, we repeat the elicitation with E = 27 E = 27 E = 27 different samples, which is the average number of times one would have to repeat the step so that each cover letter is seen at least once. We then consider the union of the features, performed with GPT-5.4 (HIGH REASONING) and Prompt 7 (see Remark 7 for why we used LLM-based taxonomy instead of semantic hierarchical clustering).
To match the cost of our approach, the scoring is repeated 20 times for Direct LLM–Multi-score, and 6 times for Local LLMs–Multi-score. Each scoring is performed by slightly modifying the prompt, changing the instruction variant (Prompt 9) and permuting the features inside the rubric.
PROMPTS
Prompt 4: Comparative method (ours) – pairwise comparison prompt.
You are helping build an interpretable scoring framework by examining how job applicants' cover letters differ. The framework works by collecting many pairwise comparisons: for each pair, the ways the two letters differ are articulated in plain terms, and aggregated across many pairs these reveal the distinct dimensions along which the applications vary. There is no fixed rubric – the dimensions are recovered from comparisons like this one, not specified in advance.
The letters below are applications for the same software-engineering role. The full job posting they respond to is:
=== JOB POSTING ===
⟨full job posting⟩
=== END JOB POSTING ===
Your task. You will be shown two cover letters, Letter A and Letter B, both written for the role above. This pair is one comparison in a larger collection.
Identify the dimensions on which the two letters meaningfully differ as applications for this role. Report the differences that genuinely stand out to you on reading this pair — bring up whatever you find salient, and do not work from a predetermined list. Favor real, noticeable differences over minor ones, but do not force the list to be short: report as many as actually distinguish these two letters.
For each dimension, first name the concrete, specific feature you actually saw in the letters — a short phrase pointing to the specific evidence. Then, grounded in that evidence, name the dimension, describe in one sentence what it captures, and indicate which letter shows more of it.
Judge each dimension on its own terms, from its own evidence. Applications have uneven profiles: it is entirely normal for one letter to be stronger on some dimensions and the other letter stronger on others, so let the letter that shows more of each dimension fall wherever the evidence for that dimension points.
Out of scope – do not treat these as dimensions of difference:
the applicant's name or any personal identifier;
the closing sentence about work authorization / visa sponsorship, which is identical in every letter;
length in itself.
How to report. List each dimension as exactly one item – do not combine dimensions with "and", "/", "&", or a comma; split related dimensions into separate items. For the evidence field, give a short phrase naming the concrete feature you saw. For the dimension field, give a concise general noun phrase naming the dimension (the kind of short label you might use as a rubric column). For the description field, give one sentence saying what that dimension captures. Indicate which letter shows more of it.
Output only JSON of the form:
{"differences": [{"evidence": "<short phrase>", "dimension": "<concise noun phrase>", "description": "<one sentence>", "winner": "A" | "B"}, ...]}
Prompt 5: Comparative method (ours) – feature taxonomy.
You are organizing a flat list of evaluation dimensions into a set of canonical features.
These dimensions were produced by judges comparing job applicants' cover letters for the same software-engineering role. Each judge, on each comparison, named the dimensions on which two letters differed. Aggregated across many comparisons, the same underlying construct was named in many slightly different ways. Your job is to group the different phrasings of the SAME underlying construct together.
Below is the full list. Each entry has an INDEX in square brackets, a short NAME, a
one-sentence DESCRIPTION of what it captures, and CITATIONS (how many times it was named —
for your awareness only).
⟨indexed dimension list⟩
Instructions:
- Group entries that denote the SAME underlying evaluative construct, judging by MEANING,
not by surface wording. Different words for the same idea belong in one group; the same
word used for different ideas belongs in different groups.
- Do NOT force everything into a small number of groups, but do NOT split one construct
into its facets either. A dimension that does not clearly share a construct with any other
does NOT become its own feature: place all such genuinely-apart, idiosyncratic dimensions
together in a single catch-all group whose canonical name is exactly "Other". Do not create
many single-member groups, and do not absorb a genuinely-apart dimension into a
loosely-related group to tidy up — send it to "Other".
- Do NOT merge two constructs just because they often co-occur or because strong applicants
tend to score well on both. Group only by shared MEANING, never by how applicants might
score.
- Citations are context only. Do not merge a frequently-named dimension into a rarely-named
one or vice versa on the basis of counts; counts do not determine grouping.
For each group, give:
- a canonical NAME (a concise noun phrase suitable as a rubric column; use exactly "Other"
for the catch-all group),
- a one-sentence DEFINITION of the construct the group captures,
- the list of members, each as its INDEX and its NAME copied verbatim from that index.
Rules on membership (these will be checked programmatically):
- Every input index must appear in exactly one group, exactly once. Place all of them; drop
none; duplicate none.
- Do not invent indices that are not in the input.
- Each member's "name" must be the exact input NAME at that index — this is a cross-check,
so copy it from the bracketed entry, do not paraphrase.
Output only JSON:
{"features": [{"name": "<canonical noun phrase>", "definition": "<one sentence>",
"members": [{"index": <int>, "name": "<verbatim input name at that index>"}, ...]}, ...]}
Prompt 6: Pointwise baselines (Direct LLM/Local LLMs) – rubric discovery prompt.
You are helping build an interpretable scoring framework by examining how job applicants'
cover letters differ. There is no fixed rubric – the dimensions are recovered from what
actually distinguishes these letters, not specified in advance.
=== JOB POSTING ===
⟨full job posting⟩
=== END JOB POSTING ===
Below is a sample of cover letters submitted for this role, selected to span the full range
of applicant quality.
=== COVER LETTERS ===
⟨cover letters⟩
Reading across them, attend to the ways the letters differ from one another – the axes
along which you can see one letter showing more, less, or something different from another.
Identify the dimensions on which these letters meaningfully differ as applications for this
role. Report whatever genuinely stands out to you; do not work from a predetermined list,
and do not force the list to be short – report as many dimensions as actually distinguish
these letters.
A dimension may be something a letter shows more or less of along a range, or something a
letter either shows or does not.
Keep genuinely distinct dimensions separate even when they are related or tend to co-occur –
do not merge two dimensions because strong letters tend to show both. List each dimension
as exactly one item – do not combine dimensions with "and", "/", "&", or a comma; split related dimensions into separate items.
For each dimension, first name the concrete, specific feature you actually saw in the letters – a short phrase pointing to the evidence. Then, grounded in that evidence, name the dimension and describe in one sentence what it captures. Then write a short assessment guideline a future reader will use to score a new letter on this dimension on a scale you need to specify: say what to look for, and describe what distinguishes each score of the scale you come up with.
Out of scope – do not treat these as dimensions of difference:
the applicant's name or any personal identifier;
the closing sentence about work authorization / visa sponsorship, which is identical in every letter;
length in itself.
Output only JSON: {"features": [{"evidence": "<short phrase>", "name": "<concise noun phrase>", "description": "<one sentence>", "levels": [{"score": 1, "descriptor": "<a letter at the weakest point of your scale>"}, {"score": 2, "descriptor": "<...>"}, ... up to the top score of the scale you chose]}, ...]} – for each dimension choose your own scale (score 1 = weakest, up to a top point N you decide) and give one descriptor per score point, from 1 to N.
Prompt 7: Local LLMs – feature fusion prompt.
You are organizing a flat list of evaluation dimensions into a set of canonical features.
These dimensions were produced by independent rubric elicitations over job applicants' cover letters for the same software-engineering role. Each elicitation read its own random sample of letters and, working from scratch, named the dimensions on which those letters meaningfully differed. Aggregated across many independent elicitations, the same underlying construct was named in many slightly different ways. Your job is to group the different phrasings of the SAME underlying construct together.
Below is the full list. Each entry has an INDEX in square brackets, a short NAME, a one-sentence DESCRIPTION of what it captures, and RUN (which elicitation produced it – for your awareness only).
⟨indexed dimension list⟩
Instructions:
Group entries that denote the SAME underlying evaluative construct, judging by MEANING, not by surface wording. Different words for the same idea belong in one group; the same word used for different ideas belongs in different groups.
Do NOT force everything into a small number of groups, but do NOT split one construct into its facets either. A dimension that does not clearly share a construct with any other does NOT become its own feature: place all such genuinely-apart, idiosyncratic dimensions together in a single catch-all group whose canonical name is exactly "Other". Do not create many single-member groups, and do not absorb a genuinely-apart dimension into a loosely-related group to tidy up – send it to "Other".
Do NOT merge two constructs just because they often co-occur or because strong applicants tend to score well on both. Group only by shared MEANING, never by how applicants might score.
Run labels are context only. Do not merge a dimension named by many runs into one named by few, or vice versa, on the basis of how often it recurs; recurrence does not determine grouping. Two entries from the SAME run were named as distinct dimensions by that run, so they are unlikely to be the same construct – but judge by MEANING, and merge them if they plainly are.
For each group, give:
a canonical NAME (a concise noun phrase suitable as a rubric column; use exactly "Other" for the catch-all group),
a one-sentence DEFINITION of the construct the group captures,
the list of members, each as its INDEX and its NAME copied verbatim from that index.
Rules on membership (these will be checked programmatically):
Every input index must appear in exactly one group, exactly once. Place all of them; drop none; duplicate none.
Do not invent indices that are not in the input.
Each member's "name" must be the exact input NAME at that index – this is a cross-check, so copy it from the bracketed entry, do not paraphrase.
Output only JSON: {"features": [{"name": "<canonical noun phrase>", "definition": "<one sentence>", "members": [{"index": <int>, "name": "<verbatim input name at that index>"}, ...]}, ...]}
Prompt 8: Pointwise baselines (Direct LLM/Local LLMs) – scoring prompt template.
You are scoring a cover letter against a fixed rubric, for the software-engineering role below. === JOB POSTING === ⟨full job posting⟩ === END JOB POSTING === === COVER LETTER === ⟨letter⟩ ⟨instruction variant⟩ A dimension you cannot fully assess from this letter should still receive your best estimate, not a default middle score. Score each dimension independently, and do not let the score on one dimension affect the scores on others. Applications have uneven profiles: it is possible for a letter to score low on some dimensions and high on others. === RUBRIC DIMENSIONS (score in the order listed; each lists the descriptors for its own score scale) === ⟨rubric, in scoring order⟩ Out of scope – do not let these affect any score:
the applicant's name or any personal identifier;
the closing sentence about work authorization / visa sponsorship, which is identical in every letter;
length in itself. ⟨output specification⟩
Prompt 9: Pointwise scoring – the four reasoning-path instruction variants.
[direct]
Score the letter on each rubric dimension below independently. Each dimension lists a descriptor for every point of its own score scale; assign the integer score whose descriptor best matches the letter on that dimension. Judge each dimension on its own.
[evidence_first]
For each rubric dimension below, briefly note what in the letter bears on that dimension, then assign the integer score, from that dimension's own listed scale, whose descriptor best matches the letter on that dimension. Treat each dimension independently.
[standard_referenced]
For each rubric dimension below, compare the letter against the score descriptors listed for that dimension, then assign the integer score whose descriptor it matches most closely. Judge each dimension on its own.
[independent]
Go through the rubric dimensions one at a time, in the order listed. For each dimension, consider only what the letter shows on that specific dimension and assign the integer score, from that dimension's own listed scale, whose descriptor best matches; do not form an overall impression of the letter and do not let the other dimensions influence the score.
Additional Results
Model Calibration
In this subsection, we study whether or not the statistical model is well-calibrated by computing its Expected Calibration Error (ECE) and its Brier skill score against various baselines.
The fitted model assigns to each (comparison, feature) cell two probability forecasts of observable binary events: the citation gate g ^ p , k = σ ( f ( Δ ^ p , k ) + γ ^ k ) \hat{g}_{p,k} = \sigma(f(\widehat{\Delta}_{p,k}) + \hat{\gamma}_{k}) g ^ p , k = σ ( f ( Δ p , k ) + γ ^ k ) , the probability that feature k k k is cited on comparison p p p , defined on all 84,000 cells (K = 21 K = 21 K = 21 features), and the direction gate d ^ p , k = σ ( Δ ^ p , k + β ^ ) \hat{d}_{p,k} = \sigma(\widehat{\Delta}_{p,k} + \hat{\beta}) d ^ p , k = σ ( Δ p , k + β ^ ) , the probability that the first-shown letter is favored given citation, defined on the 29,030 cited cells. We score both as forecasts, out of fold: 5-fold cross-validation over comparisons, so that every cell is predicted by parameters estimated without it, with 95% bootstrap intervals over comparisons throughout; in-sample counterparts quantify the optimism of the fit.
Figure 13 reports reliability: cells are grouped into 10 equal-width bins of predicted probability, the observed event rate in each bin is compared against the diagonal of perfect calibration, and the expected calibration error (ECE) is the occupancy-weighted mean absolute gap. The direction gate is calibrated essentially everywhere: its out-of-fold ECE is 0.010 [0.008, 0.014] (in sample: 0.019), and its predictions populate the full unit interval including the extreme bins, so the diagnostic is exercised where it is most demanding. The citation gate’s out-of-fold ECE is 0.046 [0.044, 0.049] against 0.030 in sample, and its reliability curve is flatter than the diagonal (observed rates exceed predictions in the low bins and fall short in the high ones) so the fitted gate is too extreme at both ends, consistent with the amplitude of f f f slightly overshooting out of fold rather than the gap dependence being absent.
Figure 14 reports discrimination through the Brier skill score, 1 − Brier ( m o d e l ) / Brier ( r e f e r e n c e ) 1 - \operatorname{Brier}(\mathrm{model})/\operatorname{Brier}(\mathrm{reference}) 1 − Brier ( model ) / Brier ( reference ) , against two references estimated on the training folds: chance (p = 1 / 2 p = 1/2 p = 1/2 , Brier 0.25) and the empirical rate (the per-feature citation rate for the citation gate, the per-feature rate of favoring the first-shown letter among cited cells for the direction gate). The second reference is the demanding one. For the citation gate it is the maximum-likelihood fit of the gap-independent submodel f = 0 f = 0 f = 0 , whose gate σ ( γ k ) \sigma(\gamma_k) σ ( γ k ) is constant within each feature, so skill over it is a direct out-of-fold test of whether the merit gap carries information about citation. It does: the gate scores 0.069 [0.063, 0.076] against this submodel, rejecting f = 0 f = 0 f = 0 , down from 0.168 in sample; its 0.266 [0.260, 0.272] against chance partly rewards knowing the citation base rate rather than per-cell discrimination. The direction gate scores 0.548 [0.538, 0.557] against chance and 0.545 [0.536, 0.554] against the empirical rate (the two coincide because slot balance keeps the first-shown rates near 1/2) and retains most of its in-sample skill (0.623). The channel that carries the scores is thus both calibrated and strongly predictive out of sample; the gap effect on citation is significant but second-order, estimated with an amplitude that overshoots out of fold.
Discovered Features
Table 4 and Table 5 (PDF p. 70) report the features discovered by each method described in Table 1 . More precisely, Table 4 presents each feature with its reliability coefficients and its fitted γ \gamma γ coefficients, Table 5 (PDF p. 70) shows the detailed rubric created by GPT-5.4 (HIGH REASONING).
Table 5 (PDF p. 70). The rubric elicited by Direct LLM. Descriptions and the per-score guidelines are reproduced verbatim as the judge produced them; the open elicitation lets the judge choose its own scale, and it selected a 1–4 scale for every feature.
As we can see in Table 4 , our approach yields fine-grained features, as the judge distinguishes the existence of side projects (“Side-project initiative”) from their magnitude (“Side-project ambition”, which almost passed the reliability thresholds). Moreover, the reliability diagnostics correctly filter features such as “Narrative framing and tone”, which is a well-identified feature (κ = 4.9 \kappa= 4.9 κ = 4.9 ) but whose citations are inconsistent with the estimated scores (ρ = 0.72 < ρ min \rho= 0.72 < \rho_{\min} ρ = 0.72 < ρ m i n ): a sign of a “boilerplate” feature that the judge cites independently from the items it is comparing. This is partly captured by the γ \gamma γ coefficient. For example, “Employer prestige” has a smaller γ \gamma γ than “Narrative framing and tone” but is reliable: a sign that the former is discriminative while the latter is a boilerplate feature.
We also notice in Table 5 (PDF p. 70) that the rubric correctly captures some of the dials’ levels. For example, the judge correctly identifies that for the Education dial, BS < < < MS < < < PhD: the rubric is correctly tailored to the samples that the judge sees.
Run-variance analysis
Per-method run-variance bands
Table 6 repeats the point estimates of Table 1 together with each method’s run-variance band. The gray bands are per-method run-variance summaries; each arm resamples the unit its own pipeline randomizes over, so band widths reflect the design as much as the method and carry no cross-method meaning.
Table 6. Per-method point estimates with run-variance bands. Tracked, Sem.R., Attrib. in %; Stat.R. and LK are correlations; bands are in the same units as the column. Below each method’s point estimates, the gray row gives that method’s own run-variance band ([2.5, 97.5] percentiles across replicates), obtained by resampling the unit named at the left of the row. These bands are not confidence intervals for the point estimates above them – the point estimates come from the single full-data run, the bands from re-runs on resampled data, so a point estimate may fall outside its band – and they must not be compared across methods , since the arms resample different units. Paired, comparable contrasts: Table 7 .
Ours resamples the 40 matchings that make up the comparison graph, and then re-runs the taxonomy on every replicate (the judging of the pairs is not rerun as it is too costly); the K K K band covers the reliable count after the κ / ρ \kappa/\rho κ / ρ filter, from 28.9 discovered features on average. Direct LLM resamples the letters and re-elicits its rubric per replicate. Direct LLM–Multi-score shares Direct LLM’s rubric but its discovery step is deterministic, so its band reflects letter resampling only; the degenerate entries (K K K , Tracked, Sem.R.) are exact. Local LLMs redraws which E = 27 E=27 E = 27 elicitations are pooled, so its band carries fusion variance but holds the corpus fixed – unlike Direct LLM, whose band varies both. Local LLMs–Multi-score is instead the direct counterpart of Direct LLM–Multi-score: a single fusion with deterministic discovery, scored with 6 repeated passes averaged before resampling, so its band too reflects letter resampling only, with K K K , Tracked and Sem.R. again exact.
Design of the Paired Comparisons
Panel (b) of Figure 15 cannot mirror panel (a). Local LLMs discovery replicates are indexed by the choice of E = 27 E=27 E = 27 pooled elicitations rather than by letters, so they admit no pairing with corpus draws, and re-eliciting on each resampled corpus would be prohibitively costly. In (a) both arms re-run discovery on every replicate, so the intervals include discovery variance on both sides; in (b) our taxonomy is held at the canonical fit while Local LLMs’ fusion is varied over all 50 draws, the 50 × 50 50 \times50 50 × 50 (fusion, corpus) differences being pooled; the intervals therefore carry corpus variance for both arms and fusion variance for Local LLMs, but no re-discovery for ours.
Conditioning instead on the canonical fusion (the first E = 27 E=27 E = 27 elicitations) gives uniformly larger, all-significant gaps (Tracked +0.177, Stat.R. +0.080, Sem.R. +0.177, Attrib. +0.466, LK −0.016), that fusion being among the weaker draws; the pooled figures in Table 7 are the conservative choice . Across fusions, the mean differences span Tracked [+0.052, +0.177], Stat.R. [−0.003, +0.093], Sem.R. [−0.010, +0.240], Attrib. [+0.362, +0.505], LK [−0.023, −0.007]. Resampling the corpus also reduces our method’s comparison budget: a replicate retains only judged pairs with both endpoints drawn (1,603 of 4,000 on average, consistent with the ≈ 0.632 2 \approx0.632^{2} ≈ 0.63 2 2 both-endpoints survival rate of an n n n -out-of-n n n bootstrap), giving an average degree of 25.3 against the designed 40, on which our method reports ≈ 16 \approx16 ≈ 16 features on average rather than 21. The advantage does not rest on the larger budget: at degree 25 the comparative arm costs ≈ \approx ≈ $2.24, against $3.45 at degree 40 and $2.35 for Local LLMs.
Dials Recovered
Figure 16 and Figure 17 compare the dial-recovery heatmap between our method and Direct LLM. As we can see, our method yields a strong diagonal recovery with only a few discovered features tracking no known dial, while Direct LLM clearly misses 3 dials and has dials tracked by several discovered features.
Figure 16. Dial-recovery heatmap yielded by the features discovered by our method. The rows are the discovered features and the columns are the dials.
Figure 17. Dial-recovery heatmap yielded by the features discovered by Direct LLM. The rows are the discovered features and the columns the dials. The scores leak more between one another and the dial recovery is less clean than in Figure 16 as some dials are not recovered.
Remark 7. As we can see in Figure 18 , after E = 27 E = 27 E = 27 rubric elicitations, recovering every dial demands a high acceptance threshold, which leaves an unusable number of near-duplicate features.
Figure 18. Left: probability that a dial is recovered as a function of the acceptance threshold used to merge similar features across the E = 27 E = 27 E = 27 elicitations of Local LLMs (dials are matched to the merged features by thresholded Hungarian assignment at the calibrated η \eta η ). Right: number of features surviving the merge, against the number of pooled elicitations, one curve per acceptance threshold. At E = 27 E = 27 E = 27 , recovering every dial demands a high threshold, which leaves an unusable number of near-duplicate features, hence the need for an LLM taxonomy.
It is the same problem that led us to use LLM taxonomy instead of semantic clustering in our own standardization step: the judge naturally merges duplicate features and excludes idiosyncratic ones.
Experiment Details on the ChangeMyView Dataset
Dataset
Samples Used
Within each discussion tree, Tan et al. [19 ] matched each Δ \Delta Δ -winning argument to the non-winning argument from the same tree whose opening reply is most topically similar, yielding a balanced task (3,456 training and 807 held-out pairs of Δ \Delta Δ -winning/non Δ \Delta Δ -winning answers) in which topic is held fixed and the goal is to identify the persuasive member of each pair.
To mitigate length bias, we restricted to pairs whose two root replies both fall within [20, 1200] words, leaving 3,411 pairs for the training set and 800 eligible held-out pairs. For our study, we used a sub-sample of 400 pairs from the training set.
Memorization Probe
Because the CMV corpus predates the judge’s training cutoff, a skeptical reading of Table 2 is that the judge recalls which replies earned a Δ \Delta Δ rather than inferring it. To test this, we use 1,600 exchanges in four cells of 400 ({ Δ , no- Δ } × { train , held-out } \{\Delta, \text{no-}\Delta\} \times\{\text{train}, \text{held-out}\} { Δ , no- Δ } × { train , held-out } , base rate 50% by construction). Each exchange is judged under two framings on identical de-identified text (Prompts 10 and 11): an unnamed framing (P1) asking whether the reply changed the poster’s view, with platform identifiers scrubbed, and a named framing (P2) that presents the same scrubbed text but identifies the source as r/ChangeMyView and explains the Δ \Delta Δ mechanic. A memorized outcome is easier to retrieve when the source is named, so the lift P2 − P1 on constant text is the recall signal.
Naming the source buys the judge nothing: the lift is −0.1 pp with a 95% confidence interval of [−2.4, +2.3] pp (Table 8 ). Finally, the residual above-chance accuracy of the unnamed framing (53.6%, CI [51.2, 56.1]) is not recall-shaped: it is driven by a strong asymmetry across the outcome split (69.4% on no-Δ \Delta Δ exchanges versus 37.9% on Δ \Delta Δ exchanges under P1), a systematic prior toward predicting “no change”. Consistently, naming the source (P2) does not sharpen either cell but shifts the threshold, raising Δ \Delta Δ -cell accuracy to 43–46% while lowering no-Δ \Delta Δ accuracy to 62–64%.
Table 8. Memorization probe on 1,600 CMV exchanges pinned to the data of Table 2 (judge: GPT-5.4 MINI; chance = 50%). The lift is assessed by a paired McNemar test on the same exchanges (b / c = 185 / 184 b/c = 185/184 b / c = 185/184 , p = 1.000 p = 1.000 p = 1.000 ).
We conclude that the predictive result of Table 2 is not explained by outcome recall: on the very exchanges it is measured on, the judge shows no recall signal under source naming.
Prompt 10: Contamination probe – shared task framing (P1; unnamed).
You are shown one online discussion. It begins with a person's stated
VIEW (a position they hold and are willing to reconsider), followed by
ONE reply from someone arguing against that view. Your job: decide
whether that reply actually changed the original person's view. Answer
strictly yes or no.
=== VIEW ===
⟨OP view, de-identified⟩
=== REPLY ===
⟨root reply, de-identified⟩
⟨output specification⟩
Prompt 11: Contamination probe – named-source preamble (P2).
This is a post from the r/ChangeMyView (CMV) forum on Reddit. There, an original poster states a view and awards a "delta" (Δ \Delta Δ ) to any reply that succeeds in changing their view.
[followed by the shared task framing of Prompt 10, on identical de-identified text]
Methods Details
This subsection gives all the details of each method.
Shared evaluation protocol. All arms score the same 2,400 exchanges (each an (OP view, root reply) pair), which come from 1,200 of Tan’s matched pairs, each contributing the delta-winning reply and its matched loser: 400 pairs (800 exchanges) drawn from the training split and all 800 eligible held-out pairs (1,600 exchanges), with replies of 20–1,200 words. Given a feature set’s per-exchange scores, the paired task is an ℓ 1 \ell_1 ℓ 1 logistic regression on the within-pair score difference sign ⋅ ( s w i n n e r − s l o s e r ) \operatorname{sign}\cdot(s_{\mathrm{winner}}-s_{\mathrm{loser}}) sign ⋅ ( s winner − s loser ) with no intercept, the sign dealt from a balanced random sign per pair so chance is exactly 50%. The penalty λ \lambda λ in the ℓ 1 \ell_1 ℓ 1 model is selected by 5-fold GROUPKFOLD cross-validation grouped by original poster over 13 log-spaced values λ ∈ { 10 − 3 , … , 10 3 } \lambda\in\{10^{-3},\ldots,10^3\} λ ∈ { 1 0 − 3 , … , 1 0 3 } , fitted on the 400 training pairs and evaluated once on the 800 held-out pairs. Selected values are λ = 10 \lambda= 10 λ = 10 (D + BT), 10 (D + PW), 31.6 (Dr _r r + BT), 31.6 (Dr _r r + PW), 31.6 (P + BT), 3.2 (P + PW), 3.2 (S + PW), 31.6 (Tan), 31.6 (Emb.) and 31.6 (#words).
D: feature discovery (shared by D + BT, D + PW, Dr _r r + BT, Dr _r r + PW). Features are mined from 150 training pairs (300 exchanges) drawn from the training split only. The 300 exchanges are arranged in a 40-regular comparison graph built as a union of 40 random perfect matchings, giving 6,000 cross-topic comparisons performed by GPT-5.4 MINI using Prompt 12; a thread’s own winner–loser edge is forbidden, so the judge never compares two replies to the same post. For each comparison the judge names, in free text, the dimensions on which the two exchanges differ in ways bearing on persuasion, with an evidence phrase and which side shows more of it (7.65 dimensions named per comparison; 45,894 citations over 17,724 distinct phrases). Phrases cited at least 10 times are grouped into canonical features by one GPT-5.4 (HIGH REASONING) call (Prompt 13), yielding 45 features. Excluding OP-side features (they describe the shared post and so cancel within a matched pair), the catch-all “Other”, and one singleton label leaves the 29 challenger-side features used in the (D) feature set. Fitting the statistical model of § 3 and keeping features having ρ ≥ ρ min : = 0.75 \rho\ge\rho_{\min} := 0.75 ρ ≥ ρ m i n := 0.75 and κ ≤ κ max : = 50 \kappa\le\kappa_{\max} := 50 κ ≤ κ m a x := 50 yields the feature set Dr _r r of 16 features. Feature set: Table 10 .
P: prior-elicited rubric (shared by P + BT and P + PW). A 16-feature set is elicited from the model’s prior, with no data. GPT-5.4 (HIGH REASONING) is given a two-sentence setting (that people state views online and others reply to change them) and asked to name dimensions for assessing persuasiveness, each a matter of degree and distinct from the others, with a name and a one-sentence definition (Prompt 15). Asked instead to choose the count itself, the model produces 15.65 ± 1.42 15.65 \pm1.42 15.65 ± 1.42 features over two independent runs of 10 draws, so for a canonical run, we asked it to produce 16 dimensions, which is hence not a binding constraint and matches the size of the Dr _r r set. Feature set: Table 11 .
Table 11. Prior-elicited features (P). The 16 features named by GPT-5.4 (HIGH REASONING) for assessing the persuasiveness of a reply, with no data shown (Prompt 15), in the order produced.
S: sample-elicited features (S + PW). A feature set is elicited from the same model after it has seen data. In a single call, GPT-5.4 (HIGH REASONING) reads 752 pairs (1,504 replies, about 860k tokens of input), each presented as an OP view followed by two replies written independently to it, and is asked to name the dimensions along which the replies differ as attempts to change the poster’s view, giving for each a short evidence phrase, a name and a one-sentence definition (Prompt 16). Features must be properties of a reply rather than of the poster’s view, so that they can vary within a matched pair. The number of features is left to the model: with the same open-count instruction but no data, it names about 16 (see P); with data, it names 42. The exchanges keep one pair per distinct original poster, cover the range of reply lengths, and are drawn from both the training and the held-out pairs, without their labels. Feature set: Table 12 .
Table 12. Sample-elicited features (S). The 42 features named by GPT-5.4 (HIGH REASONING) after reading a sample of 752 exchange pairs, without their outcomes.
D + BT and P + BT: pairwise Bradley–Terry scoring. Each feature is scored by its own Bradley–Terry model on a frozen comparison graph (Prompt 14). The 800 training exchanges form a 16-regular anchor graph (6,400 comparisons); each of the 1,600 test exchanges is then compared against 16 anchors (25,600 comparisons), for 32,000 judge calls. Every call presents two exchanges with the full feature list and forces a winner on each feature, with a short evidence phrase. Anchor scores are fitted per feature by plain Bradley–Terry with a position-bias term β \beta β (ridge 10 − 3 10^{-3} 1 0 − 3 ); each test exchange is then scored by a one-dimensional convex solve against the frozen anchor scores, so test items never influence each other. P + BT uses a graph verified identical to D + BT’s, so the two arms differ only in the features named in the prompt.
D + PW, Dᵣ + PW and P + PW: pointwise scoring. The same features are instead rated absolutely (Prompt 17): one call per exchange rates it 0–10 on every feature so the feature set costs 2,400 calls rather than 32,000. The prompt is a deliberate near-copy of the pairwise one; two differences are unavoidable and consequential. A scale must be defined, anchored on the population of replies, because ratings must be comparable across exchanges rated in separate calls — pairwise never needs this, which is precisely the property this arm gives up; and the prompt must push against central tendency, since without it ratings pile up at 5–7 and the arm loses to compression rather than to anything substantive. Because a pointwise score is a single draw, each feature set was additionally rated a second time with the features reshuffled into different order, which supplies the independent replicate the noise correction of Figure 2 requires; the same replicate exists for the BT arms for free, by refitting on disjoint halves of the 32,000 comparisons. The accuracy reported in Table 2 is the one attained by the average of these two scores.
Additional results
Additional measures
This subsection gathers additional results, notably accuracy measures (Table 9 ) and dimensionality measures (Figure 20 ).
Figure 20. Both dimensionality measures agree on the ordering of the feature sets. Noise-corrected effective dimensionality against feature set size k k k , averaged over k k k -subsets of each feature set. Left: participation ratio, as in Figure 2 . Right: effective rank ([40 ] ), exp H ( p ) \exp H(p) exp H ( p ) with p i = σ i ∑ j σ j p_i = \frac{\sigma_i}{\sum_j \sigma_j} p i = ∑ j σ j σ i the normalized singular values. Both are computed on the disattenuated correlation matrix. The order holds under both measures. Bands, shading and dotted line as in Figure 2 .
Other runs. Table 9 shows results about other runs that were made. Methods that specify “full” were fitted on the whole training set. BOW and POS are two surface baselines of Tan et al. [19 ] , built identically except for what a reply is reduced to: BOW represents each reply by its bag of lowercased word tokens (vocabulary fitted on the training replies, tokens occurring more than five times kept), while POS discards the words entirely and represents the reply by the sequence of its fine-grained Penn part-of-speech tags as assigned by spaCy, so that only syntactic style remains (42 tags).
As we can see, D + PW and Dᵣ + PW rank among the best methods despite their smaller training and feature set size, beaten only by the opaque embeddings trained on the whole training set and by Tan et al. [19 ] ’s features, while being interpretable. Additionally, all four panels of Figure 19 show that the reliable discovered features are more predictive, at matched feature count, than either the full discovered set or the sample-elicited set: the reliability filter of the statistical model selects the features that carry the signal.
Measuring effective dimensionality. We ask how many independent directions of variation a feature set resolves, using the participation ratio P R = ( ∑ i λ i ) 2 / ∑ i λ i 2 \mathrm{PR} = \left(\sum_i \lambda_i\right)^2 / \sum_i \lambda_i^2 PR = ( ∑ i λ i ) 2 / ∑ i λ i 2 of the eigenvalues λ \lambda λ of the feature set’s score correlation matrix over the 1,600 held-out exchanges. P R ∈ [ 1 , k ] \mathrm{PR} \in[1,k] PR ∈ [ 1 , k ] , equalling 1 when all k k k features are one construct and k k k when they are mutually independent.
Three choices make the comparison meaningful. a) Rank correlations. Bradley–Terry scores live on an arbitrary logit scale and pointwise ratings on a coarse 0–10 one, and a covariance-based participation ratio weights a feature by its variance; we therefore use the Spearman correlation matrix, which is invariant to scale and to any monotone reparameterization.
b) Matched feature set size. A feature set with more features trivially has more room for independent variation, so for each k k k we average the participation ratio over k k k -subsets of the feature set — enumerated exactly while ( d k ) ≤ 3,000 \binom{d}{k} \le3{,}000 ( k d ) ≤ 3 , 000 and sampled (3,000 draws) otherwise — and read comparisons at matched k k k .
c) Measurement noise correction. The participation ratio counts independent directions and cannot distinguish signal from independent measurement error, so a noisier scorer is credited with extra dimensions; since a Bradley–Terry score aggregates 16 comparisons while a pointwise score is a single coarse draw, the uncorrected comparison is biased in favor of pointwise. We therefore measure each feature set twice, independently: for the pairwise arms by refitting on disjoint halves of the 32,000 comparisons (the anchor graph split in two, and each test exchange’s 16 anchor comparisons split 8/8, so the halves share no data), and for the pointwise arms by a second scoring run with the features reshuffled into different bundles and order.
Writing C i j = corr ( A i , B j ) C_{ij} = \operatorname{corr}(A_i, B_j) C ij = corr ( A i , B j ) for the cross-measurement correlation, the independent errors cancel in expectation, so that C i i C_{ii} C ii estimates feature i i i ’s reliability and R i j = S i j / S i i S j j R_{ij} = S_{ij}/\sqrt{S_{ii}S_{jj}} R ij = S ij / S ii S j j with S = ( C + C T ) / 2 S = (C + C^{\mathsf{T}})/2 S = ( C + C T ) /2 disattenuates the correlation between the latent features. The participation ratio is then computed on R R R . The correction is substantial and asymmetric for the full feature sets: it removes 15–20% of the raw participation ratio for the pairwise arms and 25–28% for the pointwise ones; for the reliable subset Dr _{\mathrm{r}} r it is smaller and nearly symmetric (11% under BT, 13% under PW). Reliabilities are high enough for the division to be safe (median 0.83 and 0.77 for the D and P pointwise score sets; 0.76 and 0.71 for the half-data Bradley–Terry fits, i.e. 0.86 and 0.83 for the full score sets by the Spearman–Brown correction), and no disattenuated matrix carried any negative eigenvalue mass. Bands are 95% percentile bootstrap over the 1,600 exchanges, with the cross-measurement correlation rebuilt inside each of 200 replicates.
Discovered Features and Winning Argumentation Techniques
This subsection is devoted to additional results on the discovered features: Figure 21 completes Figure 3 , while Table 10 lists each discovered feature with its description, together with its effectiveness.
Figure 21. Effectiveness of the discovered argumentation techniques (D + PW). The figure is the complete version of Figure 3 and shows each discovered feature’s effectiveness , i.e. the fraction of the test pairs in which the reply that earned the Δ \Delta Δ scores higher on that feature. 0.5 is chance. Brackets give 95% Wald intervals and q q q is the Benjamini–Hochberg false-discovery rate across the 29 tests. The discovered features are gathered by category. Exact values of the BH q q q and each feature’s description can be found in Table 10 .
Prompts
Prompt 12: Comparative method feature-discovery, CMV corpus.
You are helping build an interpretable framework for what drives persuasion in debate, by examining how two attempts to change someone's mind differ. The framework works by collecting many pairwise comparisons: for each pair, the ways the two exchanges differ are articulated in plain terms, and aggregated across many pairs these reveal the distinct dimensions along which persuasion-relevant material varies. There is no fixed rubric – the dimensions are recovered from comparisons like this one, not specified in advance.
Background. Each item is an online discussion in which an original poster (the "OP") states a view they currently hold and invites others to change it; a challenger then replies with an argument aimed at changing that specific view. You will be shown two such exchanges, Exchange A and Exchange B, each presented as a block that begins with the OP's view (marked [OP VIEW]) followed by the challenger's reply (marked [CHALLENGE]). Crucially, the two exchanges concern different posters holding different views on different topics.
Your task. Identify the dimensions on which the two exchanges meaningfully differ in ways that could bear on whether the OP's view gets changed. Consider the whole exchange: both the challenger's argument and the OP's post itself. Report the differences that genuinely stand out to you on reading this pair – bring up whatever you find salient, and do not work from a predetermined list. Favor real, noticeable differences over minor ones, but do not force the list to be short: report as many as actually distinguish these two exchanges.
For each dimension, first name the concrete, specific feature you actually saw – a short phrase pointing to the specific evidence. Then, grounded in that evidence, name the dimension, describe in one sentence what it captures, and indicate which exchange shows more of it.
Judge each dimension on its own terms, from its own evidence. Exchanges have uneven profiles: it is entirely normal for one to be stronger on some dimensions and the other stronger on others, so let the exchange that shows more of each dimension fall wherever the evidence for that dimension points.
Out of scope – do not treat these as dimensions of difference:
the topic or subject matter either exchange is about, and how interesting, important, difficult, or agreeable either OP's view is: the two exchanges concern different topics by design, so topic or subject-matter difference is never a dimension here;
which side you personally agree with, or whether you think either OP's view is correct;
the username or any other identifier of either poster or challenger;
length in itself
How to report. List each dimension as exactly one item – do not combine dimensions with "and", "/", "&", or a comma; split related dimensions into separate items. For the evidence field, give a short phrase naming the concrete feature you saw. For the dimension field, give a concise general noun phrase naming the dimension (the kind of short label you might use as a rubric column). For the description field, give one sentence saying what that dimension captures. Indicate which exchange shows more of it.
Output only JSON of the form: {"differences": [{"evidence": "<short phrase>", "dimension": "<concise noun phrase>", "description": "<one sentence>", "winner": "A" | "B", ...]}
Exchange A: {arg_a} Exchange B: {arg_b} Following the standard above, list the dimensions on which these two exchanges differ in ways bearing on persuasion – whether in the challenger's argument or in the OP's post – for each giving the concrete feature you saw, the dimension, a one-sentence description, and which exchange shows more of it. Compare on persuasion-relevant qualities, not topic. Respond with the JSON object only.
Prompt 13: Comparative method, CMV corpus – feature canonicalization prompt.
You are organizing a flat list of evaluation dimensions into a set of canonical features.
These dimensions were produced by judges comparing pairs of online exchanges – in each, an original poster (the "OP") states a view and a challenger replies with an argument aimed at changing it. On each comparison a judge named the dimensions on which two exchanges differed in ways bearing on whether the OP's view gets changed; these span both the challenger's argument and the OP's own post. Aggregated across many comparisons, the same underlying construct was named in many slightly different ways. Your job is to group the different phrasings of the SAME underlying construct together.
Below is the full list. Each entry has an INDEX in square brackets, a short NAME, a one-sentence DESCRIPTION of what it captures, and CITATIONS (how many times it was named – for your awareness only).
indexed dimension list
Instructions:
Group entries that denote the SAME underlying construct, judging by MEANING, not by surface wording. Different words for the same idea belong in one group; the same word used for different ideas belongs in different groups.
A dimension describing the CHALLENGER's argument and a dimension describing the OP's POST are DIFFERENT constructs even when they share a word – group by the specific feature and whose it is, keeping OP-side and argument-side constructs in separate groups.
Do NOT force everything into a small number of groups, but do NOT split one construct into its facets either. A dimension that does not clearly share a construct with any other – including a framing or appeal specific to one debate's subject matter that does not recur as a general persuasion feature – does NOT become its own feature: place all such genuinely-apart, idiosyncratic dimensions together in a single catch-all group whose canonical name is exactly "Other". Do not create many single-member groups, and do not absorb a genuinely-apart dimension into a loosely-related group to tidy up – send it to "Other".
Do NOT merge two constructs just because they often co-occur or because persuasive exchanges tend to show both. Group only by shared MEANING, never by how exchanges might score.
Citations are context only. Do not merge a frequently-named dimension into a rarely-named one or vice versa on the basis of counts; counts do not determine grouping.
For each group, give:
a canonical NAME (a concise noun phrase suitable as a rubric column; use exactly "Other" for the catch-all group),
a one-sentence DEFINITION of the construct the group captures,
the list of members, each as its INDEX and its NAME copied verbatim from that index.
Rules on membership (these will be checked programmatically):
Every input index must appear in exactly one group, exactly once. Place all of them; drop none; duplicate none.
Do not invent indices that are not in the input.
Each member's "name" must be the exact input NAME at that index – this is a cross-check, so copy it from the bracketed entry, do not paraphrase.
Output only JSON: {"features": [{"name": "<canonical noun phrase>", "definition": "<one sentence>", "members": [{"index": <int>, "name": "<verbatim input name at that index>"}, ...]}, ...]}
BT feature scoring, CMV corpus.
You are helping apply an interpretable framework for what drives persuasion in debate. The framework scores how two attempts to change someone's mind differ along a fixed set of dimensions, listed below. Background. Each item is an online discussion in which an original poster (the "OP") states a view they hold and invites others to change it; a challenger then replies with an argument aimed at changing that specific view. You will be shown two such exchanges, Exchange A and Exchange B, each presented as a block that begins with the OP's view (marked [OP VIEW]) followed by the challenger's reply (marked [CHALLENGE]). The two exchanges concern different posters holding different views on different topics. The dimensions: [[features]] Your task. For EACH listed dimension, in order, give a short evidence phrase (the concrete feature you saw) and then indicate which exchange shows more of it — 'A' or 'B'. Some dimensions describe the challenger's argument, others describe the OP's post — assess each exactly as its definition states. Judge each dimension on its own terms: exchanges have uneven profiles, so it is entirely normal for A to show more of some dimensions and B more of others. Even when the two are close on a dimension, pick the exchange that shows more of it.
Out of scope — do not let these drive any judgment: the topic or subject matter and how interesting, important, difficult, or agreeable either OP's view is (the two differ by design); which side you personally agree with, or whether an OP's view is correct; usernames or identifiers; length in itself.
Return one item per listed dimension, in the same order, naming the dimension exactly as given, each with a short evidence phrase and the winner ('A' or 'B').
Output only JSON of the form: {"features": [{"name": "<dimension>", "evidence": "<short phrase>", "winner": "A" | "B"}, ...]}
Exchange A: {exchange_a}
Exchange B: {exchange_b}
Following the standard above, go through each listed dimension in order: give the concrete evidence you saw and then say which exchange shows more of it. Compare on the listed dimensions, not on topic. Respond with the JSON object only.
Prompt 15: Prior feature elicitation, CMV corpus.
You are helping build a framework for assessing how persuasive an argument is.
In online debates, a person states a view they hold and invites others to change it. Others reply with arguments aimed at changing that view.
Your task. Name exactly 16 dimensions for assessing how persuasive such a reply is.
Each dimension will be used as a measurement scale: a judge will be shown two replies and asked which of the two shows more of it. So each dimension must be a matter of degree rather than a yes/no fact, and the 16 must be distinct from one another.
For each dimension give:
Output only JSON of the form:
{"features": [{"name": "<short label>", "definition": "<one sentence>"}, ...]}
Name the 16 dimensions now, as a single JSON object. Respond with the JSON only.
Prompt 16: Sample-based feature elicitation, CMV corpus.
You design an interpretable scoring rubric and output only JSON.
You are helping build an interpretable scoring framework by examining how attempts to change someone's mind differ from one another. There is no fixed set of dimensions — they are recovered from what actually distinguishes these exchanges, not specified in advance.
In online debates, a person states a view they hold and invites others to change it. Others reply with arguments aimed at changing that view.
Below is a sample of such exchanges. Each begins with the poster's view (marked [OP VIEW]), followed by two separate replies to it, written independently by different people (marked [REPLY A] and [REPLY B]). The two replies answer the same view; they are not responses to each other, and their order carries no meaning.
=== EXCHANGES === 〈sample of exchanges〉 === END EXCHANGES ===
Reading across them, attend to the ways the replies differ from one another — the axes along which you can see one reply showing more, less, or something different from another. Identify the dimensions on which these replies meaningfully differ as attempts to change the poster's view. Report whatever genuinely stands out to you; do not work from a predetermined list, and do not force the list to be short — report as many dimensions as
actually distinguish these replies. Do not pad the list either: add a dimension only when it names something none of the others already capture.
A dimension may be something a reply shows more or less of along a range, or something a reply either shows or does not.
Every dimension must be a property of a reply, not of the poster's view or of the poster: name only dimensions on which the two replies to the same view can differ.
Keep genuinely distinct dimensions separate even when they are related or tend to co-occur — do not merge two dimensions because strong replies tend to show both. List each dimension as exactly one item — do not combine dimensions with "and", "/", "&", or a comma; split related dimensions into separate items.
For each dimension, first name the concrete, specific feature you actually saw in the replies — a short phrase pointing to the evidence. Then, grounded in that evidence, name the dimension and define in one sentence what it measures. Write the definition so a rater can apply it to a new reply without having seen this sample: describe the characteristic itself, and never refer to specific exchanges or to the reply labels.
Out of scope — do not treat these as dimensions of difference:
- the topic or subject matter, and how interesting, important, difficult or agreeable the poster's view is;
- which side you personally agree with, or whether the poster's view is correct;
- usernames or identifiers;
- length in itself.
Output only JSON:
{"features": [{"evidence": "<short phrase>", "name": "<short label, 2-5 words>", "definition": "<one sentence>"}, ...]}
Prompt 17: PW feature scoring, CMV corpus.
You are helping apply an interpretable framework for what drives persuasion in debate. The framework scores an attempt to change someone's mind along a fixed set of dimensions, listed below.
Background. Each item is an online discussion in which an original poster (the "OP") states a view they hold and invites others to change it; a challenger then replies with an argument aimed at changing that specific view. You will be shown one such exchange, presented as a block that begins with the OP's view (marked [OP VIEW]) followed by the challenger's reply (marked [CHALLENGE]).
The dimensions:
[[features]]
Your task. For EACH listed dimension, in order, give a short evidence phrase (the concrete feature you saw) and then rate the exchange on that dimension from 0 to 10. Some dimensions describe the challenger's argument, others describe the OP's post – assess each exactly as its definition states.
What the numbers mean. Rate each dimension relative to online arguments of this kind in general, not relative to this exchange's other dimensions: 0 means the dimension is entirely absent, 5 means it is present to a typical degree, and 10 means it is present about as strongly as such a reply ever gets. The same rating must mean the same thing on every exchange you score, because these scores are compared across exchanges you will not see together.
Judge each dimension on its own terms; exchanges have uneven profiles, so it is entirely normal for one to score high on some dimensions and low on others. Use the full range – do not cluster every dimension in the middle.
Out of scope – do not let these drive any rating: the topic or subject matter and how interesting, important, difficult, or agreeable the OP's view is; which side you personally agree with, or whether the OP's view is correct; usernames or identifiers; length in itself.
Return one item per listed dimension, in the same order, naming the dimension exactly as given, each with a short evidence phrase and an integer rating.
Output only JSON of the form: {"features": [{"name": "<dimension>", "evidence": "<short phrase>", "score": <integer 0–10>}, ...]}
Exchange: {exchange}
Following the standard above, go through each listed dimension in order: give the concrete evidence you saw and then its rating. Respond with the JSON object only.