Introduction
In causal problems involving structured and high-dimensional outcomes, such as images, text, and graphs, a mean outcome may be undefined or poorly aligned with the structure of the data. For example, comparing protein or text sequences requires choosing a numerical representation or a notion of similarity [50 , 35 , 46 ] and comparing images with coordinate-wise averaging ignores the spatial structure of the data [32 ] . Distributional causal inference addresses this by defining the interventional distributions directly on the structured outcome space, without reducing outcomes to an arbitrary scalar summary or notion of averaging [48 , 11 , 28 , 40 ] . Kernels define similarities on these spaces [17 , 6 , 53 ] and, through kernel mean embeddings, represent interventional distributions in a reproducing kernel Hilbert space (RKHS) [57 ] .
Kernels define the geometry used to compare outcome distributions [59 , 54 , 18 ] . If this geometry is fixed a priori, the kernel may be poorly suited to detecting differences between the interventional distributions, leading to low test power [19 , 60 ] . Deep kernels address this by using neural networks to learn a representation of the outcomes and compare distributions in the resulting space [64 ] . Rather than learning this representation through a reconstruction task, the learned kernel can be optimised by directly maximising a test-power criterion [60 ] .
We consider two soft interventions π \pi π π ′ \pi' π ′ P Y π P_Y^\pi P Y π P Y π ′ P_Y^{\pi'} P Y π ′ H 0 : P Y π = P Y π ′ H_0 : P_Y^\pi= P_Y^{\pi'} H 0 : P Y π = P Y π ′ H 1 : P Y π ≠ P Y π ′ H_1 : P_Y^\pi\ne P_Y^{\pi'} H 1 : P Y π = P Y π ′
Treatment effects often need to be estimated from observational data, because experimentation can be costly, unethical or infeasible. In this setting, interventional distributions are not directly observed and must be recovered [47 , 22 , 40 , 69 ] . This can be done by estimating either treatment assignment or the conditional outcome embedding [58 , 21 ] . However, either approach alone is sensitive to model misspecification. We therefore use a doubly robust estimator, which combines both approaches and remains consistent when either model is correctly specified [3 , 36 ] .
In this paper, we introduce DR-KTE-Deep, which combines doubly robust interventional embeddings with deep kernel learning and testing. Our main contributions are: (i) we develop a method for learning deep kernels for tests of interventional distributions, estimating both the distributional discrepancy and the test-power criterion from observational data; (ii) we derive a uniform finite-sample bound for the estimated learning criterion, accounting for both kernel-class complexity and errors in estimating the treatment-assignment and outcome models; (iii) we establish conditional asymptotic validity of the held-out test after nuisance estimation and kernel training and selection; and (iv) we evaluate the method on image, text, and graph outcomes, showing substantial power gains over fixed and bandwidth-optimised doubly robust kernel tests, while maintaining Type-I error close to the nominal level.
The remainder of the paper is organised as follows. Section 3 develops the doubly robust test. Section 4 develops the deep-kernel learning objective. Section 5 evaluates its power and calibration across image, text, and graph outcomes. Section 6 concludes.
Related Work
Several works study distributional causal effects. Quantile treatment effects characterise changes across an ordered outcome distribution [1 , 10 , 16 ] , while counterfactual distribution and density methods target the full marginal law [48 , 11 , 28 ] . These methods are typically developed for scalar or finite-dimensional Euclidean outcomes. We instead consider structured outcomes whose interventional distributions must be compared without reducing them to low-dimensional summaries.
Kernel mean embeddings represent interventional outcome distributions in an RKHS. Muandet et al. [40 ] introduce counterfactual mean embeddings, while Park et al. [45 ] , Nâf et al. [41 ] , and Jain & Luedtke [25 ] use kernel methods to study conditional distributional treatment effects. For marginal effects, Fawkes et al. [15 ] and Martinez-Taboada et al. [38 ] develop doubly robust kernel tests, and Zenati et al. [69 ] extend them to general stochastic policies. Finally, Zenati et al. [70 ] further consider adaptively collected data. These methods use prescribed outcome kernels, including aggregation over kernel bandwidths, without learning a deep outcome representation. A separate literature uses kernels to estimate causal mean functions through kernel ridge regression [56 , 30 ] , instrumental variable regression [55 , 39 ] , and proxy bridge learning [7 , 8 ] . These approaches use kernels or learned representations for causal function estimation, rather than for comparing interventional outcome distributions.
Other work adapts the kernel or outcome representation to improve test power. Gretton et al. [20 ] optimise the power of linear-time MMD tests over kernel mixtures and analyse convergence of the selection criterion. Sutherland et al. [60 ] optimise kernel parameters for two-sample tests, while Liu et al. [33 ] learn characteristic deep kernels and prove uniform convergence of their learning criterion. Other approaches combine multiple kernels through aggregation or fusion [51 , 5 ] . In causal settings, Zenati & Gretton (2026) learn informative witness locations for distributional treatment effects, while Wibisono & Wang [62 ] and Durand et al. [13 ] learn outcome features or projections to detect causal effects. Our method instead learns the outcome kernel for an omnibus test of marginal interventional distributions. We estimate the learning criterion from observational data using causal nuisance models and analyse how errors in these models affect the criterion.
Doubly Robust Kernel Test
Problem Setting. We are given an observational dataset D \mathcal{D} D N N N O 1 , … , O N ∼ i . i . d . P 0 O_1,\ldots,O_N \overset{\mathrm{i.i.d.}}{\sim} \mathbb{P}_0 O 1 , … , O N ∼ i.i.d. P 0 O i = ( X i , A i , Y i ) ∈ X × A × Y O_i=(X_i,A_i,Y_i)\in\mathcal{X}\times\mathcal{A}\times\mathcal{Y} O i = ( X i , A i , Y i ) ∈ X × A × Y X \mathcal{X} X A \mathcal{A} A Y \mathcal{Y} Y P X \mathbb{P}_{X} P X X X X P 0 \mathbb{P}_0 P 0 P A , X \mathbb{P}_{A,X} P A , X ρ \rho ρ σ \sigma σ A \mathcal{A} A A ∣ X = x A\mid X=x A ∣ X = x π 0 ( ⋅ ∣ x ) \pi_0(\cdot\mid x) π 0 ( ⋅ ∣ x ) ρ \rho ρ [49 , 23 ] . We compare two soft interventions π \pi π π ′ \pi' π ′ π ( ⋅ ∣ x ) \pi(\cdot\mid x) π ( ⋅ ∣ x ) π ′ ( ⋅ ∣ x ) \pi'(\cdot\mid x) π ′ ( ⋅ ∣ x )
Identification. The observational data are generated under the logging policy π 0 \pi_0 π 0 π \pi π π ′ \pi' π ′
Assumption 3.1 (Identification). There exist a standard Borel space U \mathcal{U} U U \mathcal{U} U U U U F : A × X × U → Y F:\mathcal{A}\times\mathcal{X}\times\mathcal{U}\to\mathcal{Y} F : A × X × U → Y
Consistency and structural response: Y = F ( A , X , U ) Y=F(A,X,U) Y = F ( A , X , U ) a ∈ A a\in\mathcal{A} a ∈ A Y a : = F ( a , X , U ) Y^a:=F(a,X,U) Y a := F ( a , X , U )
Conditional exogeneity: U ⊥ ⊥ A ∣ X U\perp\!\!\!\perp A\mid X U ⊥ ⊥ A ∣ X
Positivity on intervention support: π 0 ( a ∣ x ) > 0 \pi_0(a\mid x)>0 π 0 ( a ∣ x ) > 0 π ( a ∣ x ) + π ′ ( a ∣ x ) > 0 \pi(a\mid x)+\pi'(a\mid x)>0 π ( a ∣ x ) + π ′ ( a ∣ x ) > 0 P X ⊗ ρ \mathbb{P}_X\otimes\rho P X ⊗ ρ
For p ∈ { π , π ′ } p\in\{\pi,\pi'\} p ∈ { π , π ′ } A p ∣ X A^p\mid X A p ∣ X p ( ⋅ ∣ X ) p(\cdot\mid X) p ( ⋅ ∣ X ) U U U X X X Y p : = F ( A p , X , U ) Y^p:=F(A^p,X,U) Y p := F ( A p , X , U ) P Y p : = L ( Y p ) P_Y^p:=\mathcal{L}(Y^p) P Y p := L ( Y p ) H 0 : P Y π = P Y π ′ H_0:P_Y^\pi=P_Y^{\pi'} H 0 : P Y π = P Y π ′ H 1 : P Y π ≠ P Y π ′ H_1:P_Y^\pi\ne P_Y^{\pi'} H 1 : P Y π = P Y π ′
Let Q ( ⋅ ∣ a , x ) Q(\cdot\mid a,x) Q ( ⋅ ∣ a , x ) Y Y Y ( A , X ) = ( a , x ) (A,X)=(a,x) ( A , X ) = ( a , x ) [47 ] ; Hernán & Robins [22 ] :
P Y p ( B ) = ∫ X ∫ A Q ( B ∣ a , x ) p ( a ∣ x ) ρ ( d a ) P X ( d x ) , P_Y^p(B) = \int_{\mathcal{X}} \int_{\mathcal{A}} Q(B \mid a,x)p(a \mid x)\rho(\mathrm{d}a)P_X(\mathrm{d}x), P Y p ( B ) = ∫ X ∫ A Q ( B ∣ a , x ) p ( a ∣ x ) ρ ( d a ) P X ( d x ) , for every measurable B ⊆ Y B \subseteq\mathcal{Y} B ⊆ Y
This formula averages the conditional outcome distributions over the treatments assigned by p p p
Kernel Formulation. To compare distributions over structured outcomes, we represent them as elements of an RKHS. Let Θ ⊆ R d Θ \Theta\subseteq\mathbb{R}^{d_\Theta} Θ ⊆ R d Θ θ ∈ Θ \theta\in\Theta θ ∈ Θ k θ : Y × Y → R k_\theta: \mathcal{Y} \times\mathcal{Y} \to\mathbb{R} k θ : Y × Y → R Y \mathcal{Y} Y θ \theta θ H θ \mathcal{H}_\theta H θ φ θ ( y ) : = k θ ( y , ⋅ ) \varphi_\theta(y) := k_\theta(y,\cdot) φ θ ( y ) := k θ ( y , ⋅ )
We define the interventional mean embedding [40 , 69 ] , embedding contrast, and squared maximum mean discrepancy (MMD) [19 ] respectively by
χ θ ( p ) : = ∫ Y φ θ ( y ) P Y p ( d y ) , Δ θ ( π , π ′ ) : = χ θ ( π ) − χ θ ( π ′ ) , D θ ( π , π ′ ) : = ∥ Δ θ ( π , π ′ ) ∥ H θ 2 . \chi_\theta(p) := \int_{\mathcal{Y}} \varphi_\theta(y)P_Y^p(\mathrm{d}y), \qquad\Delta_\theta(\pi,\pi') := \chi_\theta(\pi) - \chi_\theta(\pi'), \qquad D_\theta(\pi,\pi') := \|\Delta_\theta(\pi,\pi')\|_{\mathcal{H}_\theta}^2. χ θ ( p ) := ∫ Y φ θ ( y ) P Y p ( d y ) , Δ θ ( π , π ′ ) := χ θ ( π ) − χ θ ( π ′ ) , D θ ( π , π ′ ) := ∥ Δ θ ( π , π ′ ) ∥ H θ 2 . Assumption 3.2 (Embedding integrability). For every θ ∈ Θ \theta\in\Theta θ ∈ Θ p ∈ { π , π ′ } p \in\{\pi,\pi'\} p ∈ { π , π ′ } E P 0 k θ ( Y , Y ) < ∞ \mathbb{E}_{P_0}\sqrt{k_\theta(Y,Y)} < \infty E P 0 k θ ( Y , Y ) < ∞ ∫ Y k θ ( y , y ) P Y p ( d y ) < ∞ \int_{\mathcal{Y}}\sqrt{k_\theta(y,y)}P_Y^p(\mathrm{d}y) < \infty ∫ Y k θ ( y , y ) P Y p ( d y ) < ∞
If k θ k_\theta k θ H 0 H_0 H 0 D θ ( π , π ′ ) = 0 D_\theta(\pi,\pi') = 0 D θ ( π , π ′ ) = 0 [19 Section 2.2] .
Doubly Robust Representation. To estimate D θ ( π , π ′ ) D_\theta(\pi,\pi') D θ ( π , π ′ ) Δ θ ( π , π ′ ) \Delta_\theta(\pi,\pi') Δ θ ( π , π ′ ) [69 ] . We further compare the choice of estimator in Section D.4.
For a fixed θ ∈ Θ \theta\in\Theta θ ∈ Θ [21 ] by
μ θ ( a , x ) : = ∫ Y φ θ ( y ) Q ( d y ∣ a , x ) . \mu_\theta(a,x) := \int_{\mathcal{Y}} \varphi_\theta(y)Q(\mathrm{d}y \mid a,x). μ θ ( a , x ) := ∫ Y φ θ ( y ) Q ( d y ∣ a , x ) . We use the null-set convention described in Section B.1. This embedding satisfies μ θ ( A , X ) = E P 0 [ φ θ ( Y ) ∣ A , X ] \mu_\theta(A,X) = \mathbb{E}_{P_0}[\varphi_\theta(Y) \mid A,X] μ θ ( A , X ) = E P 0 [ φ θ ( Y ) ∣ A , X ] r ( a , x ) : = π ( a ∣ x ) − π ′ ( a ∣ x ) r(a,x) := \pi(a \mid x)-\pi'(a \mid x) r ( a , x ) := π ( a ∣ x ) − π ′ ( a ∣ x ) e e e ∣ r ∣ > 0 |r| > 0 ∣ r ∣ > 0 m θ m_\theta m θ r r r ∣ r ∣ |r| ∣ r ∣ r = 0 r = 0 r = 0
Define the Hilbert-valued pseudo-feature map
g θ π , π ′ ( O ; e , m θ ) : = r ( A , X ) e ( A ∣ X ) { φ θ ( Y ) − m θ ( A , X ) } + ∫ A r ( a , X ) m θ ( a , X ) ρ ( d a ) . (1) g_\theta^{\pi,\pi'}(O;e,m_\theta) := \frac{r(A,X)}{e(A \mid X)}\{\varphi_\theta(Y)-m_\theta(A,X)\}+\int_{\mathcal{A}}r(a,X)m_\theta(a,X)\rho(\mathrm{d}a). \tag*{(1)} g θ π , π ′ ( O ; e , m θ ) := e ( A ∣ X ) r ( A , X ) { φ θ ( Y ) − m θ ( A , X )} + ∫ A r ( a , X ) m θ ( a , X ) ρ ( d a ) . (1) The second term estimates the embedding contrast through the conditional outcome model, while the first corrects it using propensity-weighted outcome residuals.
We suppress the intervention superscript when no ambiguity arises, g θ ( O ; e , m θ ) : = g θ π , π ′ ( O ; e , m θ ) g_\theta(O;e,m_\theta) := g_\theta^{\pi,\pi'}(O;e,m_\theta) g θ ( O ; e , m θ ) := g θ π , π ′ ( O ; e , m θ ) g θ ( O ) : = g θ ( O ; π 0 , μ θ ) g_\theta(O) := g_\theta(O;\pi_0,\mu_\theta) g θ ( O ) := g θ ( O ; π 0 , μ θ ) E 0 \mathbb{E}_0 E 0 P 0 P_0 P 0
Proposition 3.3 (Observed-data representation and double robustness). Suppose Assumptions 3.1 and 3.2 hold and the candidate nuisances satisfy the absolute-integrability conditions of Section B.2. Write Δ θ : = Δ θ ( π , π ′ ) \Delta_\theta:= \Delta_\theta(\pi,\pi') Δ θ := Δ θ ( π , π ′ ) e e e m θ m_\theta m θ
E 0 [ g θ ( O ; e , m θ ) ] − Δ θ = E 0 [ ∫ A r ( a , X ) { π 0 ( a ∣ X ) e ( a ∣ X ) − 1 } { μ θ ( a , X ) − m θ ( a , X ) } ρ ( d a ) ] . \mathbb{E}_0[g_\theta(O;e,m_\theta)]-\Delta_\theta=\mathbb{E}_0\left[\int_{\mathcal{A}}r(a,X)\left\{\frac{\pi_0(a\mid X)}{e(a\mid X)}-1\right\}\{\mu_\theta(a,X)-m_\theta(a,X)\}\rho(da)\right]. E 0 [ g θ ( O ; e , m θ )] − Δ θ = E 0 [ ∫ A r ( a , X ) { e ( a ∣ X ) π 0 ( a ∣ X ) − 1 } { μ θ ( a , X ) − m θ ( a , X )} ρ ( d a ) ] . Thus, E 0 [ g θ ( O ; e , m θ ) ] = Δ θ \mathbb{E}_0[g_\theta(O;e,m_\theta)]=\Delta_\theta E 0 [ g θ ( O ; e , m θ )] = Δ θ e = π 0 e=\pi_0 e = π 0 m θ = μ θ m_\theta=\mu_\theta m θ = μ θ ∣ r ∣ |r| ∣ r ∣
A proof is given in Section B.2. We examine the effect of outcome-model misspecification empirically in Section D.5, using a misspecified m θ m_\theta m θ e ≡ π 0 e\equiv\pi_0 e ≡ π 0
Test Statistic. Since D θ ( π , π ′ ) = ∥ Δ θ ( π , π ′ ) ∥ H θ 2 D_\theta(\pi,\pi')=\|\Delta_\theta(\pi,\pi')\|_{\mathcal{H}_\theta}^2 D θ ( π , π ′ ) = ∥ Δ θ ( π , π ′ ) ∥ H θ 2 H 0 H_0 H 0 H θ \mathcal{H}_\theta H θ D θ ( π , π ′ ) D_\theta(\pi,\pi') D θ ( π , π ′ ) Δ θ ( π , π ′ ) \Delta_\theta(\pi,\pi') Δ θ ( π , π ′ ) [29 ] and Shekhar et al. [52 ] .
Let π ^ 0 \hat{\pi}_0 π ^ 0 μ ^ θ \hat{\mu}_\theta μ ^ θ g ^ θ , i : = g θ ( O i ; π ^ 0 , μ ^ θ ) \hat{g}_{\theta,i}:=g_\theta(O_i;\hat{\pi}_0,\hat{\mu}_\theta) g ^ θ , i := g θ ( O i ; π ^ 0 , μ ^ θ ) I 1 \mathcal{I}_1 I 1 I 2 \mathcal{I}_2 I 2 m m m g ˉ θ , ℓ : = 1 m ∑ i ∈ I ℓ g ^ θ , i \bar{g}_{\theta,\ell}:=\frac{1}{m}\sum_{i\in\mathcal{I}_\ell}\hat{g}_{\theta,i} g ˉ θ , ℓ := m 1 ∑ i ∈ I ℓ g ^ θ , i ℓ ∈ { 1 , 2 } \ell\in\{1,2\} ℓ ∈ { 1 , 2 }
D ^ θ c r o s s : = ⟨ g ˉ θ , 1 , g ˉ θ , 2 ⟩ H θ = 1 m 2 ∑ i ∈ I 1 ∑ j ∈ I 2 ⟨ g ^ θ , i , g ^ θ , j ⟩ H θ , s ^ θ 2 : = 1 m ∑ j ∈ I 2 ( ⟨ g ˉ θ , 1 , g ^ θ , j ⟩ H θ − D ^ θ c r o s s ) 2 . \hat{D}^{\mathrm{cross}}_\theta:=\langle\bar{g}_{\theta,1},\bar{g}_{\theta,2}\rangle_{\mathcal{H}_\theta}=\frac{1}{m^2}\sum_{i\in\mathcal{I}_1}\sum_{j\in\mathcal{I}_2}\langle\hat{g}_{\theta,i},\hat{g}_{\theta,j}\rangle_{\mathcal{H}_\theta},\qquad\hat{s}_\theta^2:=\frac{1}{m}\sum_{j\in\mathcal{I}_2}\left(\langle\bar{g}_{\theta,1},\hat{g}_{\theta,j}\rangle_{\mathcal{H}_\theta}-\hat{D}^{\mathrm{cross}}_\theta\right)^2. D ^ θ cross := ⟨ g ˉ θ , 1 , g ˉ θ , 2 ⟩ H θ = m 2 1 i ∈ I 1 ∑ j ∈ I 2 ∑ ⟨ g ^ θ , i , g ^ θ , j ⟩ H θ , s ^ θ 2 := m 1 j ∈ I 2 ∑ ( ⟨ g ˉ θ , 1 , g ^ θ , j ⟩ H θ − D ^ θ cross ) 2 . Our test statistic is the studentised cross-U statistic
T θ : = m D ^ θ c r o s s / s ^ θ . (2) T_\theta:=\sqrt{m}\hat{D}^{\mathrm{cross}}_\theta/\hat{s}_\theta. \tag*{(2)} T θ := m D ^ θ cross / s ^ θ . (2) We set T θ = 0 T_\theta=0 T θ = 0 s ^ θ = 0 \hat{s}_\theta=0 s ^ θ = 0
Let Φ \Phi Φ
T θ → d N ( 0 , 1 ) T_\theta\xrightarrow{d}\mathcal{N}(0,1) T θ d N ( 0 , 1 ) under H 0 H_0 H 0 D θ ( π , π ′ ) > 0 D_\theta(\pi,\pi')>0 D θ ( π , π ′ ) > 0 H 1 H_1 H 1 T θ T_\theta T θ H 0 H_0 H 0 α \alpha α p ^ = 1 − Φ ( T θ ) < α \hat{p}=1-\Phi(T_\theta)<\alpha p ^ = 1 − Φ ( T θ ) < α
Learning Deep Kernels
Learning Objective. The choice of outcome kernel affects both the population discrepancy D θ D_\theta D θ θ \theta θ [20 , 60 , 33 ] )
J θ = D θ / s θ , with D θ : = D θ ( π , π ′ ) , s θ 2 : = Var O ∼ P 0 [ ⟨ Δ θ , g θ ( O ) ⟩ H θ ] J_\theta=D_\theta/s_\theta,\qquad\text{with }D_\theta:=D_\theta(\pi,\pi'),\qquad s_\theta^2:=\operatorname{Var}_{O\sim P_0}\left[\langle\Delta_\theta,g_\theta(O)\rangle_{\mathcal{H}_\theta}\right] J θ = D θ / s θ , with D θ := D θ ( π , π ′ ) , s θ 2 := Var O ∼ P 0 [ ⟨ Δ θ , g θ ( O ) ⟩ H θ ] whenever s θ > 0 s_\theta>0 s θ > 0 J θ J_\theta J θ
Proposition 4.1 (Fixed-alternative growth rate). Suppose Assumptions 3.1 and 3.2 hold. For a fixed θ \theta θ D θ > 0 D_\theta>0 D θ > 0 s θ > 0 s_\theta>0 s θ > 0 E O ∼ P 0 [ ∥ g θ π , π ′ ( O ) ∥ H θ ] < ∞ \mathbb{E}_{O\sim P_0}[\|g_\theta^{\pi,\pi'}(O)\|_{\mathcal{H}_\theta}]<\infty E O ∼ P 0 [ ∥ g θ π , π ′ ( O ) ∥ H θ ] < ∞
D ^ θ c r o s s → P D θ , s ^ θ 2 → P s θ 2 , T θ m → P J θ . \hat{D}^{\mathrm{cross}}_\theta\xrightarrow{\mathbb{P}}D_\theta,\qquad\hat{s}_\theta^2\xrightarrow{\mathbb{P}}s_\theta^2,\qquad\frac{T_\theta}{\sqrt{m}}\xrightarrow{\mathbb{P}}J_\theta. D ^ θ cross P D θ , s ^ θ 2 P s θ 2 , m T θ P J θ . A proof is given in Section B.3. Section D.6 further studies the choice of learning criterion. This result motivates maximising J θ J_{\theta} J θ J θ J_{\theta} J θ D θ D_{\theta} D θ s θ s_{\theta} s θ [26 , 27 ]
J ^ λ , θ D R = D ^ θ c r o s s s ^ θ , λ (3) \widehat{J}^{\mathrm{DR}}_{\lambda,\theta} = \frac{\widehat{D}^{\mathrm{cross}}_{\theta}}{\widehat{s}_{\theta,\lambda}} \tag*{(3)} J λ , θ DR = s θ , λ D θ cross (3) where s ^ θ , λ 2 : = s ^ θ 2 + λ \widehat{s}^{2}_{\theta,\lambda} := \widehat{s}^{2}_{\theta} + \lambda s θ , λ 2 := s θ 2 + λ λ > 0 \lambda> 0 λ > 0 s ^ θ 2 \widehat{s}^{2}_{\theta} s θ 2
While the learning criterion can be used to optimise the parameters of a simple kernel family, like the lengthscale of a Gaussian kernel, the kernel still compares outcomes isotropically throughout the outcome space, and may have low power against localised or anisotropic perturbations at finite sample size [33 Section 3] (Figure 2 ).
Figure 2. Synthetic example where treatment assignment is driven by an indicator of disease severity X X X
Deep Kernels
We therefore learn a deep kernel combining a neural component with a base characteristic kernel. Since deep kernel optimisation is sensitive to random initialisation, we train and select an ensemble of M M M Ξ ⊆ R d 0 \Xi\subseteq\mathbb{R}^{d_0} Ξ ⊆ R d 0 θ = ( ϑ 1 , … , ϑ M ) ∈ Θ \theta= (\vartheta_1,\ldots,\vartheta_M) \in\Theta θ = ( ϑ 1 , … , ϑ M ) ∈ Θ ϑ b ∈ Ξ \vartheta_b \in\Xi ϑ b ∈ Ξ b ∈ [ 1 , M ] b \in[1,M] b ∈ [ 1 , M ]
k ϑ b ( y , y ′ ) = ( 1 − w ϑ b ) κ ϑ b ( ζ ϑ b ( y ) , ζ ϑ b ( y ′ ) ) + w ϑ b k ϑ b c h a r ( y , y ′ ) , k θ ( y , y ′ ) = 1 M ∑ b = 1 M k ϑ b ( y , y ′ ) . (4) k_{\vartheta_b}(y,y') = (1-w_{\vartheta_b})\kappa_{\vartheta_b}\bigl(\zeta_{\vartheta_b}(y),\zeta_{\vartheta_b}(y')\bigr) + w_{\vartheta_b} k^{\mathrm{char}}_{\vartheta_b}(y,y'), \qquad k_{\theta}(y,y') = \frac{1}{M}\sum_{b=1}^{M} k_{\vartheta_b}(y,y'). \tag*{(4)} k ϑ b ( y , y ′ ) = ( 1 − w ϑ b ) κ ϑ b ( ζ ϑ b ( y ) , ζ ϑ b ( y ′ ) ) + w ϑ b k ϑ b char ( y , y ′ ) , k θ ( y , y ′ ) = M 1 b = 1 ∑ M k ϑ b ( y , y ′ ) . (4) Here ζ ϑ : Y → R d \zeta_{\vartheta}:\mathcal{Y}\to\mathbb{R}^{d} ζ ϑ : Y → R d κ ϑ \kappa_{\vartheta} κ ϑ k ϑ c h a r k^{\mathrm{char}}_{\vartheta} k ϑ char Y \mathcal{Y} Y w ϑ ∈ [ w min , 1 ] w_{\vartheta} \in[w_{\min},1] w ϑ ∈ [ w m i n , 1 ] w min > 0 w_{\min} > 0 w m i n > 0
The learned component κ ϑ b ( ζ ϑ b ( y ) , ζ ϑ b ( y ′ ) ) \kappa_{\vartheta_b}(\zeta_{\vartheta_b}(y),\zeta_{\vartheta_b}(y')) κ ϑ b ( ζ ϑ b ( y ) , ζ ϑ b ( y ′ ))
Lemma 4.2 (Characteristic ensemble safeguard). For every pair of probability laws P , Q P,Q P , Q
MMD k θ 2 ( P , Q ) ≥ w min M ∑ b = 1 M MMD k ϑ b c h a r 2 ( P , Q ) . (5) \operatorname{MMD}^{2}_{k_{\theta}}(P,Q) \ge\frac{w_{\min}}{M}\sum_{b=1}^{M}\operatorname{MMD}^{2}_{k^{\mathrm{char}}_{\vartheta_b}}(P,Q). \tag*{(5)} MMD k θ 2 ( P , Q ) ≥ M w m i n b = 1 ∑ M MMD k ϑ b char 2 ( P , Q ) . (5) Consequently k θ k_{\theta} k θ θ ∈ Θ \theta\in\Theta θ ∈ Θ
A proof is given in Section B.4. For every θ ∈ Θ \theta\in\Theta θ ∈ Θ D θ ( π , π ′ ) = 0 D_{\theta}(\pi,\pi')=0 D θ ( π , π ′ ) = 0
Training and Testing Procedure. We detail the training and testing procedure in Algorithm 1 and Section C.1. D \mathcal{D} D [20 , 33 ] ). We optimise the kernel parameters on the learning sample, select a kernel on the validation sample, and compute T θ T_{\theta} T θ D f i t \mathcal{D}_{\mathrm{fit}} D fit D s e l , 1 \mathcal{D}_{\mathrm{sel},1} D sel , 1 D s e l , 2 \mathcal{D}_{\mathrm{sel},2} D sel , 2 D f i t \mathcal{D}_{\mathrm{fit}} D fit J ^ λ , ϑ b D R \widehat{J}^{\mathrm{DR}}_{\lambda,\vartheta_b} J λ , ϑ b DR D s e l , 1 \mathcal{D}_{\mathrm{sel},1} D sel , 1 D s e l , 2 \mathcal{D}_{\mathrm{sel},2} D sel , 2 ϑ b \vartheta_b ϑ b [61 ] ) and select the EMA checkpoint with the largest unregularised validation statistic.
Let F l e a r n \mathcal{F}_{\mathrm{learn}} F learn F l e a r n \mathcal{F}_{\mathrm{learn}} F learn o P ( m − 1 / 2 ) o_{\mathbb{P}}(m^{-1/2}) o P ( m − 1/2 )
Theorem 4.3 (Held-out null validity). Suppose Assumption B.4 holds, and set T ^ θ ^ = 0 \widehat{T}_{\hat{\theta}}=0 T θ ^ = 0 s ^ θ ^ = 0 \widehat{s}_{\hat{\theta}}=0 s θ ^ = 0 H 0 H_0 H 0 m m m
sup t ∈ R ∣ P ( T ^ θ ^ ≤ t ∣ F l e a r n ) − Φ ( t ) ∣ → P 0. (6) \sup_{t \in\mathbb{R}} \left|\mathbb{P}\left(\widehat{T}_{\hat{\theta}} \leq t \mid\mathcal{F}_{\mathrm{learn}}\right)-\Phi(t)\right| \xrightarrow{\mathbb{P}} 0. \tag*{(6)} t ∈ R sup P ( T θ ^ ≤ t ∣ F learn ) − Φ ( t ) P 0. (6) Consequently, T ^ θ ^ → d N ( 0 , 1 ) \widehat{T}_{\hat{\theta}} \xrightarrow{d} N(0,1) T θ ^ d N ( 0 , 1 ) α \alpha α α ∈ ( 0 , 1 ) \alpha\in(0,1) α ∈ ( 0 , 1 )
A proof is given in Section B.6. The result is conditional on the full learning procedure and does not require the optimised parameters to converge to a deterministic kernel.
Uniform Control of the Learning Criterion. We provide a uniform analysis of the learning criterion (see also Gretton et al. [20 ] , Liu et al. [33 ] ) with the fitted nuisances π ^ 0 \widehat{\pi}_0 π 0 μ ^ θ \widehat{\mu}_{\theta} μ θ
J λ , θ : = D θ s θ 2 + λ , λ > 0. (7) J_{\lambda,\theta} := \frac{D_{\theta}}{\sqrt{s_{\theta}^{2}+\lambda}}, \qquad\lambda> 0. \tag*{(7)} J λ , θ := s θ 2 + λ D θ , λ > 0. (7) Let the nuisance-fitting sample have size n n n m m m F f i t \mathcal{F}_{\mathrm{fit}} F fit F f i t \mathcal{F}_{\mathrm{fit}} F fit θ \theta θ
On a compact finite-dimensional parameter class, we impose the conditions of Assumption B.5. These require the fitted kernels, constructed from k θ k_{\theta} k θ θ \theta θ
Theorem 4.4 (Uniform feasible learning criterion). Let Assumptions B.5 and 3.1 hold and fix λ 0 > 0 \lambda_0>0 λ 0 > 0 ϵ π , n 2 \epsilon_{\pi,n}^{2} ϵ π , n 2 ϵ μ , n 2 \epsilon_{\mu,n}^{2} ϵ μ , n 2 F f i t \mathcal{F}_{\mathrm{fit}} F fit λ ≥ λ 0 \lambda\geq\lambda_0 λ ≥ λ 0 η ∈ ( 0 , 1 ) \eta\in(0,1) η ∈ ( 0 , 1 ) C C C λ 0 \lambda_0 λ 0 1 − η 1-\eta 1 − η
sup θ ∈ Θ ∣ J ^ λ , θ D R − J λ , θ ∣ ≤ C { d θ log ( 1 + 2 R Θ m ) + log ( 4 / η ) + 1 m + ϵ π , n ϵ μ , n + ϵ π , n + ϵ μ , n 2 } . (8) \sup_{\theta\in\Theta}\left|\widehat{J}^{\mathrm{DR}}_{\lambda,\theta}-J_{\lambda,\theta}\right|
\leq C\left\{\frac{\sqrt{d_{\theta}\log(1+2R_{\Theta}\sqrt{m})+\log(4/\eta)+1}}{\sqrt{m}}+\epsilon_{\pi,n}\epsilon_{\mu,n}+\epsilon_{\pi,n}+\epsilon_{\mu,n}^{2}\right\}. \tag*{(8)} θ ∈ Θ sup J λ , θ DR − J λ , θ ≤ C { m d θ log ( 1 + 2 R Θ m ) + log ( 4/ η ) + 1 + ϵ π , n ϵ μ , n + ϵ π , n + ϵ μ , n 2 } . (8) For three cyclic rotations of the learning folds satisfying the same regularity constants, the averaged criterion obeys the analogous bound with unconditional probability at least 1 − 3 η 1 - 3\eta 1 − 3 η
The uniform bound separates sampling error from nuisance-estimation error. The MMD numerator retains the doubly robust product remainder ϵ π , n ϵ μ , n \epsilon_{\pi,n}\epsilon_{\mu,n} ϵ π , n ϵ μ , n s ^ θ 2 \hat{s}_{\theta}^{2} s ^ θ 2 ϵ π , n \epsilon_{\pi,n} ϵ π , n ϵ μ , n 2 \epsilon_{\mu,n}^{2} ϵ μ , n 2 O ( log m / m ) O(\sqrt{\log m/m}) O ( log m / m ) O ( m − 1 / 3 ) O(m^{-1/3}) O ( m − 1/3 )
Theorem 4.4 controls J λ , θ J_{\lambda,\theta} J λ , θ Θ \Theta Θ λ \lambda λ
Θ s : = { θ ∈ Θ : s θ 2 ≥ s 2 } , \Theta_s := \{\theta\in\Theta: s_\theta^2 \ge s^2\}, Θ s := { θ ∈ Θ : s θ 2 ≥ s 2 } , the results in Section B allow λ m → 0 \lambda_m \to0 λ m → 0 J θ J_\theta J θ Theorem B.13 .
Since each kernel member is trained separately, we apply these results setting M = 1 M = 1 M = 1 Θ = Ξ \Theta= \Xi Θ = Ξ d Θ = d 0 d_\Theta= d_0 d Θ = d 0 M > 1 M > 1 M > 1 Proposition B.3 gives J λ , θ ≥ min b J λ , ϑ b J_{\lambda,\theta} \ge\min_b J_{\lambda,\vartheta_b} J λ , θ ≥ min b J λ , ϑ b λ ≥ 0 \lambda\ge0 λ ≥ 0 1 / M 1/M 1/ M Proposition B.3 establishes these population guarantees. Section D.8 studies the effect of ensemble size on test power. Section D.12 analyses the effect of jointly maximising the averaged criterion.
Experimental Evaluation
We evaluate our doubly robust deep kernel test, DR-KTE-Deep, on image, text and graph outcomes. Evaluations on low-dimensional outcomes are included in Section D. We compare our method with DR-KTE [38 ] , which uses a fixed Gaussian kernel, and DR-KTE-Opt, which optimises the lengthscale of a Gaussian kernel. Experimental details are given in Section C. Section D.10 additionally compares our method to linear kernels. All methods are fitted and evaluated using the same total sample size. Our code is available at https://www.github.com/robinlmn/deep-kte.
We learn the deep kernel and nuisance parameters on D l e a r n \mathcal{D}_{\mathrm{learn}} D learn D v a l \mathcal{D}_{\mathrm{val}} D val D l e a r n ∪ D v a l \mathcal{D}_{\mathrm{learn}} \cup\mathcal{D}_{\mathrm{val}} D learn ∪ D val α = 0.05 \alpha= 0.05 α = 0.05 D t e s t \mathcal{D}_{\mathrm{test}} D test ± \pm ±
Image Outcomes. We first evaluate our method on mean-preserving distributional effects on medical-image outcomes. We extend the OCTMNIST construction of Zenati & Gretton (2026). Potential outcomes are constructed as Y i a = B i + R i ( a ) Y_i^a = B_i + R_i(a) Y i a = B i + R i ( a ) B i B_i B i [66 ] and R i ( a ) R_i(a) R i ( a ) δ \delta δ E [ Y i 1 − Y i 0 ∣ X i ] = 0 \mathbb{E}[Y_i^1 - Y_i^0 \mid X_i] = 0 E [ Y i 1 − Y i 0 ∣ X i ] = 0 Figure 1 . Section C.2 details the full generative process.
Figure 1. Semi-synthetic medical-image experiments adapted from MedMNIST ([65 , 66 ] ). Each pair shows a treated potential outcome Y 1 Y^1 Y 1
DR-KTE-Deep achieves substantially higher estimated power than the baselines on all constructions (Table 1), while DR-KTE-Opt only slightly improves power over the fixed kernel. Under H 0 H_0 H 0 α = 0.05 \alpha= 0.05 α = 0.05 Figure 3 , the power of the deep kernel grows with sample size and effect strength, while Gaussian baselines remain near the nominal rejection rate across the sample-size sweep.
Figure 3. Results for OCT-Drusen. (a) Type-I error under H 0 H_0 H 0 N N N δ = 0.4 \delta= 0.4 δ = 0.4 δ \delta δ N = 1500 N = 1500 N = 1500 ± \pm ± α = 0.05 \alpha= 0.05 α = 0.05
Text Outcomes. We now evaluate our method on text outcomes, using a language model to generate synthetic data under the assumed causal structure [9 ] . Covariates and treatments generate a scalar pseudo-outcome, discretised into one of three tones used to condition the language model. We consider three scenarios: Oncology Notes, Movie Reviews and Clinical Notes. The generated text is represented as a sequence of tokens and used as input to each kernel method. Construction details are given in Section C.3. Section D.7 additionally compares each method through an alternative text embedding outcome representation.
DR-KTE-Deep achieves the highest estimated power (Table 2 ), with substantial gains on Oncology Notes and Clinical Notes and smaller improvements on Movie Reviews. Under H 0 H_0 H 0 α = 0.05 \alpha= 0.05 α = 0.05 Figure 4 , we vary the sample size N N N δ \delta δ δ \delta δ
Table 2. Power (H 1 H_1 H 1 H 0 H_0 H 0 N = 1500 N = 1500 N = 1500 α = 0.05 \alpha= 0.05 α = 0.05 H 1 / H 0 H_1/H_0 H 1 / H 0 ± \pm ±
Figure 4. Results for Movie Reviews. (a) Type-I error under H 0 H_0 H 0 N N N δ = 1.0 \delta= 1.0 δ = 1.0 δ \delta δ N = 1500 N = 1500 N = 1500 ± \pm ± α = 0.05 \alpha= 0.05 α = 0.05
Graph Outcomes. We consider two experiments in which the outcome is a labelled graph. For LLM Code Repairs, we simulate the effect of using a large language model (LLM) to fix a buggy Python function represented as its abstract syntax tree (AST). For Contact Networks, we simulate the effect of social distancing on the spread of an infection using contact networks sampled from the Infectious SocioPatterns dataset [24 ] . The graph constructions and encodings are detailed in Section C.4. In addition to Gaussian-based kernels, we evaluate Weisfeiler-Lehman (WL) subtree kernels [53 ] in Section D.9.
DR-KTE-Deep achieves higher estimated average power on both LLM Code Repairs and Contact Networks, with smaller improvements on the latter. The Gaussian baselines already have high power on both constructions. Under H 0 H_0 H 0 Table 3 ).
Table 3. Power (H 1 H_1 H 1 H 0 H_0 H 0 N = 1500 N = 1500 N = 1500 H 1 / H 0 H_1/H_0 H 1 / H 0 ± \pm ±
Discussion
In this paper, we introduced DR-KTE-Deep, a doubly robust test of interventional distributions. We learned a deep outcome kernel by maximising an estimated power criterion, allowing the kernel to adapt to the geometry of the outcome space. We established conditional asymptotic null validity after kernel learning and selection, and a uniform finite-sample bound for the regularised learning criterion. DR-KTE-Deep achieved substantially higher empirical power than fixed and bandwidth-optimised baselines, while maintaining Type-I error close to the nominal level. Section D further motivates our choice of estimator and learning objective, and examines how outcome-model misspecification, outcome representation, ensemble size and outcome dimension affect the test. We use separate samples to learn the kernel and compute the test statistic. Future work could investigate whether reusing observations can further improve power while maintaining Type-I error control. We also restrict the test to two interventions, fixed in advance. Future work could extend our method to multiple candidate interventions.
AI Use Statement
Language models were used to generate synthetic text outcomes for our experiments, as described in Section C.3. We also used generative AI for coding assistance, primarily for plotting and running scripts. We also used generative AI for proof strategies and checking theoretical results. Finally, we used generative AI to proofread and improve writing and clarity. The synthetic text datasets were manually checked by the authors for quality. LLM-generated code was verified and tested for correctness by the authors. We take responsibility for the final content of this work, including text, claims or artifacts produced with the aid of generative AI.
References
Appendix
Appendix A recalls the main notation used throughout this paper. Appendix B presents our theoretical results and proofs. Appendix C describes the training and testing procedure, data constructions, architectures and hyperparameters. Appendix D presents additional experiments and ablations.
Notations
Data, interventions and samples
O i = ( X i , A i , Y i ) ∈ X × A × Y O_i=(X_i,A_i,Y_i)\in\mathcal{X}\times\mathcal{A}\times\mathcal{Y} O i = ( X i , A i , Y i ) ∈ X × A × Y D = { O i } i = 1 N \mathcal{D}=\{O_i\}_{i=1}^{N} D = { O i } i = 1 N
P 0 \mathbb{P}_0 P 0 P X \mathbb{P}_X P X P A , X \mathbb{P}_{A,X} P A , X E 0 \mathbb{E}_0 E 0 P 0 \mathbb{P}_0 P 0
π 0 ( a ∣ x ) \pi_0(a\mid x) π 0 ( a ∣ x ) π ( a ∣ x ) \pi(a\mid x) π ( a ∣ x ) π ′ ( a ∣ x ) \pi'(a\mid x) π ′ ( a ∣ x ) σ \sigma σ ρ \rho ρ
U ∈ U U\in\mathcal{U} U ∈ U U ⊥ ⊥ A ∣ X U\perp\!\!\!\perp A\mid X U ⊥ ⊥ A ∣ X F : A × X × U → Y F:\mathcal{A}\times\mathcal{X}\times\mathcal{U}\to\mathcal{Y} F : A × X × U → Y Y a = F ( a , X , U ) Y^a=F(a,X,U) Y a = F ( a , X , U ) Y = F ( A , X , U ) Y=F(A,X,U) Y = F ( A , X , U )
A p ∣ X ∼ p ( ⋅ ∣ X ) A^p\mid X\sim p(\cdot\mid X) A p ∣ X ∼ p ( ⋅ ∣ X ) U U U X X X Y p = F ( A p , X , U ) Y^p=F(A^p,X,U) Y p = F ( A p , X , U ) P Y p \mathbb{P}^p_Y P Y p H 0 : P Y π = P Y π ′ H_0:\mathbb{P}^{\pi}_Y=\mathbb{P}^{\pi'}_Y H 0 : P Y π = P Y π ′
P U ∣ X \mathbb{P}_{U\mid X} P U ∣ X U U U X X X Q ( B ∣ a , x ) = ∫ 1 { F ( a , x , u ) ∈ B } P U ∣ X ( d u ∣ x ) Q(B\mid a,x)=\int1\{F(a,x,u)\in B\}\mathbb{P}_{U\mid X}(\mathrm{d}u\mid x) Q ( B ∣ a , x ) = ∫ 1 { F ( a , x , u ) ∈ B } P U ∣ X ( d u ∣ x )
r ( a , x ) = π ( a ∣ x ) − π ′ ( a ∣ x ) r(a,x)=\pi(a\mid x)-\pi'(a\mid x) r ( a , x ) = π ( a ∣ x ) − π ′ ( a ∣ x ) r r r ∣ r ∣ \lvert r\rvert ∣ r ∣ r = 0 r=0 r = 0
D l e a r n \mathcal{D}_{\mathrm{learn}} D learn D v a l \mathcal{D}_{\mathrm{val}} D val D t e s t \mathcal{D}_{\mathrm{test}} D test D ( 1 ) \mathcal{D}^{(1)} D ( 1 ) D ( 2 ) \mathcal{D}^{(2)} D ( 2 ) D ( 3 ) \mathcal{D}^{(3)} D ( 3 )
n n n I 1 , I 2 \mathcal{I}_1,\mathcal{I}_2 I 1 , I 2 m m m m m m m t e s t m_{\mathrm{test}} m test
Kernels and interventional mean embeddings
k θ k_\theta k θ θ ∈ Θ ⊆ R d Θ \theta\in\Theta\subseteq\mathbb{R}^{d_\Theta} θ ∈ Θ ⊆ R d Θ H θ \mathcal{H}_\theta H θ φ θ ( y ) = k θ ( y , ⋅ ) \varphi_\theta(y)=k_\theta(y,\cdot) φ θ ( y ) = k θ ( y , ⋅ ) H θ \mathcal{H}_\theta H θ
χ θ ( p ) = ∫ φ θ ( y ) P Y p ( d y ) \chi_\theta(p)=\int\varphi_\theta(y)\mathbb{P}^p_Y(\mathrm{d}y) χ θ ( p ) = ∫ φ θ ( y ) P Y p ( d y ) p ∈ { π , π ′ } p\in\{\pi,\pi'\} p ∈ { π , π ′ }
Δ θ = χ θ ( π ) − χ θ ( π ′ ) \Delta_\theta=\chi_\theta(\pi)-\chi_\theta(\pi') Δ θ = χ θ ( π ) − χ θ ( π ′ ) D θ = ∥ Δ θ ∥ H θ 2 D_\theta=\lVert\Delta_\theta\rVert^2_{\mathcal{H}_\theta} D θ = ∥ Δ θ ∥ H θ 2 k θ k_\theta k θ H 0 H_0 H 0 D θ = 0 D_\theta=0 D θ = 0
μ θ ( a , x ) = ∫ φ θ ( y ) Q ( d y ∣ a , x ) \mu_\theta(a,x)=\int\varphi_\theta(y)Q(\mathrm{d}y\mid a,x) μ θ ( a , x ) = ∫ φ θ ( y ) Q ( d y ∣ a , x ) P A , X \mathbb{P}_{A,X} P A , X
{ k ϑ : ϑ ∈ Ξ } \{k_\vartheta:\vartheta\in\Xi\} { k ϑ : ϑ ∈ Ξ } Ξ ⊆ R d 0 \Xi\subseteq\mathbb{R}^{d_0} Ξ ⊆ R d 0 M M M θ = ( ϑ 1 , … , ϑ M ) ∈ Θ = Ξ M \theta=(\vartheta_1,\ldots,\vartheta_M)\in\Theta=\Xi^M θ = ( ϑ 1 , … , ϑ M ) ∈ Θ = Ξ M d Θ = M d 0 d_\Theta=Md_0 d Θ = M d 0
ζ ϑ b : Y → R d \zeta_{\vartheta_b}:\mathcal{Y}\to\mathbb{R}^d ζ ϑ b : Y → R d b b b κ ϑ b \kappa_{\vartheta_b} κ ϑ b
κ ϑ b c h a r \kappa^{\mathrm{char}}_{\vartheta_b} κ ϑ b char w b ∈ [ w min , 1 ] w_b\in[w_{\min},1] w b ∈ [ w m i n , 1 ] w min > 0 w_{\min}>0 w m i n > 0 k θ = M − 1 ∑ b = 1 M k ϑ b k_\theta=M^{-1}\sum_{b=1}^{M}k_{\vartheta_b} k θ = M − 1 ∑ b = 1 M k ϑ b
Nuisance estimation, kernel learning and testing
e e e m θ m_\theta m θ π ^ 0 \widehat{\pi}_0 π 0 μ ^ θ \widehat{\mu}_\theta μ θ π 0 \pi_0 π 0 μ θ \mu_\theta μ θ
g θ ( O ; e , m θ ) g_\theta(O;e,m_\theta) g θ ( O ; e , m θ ) g θ ( O ) = g θ ( O ; π 0 , μ θ ) g_\theta(O)=g_\theta(O;\pi_0,\mu_\theta) g θ ( O ) = g θ ( O ; π 0 , μ θ ) E 0 [ g θ ( O ) ] = Δ θ \mathbb{E}_0[g_\theta(O)]=\Delta_\theta E 0 [ g θ ( O )] = Δ θ
g ~ θ = g θ ( ⋅ ; π ^ 0 , μ ^ θ ) \widetilde{g}_\theta=\ g_\theta(\cdot;\widehat{\pi}_0,\widehat{\mu}_\theta) g θ = g θ ( ⋅ ; π 0 , μ θ ) g ^ θ , i = g ~ θ ( O i ) \widehat{g}_{\theta,i}=\widetilde{g}_\theta(O_i) g θ , i = g θ ( O i ) g ^ θ , ℓ = m − 1 ∑ i ∈ I ℓ g ^ θ , i \widehat{g}_{\theta,\ell}=m^{-1}\sum_{i\in\mathcal{I}_\ell}\widehat{g}_{\theta,i} g θ , ℓ = m − 1 ∑ i ∈ I ℓ g θ , i
h θ ( o , o ′ ) = ⟨ g θ ( o ) , g θ ( o ′ ) ⟩ H θ h_\theta(o,o')=\langle g_\theta(o),g_\theta(o')\rangle_{\mathcal{H}_\theta} h θ ( o , o ′ ) = ⟨ g θ ( o ) , g θ ( o ′ ) ⟩ H θ h ~ θ \widetilde{h}_\theta h θ
D ^ θ c r o s s = ⟨ g ˉ θ , 1 , g ˉ θ , 2 ⟩ H θ \widehat{D}^{\mathrm{cross}}_\theta=\langle\bar{g}_{\theta,1},\bar{g}_{\theta,2}\rangle_{\mathcal{H}_\theta} D θ cross = ⟨ g ˉ θ , 1 , g ˉ θ , 2 ⟩ H θ s ^ θ 2 \widehat{s}^2_\theta s θ 2 U θ , j = ⟨ g ˉ θ , 1 , g ^ θ , j ⟩ H θ U_{\theta,j}=\langle\bar{g}_{\theta,1},\widehat{g}_{\theta,j}\rangle_{\mathcal{H}_\theta} U θ , j = ⟨ g ˉ θ , 1 , g θ , j ⟩ H θ j ∈ I 2 j\in\mathcal{I}_2 j ∈ I 2 m m m
s θ 2 = Var P 0 ( ⟨ Δ θ , g θ ( O ) ⟩ H θ ) s^2_\theta=\operatorname{Var}_{P_0}(\langle\Delta_\theta,g_\theta(O)\rangle_{\mathcal{H}_\theta}) s θ 2 = Var P 0 (⟨ Δ θ , g θ ( O ) ⟩ H θ ) J θ = D θ / s θ J_\theta=D_\theta/s_\theta J θ = D θ / s θ s θ > 0 s_\theta>0 s θ > 0
λ > 0 \lambda>0 λ > 0 J λ , θ = D θ / s θ 2 + λ J_{\lambda,\theta}=D_\theta/\sqrt{s^2_\theta+\lambda} J λ , θ = D θ / s θ 2 + λ J ^ λ , θ D R = D ^ θ c r o s s / s ^ θ , λ \widehat{J}^{\mathrm{DR}}_{\lambda,\theta}=\widehat{D}^{\mathrm{cross}}_\theta/\widehat{s}_{\theta,\lambda} J λ , θ DR = D θ cross / s θ , λ s ^ θ , λ 2 = s ^ θ 2 + λ \widehat{s}^2_{\theta,\lambda}=\widehat{s}^2_\theta+\lambda s θ , λ 2 = s θ 2 + λ
ϑ ~ b \widetilde{\vartheta}_b ϑ b ϑ ^ b \widehat{\vartheta}_b ϑ b b b b θ ^ = ( ϑ ^ 1 , … , ϑ ^ M ) \widehat{\theta}=(\widehat{\vartheta}_1,\ldots,\widehat{\vartheta}_M) θ = ( ϑ 1 , … , ϑ M )
T θ = m D ^ θ c r o s s / s ^ θ T_\theta=\sqrt{m}\,\widehat{D}^{\mathrm{cross}}_\theta/\widehat{s}_\theta T θ = m D θ cross / s θ H 0 H_0 H 0 α \alpha α p ^ = 1 − Φ ( T θ ) < α \widehat{p}=1-\Phi(T_\theta)<\alpha p = 1 − Φ ( T θ ) < α Φ \Phi Φ
Notation for the theoretical analysis
F f i t \mathcal{F}_{\mathrm{fit}} F fit F l e a r n \mathcal{F}_{\mathrm{learn}} F learn F d a t a \mathcal{F}_{\mathrm{data}} F data
Δ ~ θ \widetilde{\Delta}_\theta Δ θ D ~ θ \widetilde{D}_\theta D θ s ~ θ 2 \widetilde{s}^2_\theta s θ 2 Δ θ \Delta_\theta Δ θ D θ D_\theta D θ s θ 2 s^2_\theta s θ 2 g ~ θ \widetilde{g}_\theta g θ
ϵ π , n \epsilon_{\pi,n} ϵ π , n ϵ μ , n \epsilon_{\mu,n} ϵ μ , n ∣ r ∣ |r| ∣ r ∣ Θ \Theta Θ
R Θ R_\Theta R Θ Θ \Theta Θ c m ( η ) c_m(\eta) c m ( η ) η ∈ ( 0 , 1 ) \eta\in(0,1) η ∈ ( 0 , 1 )
κ \kappa κ W W W A A A B μ B_\mu B μ ∣ r ∣ / π 0 |r|/\pi_0 ∣ r ∣/ π 0 L L L L h L_h L h
Θ s = { θ ∈ Θ : s θ 2 ≥ s 2 } \Theta_s=\{\theta\in\Theta:s^2_\theta\ge s^2\} Θ s = { θ ∈ Θ : s θ 2 ≥ s 2 } s > 0 s>0 s > 0 λ m \lambda_m λ m
b t e s t = E 0 [ g ^ θ ^ ( O ) ∣ F l e a r n ] b_{\mathrm{test}}=\mathbb{E}_0[\widehat{g}_{\widehat{\theta}}(O)\mid\mathcal{F}_{\mathrm{learn}}] b test = E 0 [ g θ ( O ) ∣ F learn ] H 0 H_0 H 0 ξ ( O ) = g ^ θ ^ ( O ) − b t e s t \xi(O)=\widehat{g}_{\widehat{\theta}}(O)-b_{\mathrm{test}} ξ ( O ) = g θ ( O ) − b test
a π , t e s t a_{\pi,\mathrm{test}} a π , test b μ , t e s t b_{\mu,\mathrm{test}} b μ , test ∥ b t e s t ∥ H θ ^ \|b_{\mathrm{test}}\|_{\mathcal{H}_{\widehat{\theta}}} ∥ b test ∥ H θ H 0 H_0 H 0
Theoretical Results and Proofs
We prove the results of Sections 3 and 4, then give further results on nuisance errors and the learned test under H 1 H_1 H 1 g θ g_\theta g θ O P ( 1 ) O_P(1) O P ( 1 ) o P ( 1 ) o_P(1) o P ( 1 )
Identification of Soft-Intervention Outcome Laws
We show that the observational conditional outcome law can be used to recover the outcome distribution under a soft intervention.
Proposition B.1 (Soft-intervention identification). Under Assumption 3.1, there exists a measurable probability kernel Q Q Q A × X \mathcal{A}\times\mathcal{X} A × X Y \mathcal{Y} Y Y Y Y ( A , X ) (A,X) ( A , X ) a ∈ A a\in\mathcal{A} a ∈ A Y a Y^a Y a X X X
For any conditional treatment density p p p ρ \rho ρ A p ∣ X A^p\mid X A p ∣ X p ( ⋅ ∣ X ) p(\cdot\mid X) p ( ⋅ ∣ X )
U U U X X X Y p = F ( A p , X , U ) Y^p = F(A^p, X, U) Y p = F ( A p , X , U ) B ⊆ Y B \subseteq\mathcal{Y} B ⊆ Y
P Y p ( B ) = ∫ X ∫ A Q ( B ∣ a , x ) p ( a ∣ x ) ρ ( d a ) P X ( d x ) . P_Y^p(B) = \int_{\mathcal{X}} \int_{\mathcal{A}} Q(B \mid a, x)p(a \mid x)\rho(da)P_X(dx). P Y p ( B ) = ∫ X ∫ A Q ( B ∣ a , x ) p ( a ∣ x ) ρ ( d a ) P X ( d x ) . If p = 0 p = 0 p = 0 π 0 = 0 \pi_0 = 0 π 0 = 0 P X ⊗ ρ P_X \otimes\rho P X ⊗ ρ P Y π P_Y^\pi P Y π P Y π ′ P_Y^{\pi'} P Y π ′
Proof. Let P U ∣ X ( ⋅ ∣ x ) P_{U\mid X}(\cdot\mid x) P U ∣ X ( ⋅ ∣ x ) U U U X = x X = x X = x B ⊆ Y B \subseteq\mathcal{Y} B ⊆ Y
Q ( B ∣ a , x ) : = ∫ U 1 { F ( a , x , u ) ∈ B } P U ∣ X ( d u ∣ x ) . Q(B \mid a, x) := \int_{\mathcal{U}} \mathbf{1}\{F(a,x,u) \in B\}P_{U\mid X}(du \mid x). Q ( B ∣ a , x ) := ∫ U 1 { F ( a , x , u ) ∈ B } P U ∣ X ( d u ∣ x ) . Since F F F Q Q Q
First, for any measurable C ⊆ A × X C \subseteq\mathcal{A} \times\mathcal{X} C ⊆ A × X U ⊥ ⊥ A ∣ X U \perp\!\!\!\perp A \mid X U ⊥ ⊥ A ∣ X
P 0 ( Y ∈ B , ( A , X ) ∈ C ) = ∫ C ∫ U 1 { F ( a , x , u ) ∈ B } P U ∣ X ( d u ∣ x ) P A , X ( d a , d x ) = ∫ C Q ( B ∣ a , x ) P A , X ( d a , d x ) . \begin{aligned}
\mathbb{P}_0(Y \in B, (A,X) \in C) = \int_C \int_{\mathcal{U}} \mathbf{1}\{F(a,x,u) \in B\}P_{U\mid X}(du \mid x)P_{A,X}(da,dx) \\
&= \int_C Q(B \mid a,x)P_{A,X}(da,dx).
\end{aligned} P 0 ( Y ∈ B , ( A , X ) ∈ C ) = ∫ C ∫ U 1 { F ( a , x , u ) ∈ B } P U ∣ X ( d u ∣ x ) P A , X ( d a , d x ) = ∫ C Q ( B ∣ a , x ) P A , X ( d a , d x ) . Hence Q Q Q Y Y Y ( A , X ) (A,X) ( A , X ) a ∈ A a \in\mathcal{A} a ∈ A Y a = F ( a , X , U ) Y^a = F(a,X,U) Y a = F ( a , X , U ) D ⊆ X D \subseteq\mathcal{X} D ⊆ X
P ( Y a ∈ B , X ∈ D ) = ∫ D Q ( B ∣ a , x ) P X ( d x ) . \mathbb{P}(Y^a \in B, X \in D) = \int_D Q(B \mid a,x)P_X(dx). P ( Y a ∈ B , X ∈ D ) = ∫ D Q ( B ∣ a , x ) P X ( d x ) . Thus Q ( ⋅ ∣ a , X ) Q(\cdot\mid a,X) Q ( ⋅ ∣ a , X ) Y a Y^a Y a X X X
We next consider a soft intervention p p p A p A^p A p p ( ⋅ ∣ X ) p(\cdot\mid X) p ( ⋅ ∣ X ) U U U X X X ( X , U ) (X,U) ( X , U )
P Y p ( B ) = E [ ∫ A 1 { F ( a , X , U ) ∈ B } p ( a ∣ X ) ρ ( d a ) ] = ∫ X ∫ A Q ( B ∣ a , x ) p ( a ∣ x ) ρ ( d a ) P X ( d x ) , \begin{aligned}
P_Y^p(B) = \mathbb{E}\left[\int_{\mathcal{A}} \mathbf{1}\{F(a,X,U) \in B\}p(a \mid X)\rho(da)\right] \\
&= \int_{\mathcal{X}} \int_{\mathcal{A}} Q(B \mid a,x)p(a \mid x)\rho(da)P_X(dx),
\end{aligned} P Y p ( B ) = E [ ∫ A 1 { F ( a , X , U ) ∈ B } p ( a ∣ X ) ρ ( d a ) ] = ∫ X ∫ A Q ( B ∣ a , x ) p ( a ∣ x ) ρ ( d a ) P X ( d x ) , where the second equality follows by conditioning on X X X
Finally, P A , X ( d a , d x ) = π 0 ( a ∣ x ) ρ ( d a ) P X ( d x ) P_{A,X}(da,dx) = \pi_0(a \mid x)\rho(da)P_X(dx) P A , X ( d a , d x ) = π 0 ( a ∣ x ) ρ ( d a ) P X ( d x ) p = 0 p = 0 p = 0 π 0 = 0 \pi_0 = 0 π 0 = 0 P X ⊗ ρ P_X \otimes\rho P X ⊗ ρ ( A p , X ) (A^p, X) ( A p , X ) P A , X P_{A,X} P A , X B B B Q ( B ∣ a , x ) Q(B \mid a,x) Q ( B ∣ a , x ) P A , X P_{A,X} P A , X π \pi π π ′ \pi' π ′
Mean embeddings and integrability. For fixed θ \theta θ μ θ ( a , x ) = ∫ φ θ ( y ) Q ( d y ∣ a , x ) \mu_\theta(a,x) = \int\varphi_\theta(y)Q(dy \mid a,x) μ θ ( a , x ) = ∫ φ θ ( y ) Q ( d y ∣ a , x ) ∫ k θ ( y , y ) Q ( d y ∣ a , x ) < ∞ \int\sqrt{k_\theta(y,y)}Q(dy \mid a,x) < \infty ∫ k θ ( y , y ) Q ( d y ∣ a , x ) < ∞ P A , X P_{A,X} P A , X μ θ \mu_\theta μ θ μ θ ( A , X ) \mu_\theta(A,X) μ θ ( A , X )
χ θ ( p ) = ∫ X ∫ A μ θ ( a , x ) p ( a ∣ x ) ρ ( d a ) P X ( d x ) . \chi_\theta(p) = \int_{\mathcal{X}} \int_{\mathcal{A}} \mu_\theta(a,x)p(a \mid x)\rho(da)P_X(dx). χ θ ( p ) = ∫ X ∫ A μ θ ( a , x ) p ( a ∣ x ) ρ ( d a ) P X ( d x ) . The observed-law first moment in Assumption 3.2 is needed for the observational conditional mean, since target moments need not control actions outside the target support. Uniformly bounded kernels satisfy all these moment conditions.
For finite or countable A \mathcal{A} A ρ \rho ρ
Observed-Data Representation and Double Robustness
For Proposition 3.3 , the candidate nuisances are assumed to satisfy
E 0 [ ∣ r ( A , X ) ∣ e ( A ∣ X ) ∥ φ θ ( Y ) − m θ ( A , X ) ∥ H θ ] < ∞ , \mathbb{E}_0\left[\frac{\lvert r(A,X)\rvert}{e(A\mid X)}\left\|\varphi_\theta(Y)-m_\theta(A,X)\right\|_{\mathcal{H}_\theta}\right]<\infty, E 0 [ e ( A ∣ X ) ∣ r ( A , X )∣ ∥ φ θ ( Y ) − m θ ( A , X ) ∥ H θ ] < ∞ , E 0 [ ∫ A ∣ r ( a , X ) ∣ ∥ m θ ( a , X ) ∥ H θ ρ ( d a ) ] < ∞ . \mathbb{E}_0\left[\int_{\mathcal{A}}\lvert r(a,X)\rvert\left\|m_\theta(a,X)\right\|_{\mathcal{H}_\theta}\rho(\mathrm{d}a)\right]<\infty. E 0 [ ∫ A ∣ r ( a , X )∣ ∥ m θ ( a , X ) ∥ H θ ρ ( d a ) ] < ∞. These conditions make both terms of the pseudo-feature Bochner integrable and justify the conditional integration below. The target moments also ensure E 0 ∫ A ∣ r ( a , X ) ∣ ∥ μ θ ( a , X ) ∥ H θ ρ ( d a ) < ∞ \mathbb{E}_0\int_{\mathcal{A}}\lvert r(a,X)\rvert\left\|\mu_\theta(a,X)\right\|_{\mathcal{H}_\theta}\rho(\mathrm{d}a)<\infty E 0 ∫ A ∣ r ( a , X )∣ ∥ μ θ ( a , X ) ∥ H θ ρ ( d a ) < ∞
Proof of Proposition 3.3. Recall that r ( a , x ) = π ( a ∣ x ) − π ′ ( a ∣ x ) r(a,x)=\pi(a\mid x)-\pi'(a\mid x) r ( a , x ) = π ( a ∣ x ) − π ′ ( a ∣ x ) § 3 , ratios and integrands weighted by r r r ∣ r ∣ \lvert r\rvert ∣ r ∣ r = 0 r=0 r = 0
Under Assumptions 3.1 and 3.2, Proposition B.1 and Bochner Fubini identify each target embedding through the common response kernel. Subtracting the two embeddings gives
Δ θ ( π , π ′ ) = E P 0 [ ∫ A r ( a , X ) μ θ ( a , X ) ρ ( d a ) ] . \Delta_\theta(\pi,\pi')=\mathbb{E}_{P_0}\left[\int_{\mathcal{A}}r(a,X)\mu_\theta(a,X)\rho(\mathrm{d}a)\right]. Δ θ ( π , π ′ ) = E P 0 [ ∫ A r ( a , X ) μ θ ( a , X ) ρ ( d a ) ] . The weight r ( A , X ) / e ( A ∣ X ) r(A,X)/e(A\mid X) r ( A , X ) / e ( A ∣ X ) ( A , X ) (A,X) ( A , X ) μ θ ( A , X ) \mu_\theta(A,X) μ θ ( A , X )
E P 0 [ r ( A , X ) e ( A ∣ X ) { φ θ ( Y ) − m θ ( A , X ) } ∣ A , X ] = r ( A , X ) e ( A ∣ X ) { μ θ ( A , X ) − m θ ( A , X ) } . \mathbb{E}_{P_0}\left[\left.\frac{r(A,X)}{e(A\mid X)}\{\varphi_\theta(Y)-m_\theta(A,X)\}\right|A,X\right]=\frac{r(A,X)}{e(A\mid X)}\{\mu_\theta(A,X)-m_\theta(A,X)\}. E P 0 [ e ( A ∣ X ) r ( A , X ) { φ θ ( Y ) − m θ ( A , X )} A , X ] = e ( A ∣ X ) r ( A , X ) { μ θ ( A , X ) − m θ ( A , X )} . Averaging over A ∣ X A\mid X A ∣ X π 0 \pi_0 π 0
E P 0 [ g θ π , π ′ ( O ; e , m θ ) ] = E P 0 [ ∫ A π 0 ( a ∣ X ) r ( a , X ) e ( a ∣ X ) { μ θ ( a , X ) − m θ ( a , X ) } ρ ( d a ) ] + E P 0 [ ∫ A r ( a , X ) m θ ( a , X ) ρ ( d a ) ] . \mathbb{E}_{P_0}\left[g_\theta^{\pi,\pi'}(O;e,m_\theta)\right]
=\mathbb{E}_{P_0}\left[\int_{\mathcal{A}}\pi_0(a\mid X)\frac{r(a,X)}{e(a\mid X)}\{\mu_\theta(a,X)-m_\theta(a,X)\}\rho(\mathrm{d}a)\right]
+\mathbb{E}_{P_0}\left[\int_{\mathcal{A}}r(a,X)m_\theta(a,X)\rho(\mathrm{d}a)\right]. E P 0 [ g θ π , π ′ ( O ; e , m θ ) ] = E P 0 [ ∫ A π 0 ( a ∣ X ) e ( a ∣ X ) r ( a , X ) { μ θ ( a , X ) − m θ ( a , X )} ρ ( d a ) ] + E P 0 [ ∫ A r ( a , X ) m θ ( a , X ) ρ ( d a ) ] . We now subtract the expression for Δ θ ( π , π ′ ) \Delta_\theta(\pi,\pi') Δ θ ( π , π ′ )
E P 0 [ g θ π , π ′ ( O ; e , m θ ) ] − Δ θ ( π , π ′ ) = E P 0 [ ∫ A r ( a , X ) { π 0 ( a ∣ X ) e ( a ∣ X ) − 1 } { μ θ ( a , X ) − m θ ( a , X ) } ρ ( d a ) ] . \mathbb{E}_{P_0}\left[g_\theta^{\pi,\pi'}(O;e,m_\theta)\right]-\Delta_\theta(\pi,\pi')
=\mathbb{E}_{P_0}\left[\int_{\mathcal{A}}r(a,X)\left\{\frac{\pi_0(a\mid X)}{e(a\mid X)}-1\right\}\{\mu_\theta(a,X)-m_\theta(a,X)\}\rho(\mathrm{d}a)\right]. E P 0 [ g θ π , π ′ ( O ; e , m θ ) ] − Δ θ ( π , π ′ ) = E P 0 [ ∫ A r ( a , X ) { e ( a ∣ X ) π 0 ( a ∣ X ) − 1 } { μ θ ( a , X ) − m θ ( a , X )} ρ ( d a ) ] . This expression is zero if either e = π 0 e=\pi_0 e = π 0 m θ = μ θ m_\theta=\mu_\theta m θ = μ θ ∣ r ∣ > 0 \lvert r\rvert>0 ∣ r ∣ > 0
E P 0 [ g θ π , π ′ ( O ; e , m θ ) ] = Δ θ ( π , π ′ ) . \mathbb{E}_{P_0}\left[g_\theta^{\pi,\pi'}(O;e,m_\theta)\right]=\Delta_\theta(\pi,\pi'). E P 0 [ g θ π , π ′ ( O ; e , m θ ) ] = Δ θ ( π , π ′ ) . Using the true nuisance functions gives E P 0 [ g θ π , π ′ ( O ) ] = Δ θ ( π , π ′ ) \mathbb{E}_{P_0}[g_\theta^{\pi,\pi'}(O)]=\Delta_\theta(\pi,\pi') E P 0 [ g θ π , π ′ ( O )] = Δ θ ( π , π ′ ) □ \square □
Remark B.2. The representation above covers continuous actions for soft interventions π , π ′ \pi, \pi^{\prime} π , π ′ [12 , 68 , 4 ] ). This remains true when the outcome kernel is learned.
Fixed-Alternative Growth Rate
Proof of Proposition 4.1. Fix θ \theta θ H θ \mathcal{H}_{\theta} H θ g ^ θ , i = g θ π , π ′ ( O i ) \widehat{g}_{\theta,i}=g_{\theta}^{\pi,\pi^{\prime}}(O_i) g θ , i = g θ π , π ′ ( O i )
g ‾ θ , ℓ = 1 m ∑ i ∈ I ℓ g θ π , π ′ ( O i ) , ℓ ∈ { 1 , 2 } . \overline{g}_{\theta,\ell}=\frac{1}{m}\sum_{i\in\mathcal{I}_{\ell}}g_{\theta}^{\pi,\pi^{\prime}}(O_i),\qquad\ell\in\{1,2\}. g θ , ℓ = m 1 i ∈ I ℓ ∑ g θ π , π ′ ( O i ) , ℓ ∈ { 1 , 2 } . By Proposition 3.3, E P 0 [ g θ π , π ′ ( O i ) ] = Δ θ ( π , π ′ ) \mathbb{E}_{P_0}[g_{\theta}^{\pi,\pi^{\prime}}(O_i)]=\Delta_{\theta}(\pi,\pi^{\prime}) E P 0 [ g θ π , π ′ ( O i )] = Δ θ ( π , π ′ ) g ‾ θ , ℓ → P Δ θ ( π , π ′ ) \overline{g}_{\theta,\ell}\xrightarrow{\mathbb{P}}\Delta_{\theta}(\pi,\pi^{\prime}) g θ , ℓ P Δ θ ( π , π ′ ) ℓ ∈ { 1 , 2 } \ell\in\{1,2\} ℓ ∈ { 1 , 2 }
D ^ θ c r o s s = ⟨ g ‾ θ , 1 , g ‾ θ , 2 ⟩ → P ∥ Δ θ ( π , π ′ ) ∥ 2 = D θ . \widehat{D}_{\theta}^{\mathrm{cross}}=\langle\overline{g}_{\theta,1},\overline{g}_{\theta,2}\rangle\xrightarrow{\mathbb{P}}\|\Delta_{\theta}(\pi,\pi^{\prime})\|^{2}=D_{\theta}. D θ cross = ⟨ g θ , 1 , g θ , 2 ⟩ P ∥ Δ θ ( π , π ′ ) ∥ 2 = D θ . For the variance estimator, define the real-valued variables
V j : = ⟨ g ‾ θ , 1 , g θ π , π ′ ( O j ) ⟩ , W j : = ⟨ Δ θ ( π , π ′ ) , g θ π , π ′ ( O j ) ⟩ , j ∈ I 2 . V_j:=\langle\overline{g}_{\theta,1},g_{\theta}^{\pi,\pi^{\prime}}(O_j)\rangle,\qquad W_j:=\langle\Delta_{\theta}(\pi,\pi^{\prime}),g_{\theta}^{\pi,\pi^{\prime}}(O_j)\rangle,\qquad j\in\mathcal{I}_2. V j := ⟨ g θ , 1 , g θ π , π ′ ( O j )⟩ , W j := ⟨ Δ θ ( π , π ′ ) , g θ π , π ′ ( O j )⟩ , j ∈ I 2 . The V j V_j V j D ^ θ c r o s s \widehat{D}_{\theta}^{\mathrm{cross}} D θ cross
s ^ θ 2 = 1 m ∑ j ∈ I 2 V j 2 − ( D ^ θ c r o s s ) 2 . \widehat{s}_{\theta}^{2}=\frac{1}{m}\sum_{j\in\mathcal{I}_2}V_j^{2}-\left(\widehat{D}_{\theta}^{\mathrm{cross}}\right)^{2}. s θ 2 = m 1 j ∈ I 2 ∑ V j 2 − ( D θ cross ) 2 . We have already shown that ( D ^ θ c r o s s ) 2 → P D θ 2 \left(\widehat{D}_{\theta}^{\mathrm{cross}}\right)^{2}\xrightarrow{\mathbb{P}}D_{\theta}^{2} ( D θ cross ) 2 P D θ 2 1 m ∑ j ∈ I 2 V j 2 \frac{1}{m}\sum_{j\in\mathcal{I}_2}V_j^{2} m 1 ∑ j ∈ I 2 V j 2 V j V_j V j W j W_j W j ∣ u 2 − w 2 ∣ = ∣ u − w ∣ ∣ u + w ∣ |u^{2}-w^{2}|=|u-w||u+w| ∣ u 2 − w 2 ∣ = ∣ u − w ∣∣ u + w ∣
∣ 1 m ∑ j ∈ I 2 ( V j 2 − W j 2 ) ∣ ≤ ∥ g ‾ θ , 1 − Δ θ ( π , π ′ ) ∥ ( ∥ g ‾ θ , 1 ∥ + ∥ Δ θ ( π , π ′ ) ∥ ) 1 m ∑ j ∈ I 2 ∥ g θ π , π ′ ( O j ) ∥ 2 = o P ( 1 ) . \left|\frac{1}{m}\sum_{j\in\mathcal{I}_2}(V_j^{2}-W_j^{2})\right|
\leq\|\overline{g}_{\theta,1}-\Delta_{\theta}(\pi,\pi^{\prime})\|\left(\|\overline{g}_{\theta,1}\|+\|\Delta_{\theta}(\pi,\pi^{\prime})\|\right)\frac{1}{m}\sum_{j\in\mathcal{I}_2}\|g_{\theta}^{\pi,\pi^{\prime}}(O_j)\|^{2}
=o_{\mathbb{P}}(1). m 1 j ∈ I 2 ∑ ( V j 2 − W j 2 ) ≤ ∥ g θ , 1 − Δ θ ( π , π ′ ) ∥ ( ∥ g θ , 1 ∥ + ∥ Δ θ ( π , π ′ ) ∥ ) m 1 j ∈ I 2 ∑ ∥ g θ π , π ′ ( O j ) ∥ 2 = o P ( 1 ) . Here the first factor converges to zero, the second is bounded in probability, and the last converges to E P 0 ∥ g θ π , π ′ ( O ) ∥ 2 < ∞ \mathbb{E}_{P_0}\|g_{\theta}^{\pi,\pi^{\prime}}(O)\|^{2}<\infty E P 0 ∥ g θ π , π ′ ( O ) ∥ 2 < ∞ W j 2 W_j^{2} W j 2
1 m ∑ j ∈ I 2 W j 2 → P E P 0 [ ⟨ Δ θ ( π , π ′ ) , g θ π , π ′ ( O ) ⟩ 2 ] . \frac{1}{m}\sum_{j\in\mathcal{I}_2}W_j^{2}\xrightarrow{\mathbb{P}}\mathbb{E}_{P_0}\left[\langle\Delta_{\theta}(\pi,\pi^{\prime}),g_{\theta}^{\pi,\pi^{\prime}}(O)\rangle^{2}\right]. m 1 j ∈ I 2 ∑ W j 2 P E P 0 [ ⟨ Δ θ ( π , π ′ ) , g θ π , π ′ ( O ) ⟩ 2 ] . We also have E P 0 [ W j ] = ⟨ Δ θ ( π , π ′ ) , Δ θ ( π , π ′ ) ⟩ = D θ \mathbb{E}_{P_0}[W_j]=\langle\Delta_{\theta}(\pi,\pi^{\prime}),\Delta_{\theta}(\pi,\pi^{\prime})\rangle=D_{\theta} E P 0 [ W j ] = ⟨ Δ θ ( π , π ′ ) , Δ θ ( π , π ′ )⟩ = D θ
s ^ θ 2 → P E P 0 [ ⟨ Δ θ ( π , π ′ ) , g θ π , π ′ ( O ) ⟩ 2 ] − D θ 2 = s θ 2 . \widehat{s}_{\theta}^{2}\xrightarrow{\mathbb{P}}\mathbb{E}_{P_0}\left[\langle\Delta_{\theta}(\pi,\pi^{\prime}),g_{\theta}^{\pi,\pi^{\prime}}(O)\rangle^{2}\right]-D_{\theta}^{2}=s_{\theta}^{2}. s θ 2 P E P 0 [ ⟨ Δ θ ( π , π ′ ) , g θ π , π ′ ( O ) ⟩ 2 ] − D θ 2 = s θ 2 . Since s θ > 0 s_{\theta}>0 s θ > 0 s ^ θ → P s θ \widehat{s}_{\theta}\xrightarrow{\mathbb{P}}s_{\theta} s θ P s θ (2) and Slutsky’s theorem,
T θ m = D ^ θ c r o s s s ^ θ → P D θ s θ = J θ . \frac{T_{\theta}}{\sqrt{m}}=\frac{\widehat{D}_{\theta}^{\mathrm{cross}}}{\widehat{s}_{\theta}}\xrightarrow{\mathbb{P}}\frac{D_{\theta}}{s_{\theta}}=J_{\theta}. m T θ = s θ D θ cross P s θ D θ = J θ . We set T θ = 0 T_{\theta}=0 T θ = 0 s ^ θ = 0 \widehat{s}_{\theta}=0 s θ = 0
Characteristic Ensemble Safeguard
This result is the analogue of Liu et al. [33 ] . Their kernel in Eq. (1) , however, additionally factors the characteristic component and the deep neural component.
Proof of Lemma 4.2. Let Y , Y ′ ∼ P Y,Y' \sim P Y , Y ′ ∼ P Z , Z ′ ∼ Q Z,Z' \sim Q Z , Z ′ ∼ Q
MMD k θ 2 ( P , Q ) = E [ k ( Y , Y ′ ) ] + E [ k ( Z , Z ′ ) ] − 2 E [ k ( Y , Z ) ] . \operatorname{MMD}_{k_\theta}^{2}(P,Q)=\mathbb{E}[k(Y,Y')]+\mathbb{E}[k(Z,Z')]-2\mathbb{E}[k(Y,Z)]. MMD k θ 2 ( P , Q ) = E [ k ( Y , Y ′ )] + E [ k ( Z , Z ′ )] − 2 E [ k ( Y , Z )] . The squared MMD is therefore linear in the kernel. For each ensemble member b b b
k ϑ b d e e p ( y , y ′ ) : = κ ϑ b ( ζ ϑ b ( y ) , ζ ϑ b ( y ′ ) ) . k_{\vartheta_b}^{\mathrm{deep}}(y,y'):=\kappa_{\vartheta_b}\left(\zeta_{\vartheta_b}(y),\zeta_{\vartheta_b}(y')\right). k ϑ b deep ( y , y ′ ) := κ ϑ b ( ζ ϑ b ( y ) , ζ ϑ b ( y ′ ) ) . Substituting the ensemble kernel form in (4) yields
MMD k θ 2 ( P , Q ) = 1 M ∑ b = 1 M [ ( 1 − w b ) MMD k ϑ b d e e p 2 ( P , Q ) + w b MMD k ϑ b c h a r 2 ( P , Q ) ] . \operatorname{MMD}_{k_\theta}^{2}(P,Q)=\frac{1}{M}\sum_{b=1}^{M}\left[(1-w_b)\operatorname{MMD}_{k_{\vartheta_b}^{\mathrm{deep}}}^{2}(P,Q)+w_b\operatorname{MMD}_{k_{\vartheta_b}^{\mathrm{char}}}^{2}(P,Q)\right]. MMD k θ 2 ( P , Q ) = M 1 b = 1 ∑ M [ ( 1 − w b ) MMD k ϑ b deep 2 ( P , Q ) + w b MMD k ϑ b char 2 ( P , Q ) ] . Every squared MMD is nonnegative, and 1 − w b ≥ 0 1-w_b\ge0 1 − w b ≥ 0 w b ≥ w min w_b\ge w_{\min} w b ≥ w m i n
MMD k θ 2 ( P , Q ) ≥ w min M ∑ b = 1 M MMD k ϑ b c h a r 2 ( P , Q ) . \operatorname{MMD}_{k_\theta}^{2}(P,Q)\ge\frac{w_{\min}}{M}\sum_{b=1}^{M}\operatorname{MMD}_{k_{\vartheta_b}^{\mathrm{char}}}^{2}(P,Q). MMD k θ 2 ( P , Q ) ≥ M w m i n b = 1 ∑ M MMD k ϑ b char 2 ( P , Q ) . Each base kernel is characteristic and w min > 0 w_{\min}>0 w m i n > 0 P ≠ Q P\ne Q P = Q P = Q P=Q P = Q k θ k_\theta k θ
Memberwise Ensemble Aggregation
Proposition B.3 (Memberwise ensemble aggregation). Suppose Assumptions 3.1 and 3.2 hold. Let k ϑ 1 , … , k ϑ M k_{\vartheta_1},\ldots,k_{\vartheta_M} k ϑ 1 , … , k ϑ M k θ = M − 1 ∑ b k ϑ b k_\theta=M^{-1}\sum_b k_{\vartheta_b} k θ = M − 1 ∑ b k ϑ b g ϑ b = g ϑ b ( ⋅ ; π 0 , μ ϑ b ) g_{\vartheta_b}=g_{\vartheta_b}(\cdot;\pi_0,\mu_{\vartheta_b}) g ϑ b = g ϑ b ( ⋅ ; π 0 , μ ϑ b ) Q Q Q ⟨ Δ ϑ b , g ϑ b ( O ) ⟩ H ϑ b \langle\Delta_{\vartheta_b},g_{\vartheta_b}(O)\rangle_{\mathcal{H}_{\vartheta_b}} ⟨ Δ ϑ b , g ϑ b ( O ) ⟩ H ϑ b λ ≥ 0 \lambda\ge0 λ ≥ 0 J λ , ϑ b = D ϑ b / s ϑ b 2 + λ J_{\lambda,\vartheta_b}=D_{\vartheta_b}/\sqrt{s_{\vartheta_b}^{2}+\lambda} J λ , ϑ b = D ϑ b / s ϑ b 2 + λ
D θ = 1 M ∑ b = 1 M D ϑ b , s θ 2 + λ ≤ 1 M ∑ b = 1 M s ϑ b 2 + λ . D_\theta=\frac{1}{M}\sum_{b=1}^{M}D_{\vartheta_b},\qquad
\sqrt{s_\theta^{2}+\lambda}\le\frac{1}{M}\sum_{b=1}^{M}\sqrt{s_{\vartheta_b}^{2}+\lambda}. D θ = M 1 b = 1 ∑ M D ϑ b , s θ 2 + λ ≤ M 1 b = 1 ∑ M s ϑ b 2 + λ . If all member and aggregate denominators are positive, then
J λ , θ ≥ ∑ b = 1 M D ϑ b ∑ b = 1 M s ϑ b 2 + λ ≥ min 1 ≤ b ≤ M J λ , ϑ b . J_{\lambda,\theta}\ge\frac{\sum_{b=1}^{M}D_{\vartheta_b}}{\sum_{b=1}^{M}\sqrt{s_{\vartheta_b}^{2}+\lambda}}\ge\min_{1\le b\le M}J_{\lambda,\vartheta_b}. J λ , θ ≥ ∑ b = 1 M s ϑ b 2 + λ ∑ b = 1 M D ϑ b ≥ 1 ≤ b ≤ M min J λ , ϑ b . For learned parameters θ ^ = ( ϑ ^ 1 , … , ϑ ^ M ) \hat{\theta}=(\hat{\vartheta}_1,\ldots,\hat{\vartheta}_M) θ ^ = ( ϑ ^ 1 , … , ϑ ^ M ) F d a t a \mathcal{F}_{\mathrm{data}} F data F d a t a \mathcal{F}_{\mathrm{data}} F data D ϑ ^ b D_{\hat{\vartheta}_b} D ϑ ^ b
Var ( D θ ^ ∣ F d a t a ) = 1 M 2 ∑ b = 1 M Var ( D ϑ ^ b ∣ F d a t a ) . \operatorname{Var}(D_{\hat{\theta}}\mid\mathcal{F}_{\mathrm{data}})=\frac{1}{M^{2}}\sum_{b=1}^{M}\operatorname{Var}(D_{\hat{\vartheta}_b}\mid\mathcal{F}_{\mathrm{data}}). Var ( D θ ^ ∣ F data ) = M 2 1 b = 1 ∑ M Var ( D ϑ ^ b ∣ F data ) . If the members are also conditionally identically distributed, this equals M − 1 Var ( D ϑ ^ 1 ∣ F d a t a ) M^{-1}\operatorname{Var}(D_{\hat{\vartheta}_1}\mid\mathcal{F}_{\mathrm{data}}) M − 1 Var ( D ϑ ^ 1 ∣ F data )
Proof. In the Hilbert direct sum ⨁ b = 1 M H ϑ b \bigoplus_{b=1}^{M}\mathcal{H}_{\vartheta_b} ⨁ b = 1 M H ϑ b
φ θ ( y ) = M − 1 / 2 ( φ ϑ 1 ( y ) , … , φ ϑ M ( y ) ) . \varphi_{\theta}(y)=M^{-1/2}\left(\varphi_{\vartheta_1}(y),\ldots,\varphi_{\vartheta_M}(y)\right). φ θ ( y ) = M − 1/2 ( φ ϑ 1 ( y ) , … , φ ϑ M ( y ) ) . Linearity of Bochner integration gives Δ θ = M − 1 / 2 ( Δ ϑ 1 , … , Δ ϑ M ) \Delta_{\theta}=M^{-1/2}(\Delta_{\vartheta_1},\ldots,\Delta_{\vartheta_M}) Δ θ = M − 1/2 ( Δ ϑ 1 , … , Δ ϑ M ) π 0 \pi_0 π 0 Q Q Q (1) therefore gives g θ ( O ) = M − 1 / 2 ( g ϑ 1 ( O ) , … , g ϑ M ( O ) ) g_{\theta}(O)=M^{-1/2}(g_{\vartheta_1}(O),\ldots,g_{\vartheta_M}(O)) g θ ( O ) = M − 1/2 ( g ϑ 1 ( O ) , … , g ϑ M ( O ))
⟨ Δ θ , g θ ( O ) ⟩ H θ = 1 M ∑ b = 1 M ⟨ Δ ϑ b , g ϑ b ( O ) ⟩ H ϑ b . \langle\Delta_{\theta},g_{\theta}(O)\rangle_{\mathcal{H}_{\theta}}=\frac{1}{M}\sum_{b=1}^{M}\langle\Delta_{\vartheta_b},g_{\vartheta_b}(O)\rangle_{\mathcal{H}_{\vartheta_b}}. ⟨ Δ θ , g θ ( O ) ⟩ H θ = M 1 b = 1 ∑ M ⟨ Δ ϑ b , g ϑ b ( O ) ⟩ H ϑ b . Let Z b : = ⟨ Δ ϑ b , g ϑ b ( O ) ⟩ H ϑ b − D ϑ b Z_b:=\langle\Delta_{\vartheta_b},g_{\vartheta_b}(O)\rangle_{\mathcal{H}_{\vartheta_b}}-D_{\vartheta_b} Z b := ⟨ Δ ϑ b , g ϑ b ( O ) ⟩ H ϑ b − D ϑ b E 0 Z b = 0 \mathbb{E}_0Z_b=0 E 0 Z b = 0 s ϑ b = ∥ Z b ∥ L 2 ( P 0 ) s_{\vartheta_b}=\|Z_b\|_{L^2(\mathbb{P}_0)} s ϑ b = ∥ Z b ∥ L 2 ( P 0 ) s θ = ∥ M − 1 ∑ b Z b ∥ L 2 ( P 0 ) s_{\theta}=\|M^{-1}\sum_b Z_b\|_{L^2(\mathbb{P}_0)} s θ = ∥ M − 1 ∑ b Z b ∥ L 2 ( P 0 ) L 2 ( P 0 ) ⊕ R L^2(\mathbb{P}_0)\oplus\mathbb{R} L 2 ( P 0 ) ⊕ R
s θ 2 + λ = ∥ 1 M ∑ b = 1 M ( Z b , λ ) ∥ ≤ 1 M ∑ b = 1 M ∥ ( Z b , λ ) ∥ = 1 M ∑ b = 1 M s ϑ b 2 + λ . \sqrt{s_{\theta}^2+\lambda}=\left\|\frac{1}{M}\sum_{b=1}^{M}(Z_b,\sqrt{\lambda})\right\|\leq\frac{1}{M}\sum_{b=1}^{M}\|(Z_b,\sqrt{\lambda})\|=\frac{1}{M}\sum_{b=1}^{M}\sqrt{s_{\vartheta_b}^2+\lambda}. s θ 2 + λ = M 1 b = 1 ∑ M ( Z b , λ ) ≤ M 1 b = 1 ∑ M ∥ ( Z b , λ ) ∥ = M 1 b = 1 ∑ M s ϑ b 2 + λ . Since D ϑ b ≥ 0 D_{\vartheta_b}\geq0 D ϑ b ≥ 0 s ϑ b 2 + λ \sqrt{s_{\vartheta_b}^2+\lambda} s ϑ b 2 + λ 1 / M 1/M 1/ M
Held-Out Null Validity
The sample-splitting principle detailed below also appears in Gretton et al. [20 ] ; Liu et al. [33 ] . Here we additionally control bias from nuisance estimation.
Let F l e a r n \mathcal{F}_{\mathrm{learn}} F learn F l e a r n \mathcal{F}_{\mathrm{learn}} F learn g ^ θ ^ ( O ) : = g θ ^ π ^ , π ^ ′ ( O ; π ^ 0 , μ ^ θ ^ ) \hat{g}_{\hat{\theta}}(O):=g^{\hat{\pi},\hat{\pi}'}_{\hat{\theta}}(O;\hat{\pi}_0,\hat{\mu}_{\hat{\theta}}) g ^ θ ^ ( O ) := g θ ^ π ^ , π ^ ′ ( O ; π ^ 0 , μ ^ θ ^ ) m m m O ∼ P 0 O\sim\mathbb{P}_0 O ∼ P 0 F l e a r n \mathcal{F}_{\mathrm{learn}} F learn
b t e s t : = E P 0 [ g ^ θ ^ ( O ) ∣ F l e a r n ] , b_{\mathrm{test}}:=\mathbb{E}_{\mathbb{P}_0}[\hat{g}_{\hat{\theta}}(O)\mid\mathcal{F}_{\mathrm{learn}}], b test := E P 0 [ g ^ θ ^ ( O ) ∣ F learn ] , ξ ( O ) : = g ^ θ ^ ( O ) − b t e s t , ξ ˉ ℓ : = 1 m ∑ i ∈ I ℓ ξ ( O i ) , ℓ ∈ { 1 , 2 } . \xi(O):=\hat{g}_{\hat{\theta}}(O)-b_{\mathrm{test}},\qquad\bar{\xi}_{\ell}:=\frac{1}{m}\sum_{i\in I_{\ell}}\xi(O_i),\qquad\ell\in\{1,2\}. ξ ( O ) := g ^ θ ^ ( O ) − b test , ξ ˉ ℓ := m 1 i ∈ I ℓ ∑ ξ ( O i ) , ℓ ∈ { 1 , 2 } . b t e s t b_{\mathrm{test}} b test H 0 H_0 H 0
Let G m \mathcal{G}_m G m F l e a r n \mathcal{F}_{\mathrm{learn}} F learn ξ ˉ 1 \bar{\xi}_1 ξ ˉ 1 G m \mathcal{G}_m G m O ∼ P 0 O\sim\mathbb{P}_0 O ∼ P 0 G m \mathcal{G}_m G m
V ( O ) : = ⟨ ξ ˉ 1 , ξ ( O ) ⟩ H θ ^ , v t e s t : = E P 0 [ V ( O ) 2 ∣ G m ] . V(O):=\langle\bar{\xi}_1,\xi(O)\rangle_{\mathcal{H}_{\hat{\theta}}},\qquad v_{\mathrm{test}}:=\mathbb{E}_{\mathbb{P}_0}[V(O)^2\mid\mathcal{G}_m]. V ( O ) := ⟨ ξ ˉ 1 , ξ ( O ) ⟩ H θ ^ , v test := E P 0 [ V ( O ) 2 ∣ G m ] . Since V ( O ) V(O) V ( O ) v t e s t v_{\mathrm{test}} v test G m \mathcal{G}_m G m
Assumption B.4 (Held-out regularity). Let m → ∞ m\to\infty m → ∞ P 0 \mathbb{P}_0 P 0 F l e a r n \mathcal{F}_{\mathrm{learn}} F learn g ^ θ ^ \hat{g}_{\hat{\theta}} g ^ θ ^ H 0 H_0 H 0
The conditional fourth moment of the centred pseudo-feature is bounded in probability:
E P 0 [ ∥ ξ ( O ) ∥ H θ ^ 4 ∣ F l e a r n ] = O P ( 1 ) . \mathbb{E}_{\mathbb{P}_0}\left[\|\xi(O)\|_{\mathcal{H}_{\hat{\theta}}}^{4}\mid\mathcal{F}_{\mathrm{learn}}\right]=O_{\mathbb{P}}(1). E P 0 [ ∥ ξ ( O ) ∥ H θ ^ 4 ∣ F learn ] = O P ( 1 ) . The projected variance is nonzero with probability tending to one, P ( v t e s t > 0 ) → 1 \mathbb{P}(v_{\mathrm{test}}>0)\to1 P ( v test > 0 ) → 1
( m v t e s t ) − 1 = O P ( 1 ) , E P 0 [ V ( O ) 4 ∣ G m ] m v t e s t 2 → P 0. (mv_{\mathrm{test}})^{-1}=O_{\mathbb{P}}(1),\qquad\frac{\mathbb{E}_{P_0}[V(O)^4\mid\mathcal{G}_m]}{mv_{\mathrm{test}}^2}\xrightarrow{\mathbb{P}}0. ( m v test ) − 1 = O P ( 1 ) , m v test 2 E P 0 [ V ( O ) 4 ∣ G m ] P 0. The inverse and ratio are defined to be zero when v t e s t = 0 v_{\mathrm{test}}=0 v test = 0
The conditional mean tends to zero faster than m − 1 / 2 m^{-1/2} m − 1/2
m ∥ b t e s t ∥ H θ ^ → P 0. \sqrt{m}\|b_{\mathrm{test}}\|_{\mathcal{H}_{\widehat{\theta}}}\xrightarrow{\mathbb{P}}0. m ∥ b test ∥ H θ P 0. Double robustness gives a sufficient nuisance-rate condition for condition 3 of Assumption B.4. Under Assumptions 3.1 and 3.2, define the weighted nuisance errors
a π , t e s t 2 : = E P 0 [ ∫ A ∣ r ( a , X ) ∣ { π 0 ( a ∣ X ) π ^ 0 ( a ∣ X ) − 1 } 2 ρ ( d a ) | F l e a r n ] , a_{\pi,\mathrm{test}}^2:=\mathbb{E}_{P_0}\left[\int_{\mathcal{A}}|r(a,X)|\left\{\frac{\pi_0(a\mid X)}{\widehat{\pi}_0(a\mid X)}-1\right\}^2\rho(da)\,\middle|\,\mathcal{F}_{\mathrm{learn}}\right], a π , test 2 := E P 0 [ ∫ A ∣ r ( a , X ) ∣ { π 0 ( a ∣ X ) π 0 ( a ∣ X ) − 1 } 2 ρ ( d a ) F learn ] , b μ , t e s t 2 : = E P 0 [ ∫ A ∣ r ( a , X ) ∣ ∥ μ ^ θ ^ ( a , X ) − μ θ ^ ( a , X ) ∥ H θ ^ 2 ρ ( d a ) | F l e a r n ] . b_{\mu,\mathrm{test}}^2:=\mathbb{E}_{P_0}\left[\int_{\mathcal{A}}|r(a,X)|\|\widehat{\mu}_{\widehat{\theta}}(a,X)-\mu_{\widehat{\theta}}(a,X)\|_{\mathcal{H}_{\widehat{\theta}}}^2\rho(da)\,\middle|\,\mathcal{F}_{\mathrm{learn}}\right]. b μ , test 2 := E P 0 [ ∫ A ∣ r ( a , X ) ∣∥ μ θ ( a , X ) − μ θ ( a , X ) ∥ H θ 2 ρ ( d a ) F learn ] . Under H 0 H_0 H 0 Δ θ ^ ( π , π ′ ) = 0 \Delta_{\widehat{\theta}}(\pi,\pi')=0 Δ θ ( π , π ′ ) = 0 π ^ 0 \widehat{\pi}_0 π 0 ∣ r ∣ |r| ∣ r ∣ ∣ r ( a , X ) ∣ |r(a,X)| ∣ r ( a , X ) ∣
∥ b t e s t ∥ H θ ^ ≤ a π , t e s t b μ , t e s t . (9) \|b_{\mathrm{test}}\|_{\mathcal{H}_{\widehat{\theta}}}\leq a_{\pi,\mathrm{test}}b_{\mu,\mathrm{test}}. \tag*{(9)} ∥ b test ∥ H θ ≤ a π , test b μ , test . (9) a π , t e s t b μ , t e s t = o P ( m − 1 / 2 ) a_{\pi,\mathrm{test}}b_{\mu,\mathrm{test}}=o_{\mathbb{P}}(m^{-1/2}) a π , test b μ , test = o P ( m − 1/2 )
Proof of Theorem 4.3. Write b = b t e s t b=b_{\mathrm{test}} b = b test v = v t e s t v=v_{\mathrm{test}} v = v test H θ ^ \mathcal{H}_{\widehat{\theta}} H θ V j : = V ( O j ) V_j:=V(O_j) V j := V ( O j ) j ∈ I 2 j\in\mathcal{I}_2 j ∈ I 2
D ^ c : = 1 m ∑ j ∈ I 2 V j = ⟨ ξ ‾ 1 , ξ ‾ 2 ⟩ , \widehat{D}^{c}:=\frac{1}{m}\sum_{j\in\mathcal{I}_2}V_j=\langle\overline{\xi}_1,\overline{\xi}_2\rangle, D c := m 1 j ∈ I 2 ∑ V j = ⟨ ξ 1 , ξ 2 ⟩ , ( s ^ c ) 2 : = 1 m ∑ j ∈ I 2 V j 2 − ( D ^ c ) 2 , T c : = m D ^ c s ^ c . (\widehat{s}^{c})^2:=\frac{1}{m}\sum_{j\in\mathcal{I}_2}V_j^2-(\widehat{D}^{c})^2,\qquad T^c:=\frac{\sqrt{m}\widehat{D}^{c}}{\widehat{s}^{c}}. ( s c ) 2 := m 1 j ∈ I 2 ∑ V j 2 − ( D c ) 2 , T c := s c m D c . Set T c = 0 T^c=0 T c = 0 s ^ c = 0 \widehat{s}^{c}=0 s c = 0
V j V_j V j G m \mathcal{G}_m G m v v v { v > 0 } \{v>0\} { v > 0 } m D ^ c / v \sqrt{m}\widehat{D}^{c}/\sqrt{v} m D c / v δ > 0 \delta>0 δ > 0
P ( ∣ ( s ^ c ) 2 v − 1 ∣ > δ ∣ G m ) ≤ 4 E P 0 [ V ( O ) 4 ∣ G m ] δ 2 m v 2 + 2 m δ → P 0. \mathbb{P}\left(\left.\left|\frac{(\widehat{s}^{c})^2}{v}-1\right|>\delta\right|\mathcal{G}_m\right)\leq\frac{4\mathbb{E}_{P_0}[V(O)^4\mid\mathcal{G}_m]}{\delta^2mv^2}+\frac{2}{m\delta}\xrightarrow{\mathbb{P}}0. P ( v ( s c ) 2 − 1 > δ G m ) ≤ δ 2 m v 2 4 E P 0 [ V ( O ) 4 ∣ G m ] + m δ 2 P 0. Here the two bounds control the sample second moment and the squared sample mean, using E P 0 [ ( D ^ c ) 2 / v ∣ G m ] = 1 / m \mathbb{E}_{P_0}[(\widehat{D}^{c})^2/v\mid\mathcal{G}_m]=1/m E P 0 [( D c ) 2 / v ∣ G m ] = 1/ m
d m : = sup t ∈ R ∣ P ( T c ≤ t ∣ G m ) − Φ ( t ) ∣ → P 0. d_m:=\sup_{t\in\mathbb{R}}\left|\mathbb{P}(T^c\leq t\mid\mathcal{G}_m)-\Phi(t)\right|\xrightarrow{\mathbb{P}}0. d m := t ∈ R sup ∣ P ( T c ≤ t ∣ G m ) − Φ ( t ) ∣ P 0. Since P ( v > 0 ) → 1 \mathbb{P}(v>0)\to1 P ( v > 0 ) → 1 ( s ^ c ) 2 / v → P 1 (\widehat{s}^{c})^2/v\xrightarrow{\mathbb{P}}1 ( s c ) 2 / v P 1 P ( s ^ c > 0 ) → 1 \mathbb{P}(\widehat{s}^{c}>0)\to1 P ( s c > 0 ) → 1 { s ^ c > 0 } \{\widehat{s}^{c}>0\} { s c > 0 } ( m v ) − 1 = O P ( 1 ) (mv)^{-1}=O_{\mathbb{P}}(1) ( m v ) − 1 = O P ( 1 ) ( s ^ c ) − 1 = O P ( m ) (\widehat{s}^{c})^{-1}=O_{\mathbb{P}}(\sqrt{m}) ( s c ) − 1 = O P ( m )
E P [ ∥ ξ ˉ ℓ ∥ 2 ∣ F l e a r n ] = 1 m E P [ ∥ ξ ( O ) ∥ 2 ∣ F l e a r n ] = O P ( m − 1 ) . \mathbb{E}_{\mathbb{P}}\left[\|\bar{\xi}_{\ell}\|^2 \mid\mathcal{F}_{\mathrm{learn}}\right] = \frac{1}{m}\mathbb{E}_{\mathbb{P}}\left[\|\xi(O)\|^2 \mid\mathcal{F}_{\mathrm{learn}}\right] = O_{\mathbb{P}}(m^{-1}). E P [ ∥ ξ ˉ ℓ ∥ 2 ∣ F learn ] = m 1 E P [ ∥ ξ ( O ) ∥ 2 ∣ F learn ] = O P ( m − 1 ) . By condition 1 and conditional Markov’s inequality,
∥ ξ ˉ ℓ ∥ = O P ( m − 1 / 2 ) , 1 m ∑ j ∈ I 2 ∥ ξ ( O j ) ∥ 2 = O P ( 1 ) . \|\bar{\xi}_{\ell}\| = O_{\mathbb{P}}(m^{-1/2}), \qquad\frac{1}{m}\sum_{j\in\mathcal{I}_2}\|\xi(O_j)\|^2 = O_{\mathbb{P}}(1). ∥ ξ ˉ ℓ ∥ = O P ( m − 1/2 ) , m 1 j ∈ I 2 ∑ ∥ ξ ( O j ) ∥ 2 = O P ( 1 ) . Since the fitted half-sample means are ξ ˉ ℓ + b \bar{\xi}_{\ell}+b ξ ˉ ℓ + b
∣ D ^ θ ^ c r o s s − D ^ c ∣ = ∣ ⟨ b , ξ ˉ 1 + ξ ˉ 2 ⟩ + ∥ b ∥ 2 ∣ ≤ ∥ b ∥ ( ∥ ξ ˉ 1 ∥ + ∥ ξ ˉ 2 ∥ ) + ∥ b ∥ 2 = o P ( m − 1 ) . \left|\widehat{D}_{\hat{\theta}}^{\mathrm{cross}}-\widehat{D}^{c}\right| = \left|\langle b,\bar{\xi}_1+\bar{\xi}_2\rangle+\|b\|^2\right|
\leq\|b\|(\|\bar{\xi}_1\|+\|\bar{\xi}_2\|)+\|b\|^2 = o_{\mathbb{P}}(m^{-1}). D θ ^ cross − D c = ⟨ b , ξ ˉ 1 + ξ ˉ 2 ⟩ + ∥ b ∥ 2 ≤ ∥ b ∥ ( ∥ ξ ˉ 1 ∥ + ∥ ξ ˉ 2 ∥ ) + ∥ b ∥ 2 = o P ( m − 1 ) . Applying the reverse triangle inequality to the residual vectors and Cauchy-Schwarz gives
∣ s ^ θ ^ − s ^ c ∣ ≤ ∥ b ∥ { 1 m ∑ j ∈ I 2 ∥ ξ ( O j ) − ξ ˉ 2 ∥ 2 } 1 / 2 ≤ ∥ b ∥ { 1 m ∑ j ∈ I 2 ∥ ξ ( O j ) ∥ 2 } 1 / 2 = o P ( m − 1 / 2 ) . \begin{aligned}
|\hat{s}_{\hat{\theta}}-\hat{s}^{c}| \leq\|b\|\left\{\frac{1}{m}\sum_{j\in\mathcal{I}_2}\|\xi(O_j)-\bar{\xi}_2\|^2\right\}^{1/2} \\
&\leq\|b\|\left\{\frac{1}{m}\sum_{j\in\mathcal{I}_2}\|\xi(O_j)\|^2\right\}^{1/2}=o_{\mathbb{P}}(m^{-1/2}).
\end{aligned} ∣ s ^ θ ^ − s ^ c ∣ ≤ ∥ b ∥ ⎩ ⎨ ⎧ m 1 j ∈ I 2 ∑ ∥ ξ ( O j ) − ξ ˉ 2 ∥ 2 ⎭ ⎬ ⎫ 1/2 ≤ ∥ b ∥ ⎩ ⎨ ⎧ m 1 j ∈ I 2 ∑ ∥ ξ ( O j ) ∥ 2 ⎭ ⎬ ⎫ 1/2 = o P ( m − 1/2 ) . On { s ^ c > 0 } \{\hat{s}^{c}>0\} { s ^ c > 0 } ( s ^ c ) − 1 = O P ( m ) (\hat{s}^{c})^{-1}=O_{\mathbb{P}}(\sqrt{m}) ( s ^ c ) − 1 = O P ( m )
m ( D ^ θ ^ c r o s s − D ^ c ) s ^ c = o P ( 1 ) , s ^ θ ^ s ^ c → P 1. \frac{\sqrt{m}(\widehat{D}_{\hat{\theta}}^{\mathrm{cross}}-\widehat{D}^{c})}{\hat{s}^{c}}=o_{\mathbb{P}}(1), \qquad\frac{\hat{s}_{\hat{\theta}}}{\hat{s}^{c}}\xrightarrow{\mathbb{P}}1. s ^ c m ( D θ ^ cross − D c ) = o P ( 1 ) , s ^ c s ^ θ ^ P 1. Hence P ( s ^ θ ^ > 0 ) → 1 \mathbb{P}(\hat{s}_{\hat{\theta}}>0)\to1 P ( s ^ θ ^ > 0 ) → 1 T c = O P ( 1 ) T^{c}=O_{\mathbb{P}}(1) T c = O P ( 1 ) T θ ^ − T c = o P ( 1 ) T_{\hat{\theta}}-T^{c}=o_{\mathbb{P}}(1) T θ ^ − T c = o P ( 1 )
Since 0 ≤ d m ≤ 1 0\leq d_m\leq1 0 ≤ d m ≤ 1 d m → P 0 d_m\xrightarrow{\mathbb{P}}0 d m P 0 E [ d m ] → 0 \mathbb{E}[d_m]\to0 E [ d m ] → 0 E [ d m ∣ F l e a r n ] → P 0 \mathbb{E}[d_m\mid\mathcal{F}_{\mathrm{learn}}]\xrightarrow{\mathbb{P}}0 E [ d m ∣ F learn ] P 0 δ > 0 \delta>0 δ > 0
q m ( δ ) : = P ( ∣ T θ ^ − T c ∣ > δ ∣ F l e a r n ) → P 0 , q_m(\delta):=\mathbb{P}\left(|T_{\hat{\theta}}-T^{c}|>\delta\mid\mathcal{F}_{\mathrm{learn}}\right)\xrightarrow{\mathbb{P}}0, q m ( δ ) := P ( ∣ T θ ^ − T c ∣ > δ ∣ F learn ) P 0 , by Markov’s inequality, since E [ q m ( δ ) ] = P ( ∣ T θ ^ − T c ∣ > δ ) → 0 \mathbb{E}[q_m(\delta)]=\mathbb{P}(|T_{\hat{\theta}}-T^{c}|>\delta)\to0 E [ q m ( δ )] = P ( ∣ T θ ^ − T c ∣ > δ ) → 0
Averaging over the first test half gives the bound E [ d m ∣ F l e a r n ] \mathbb{E}[d_m\mid\mathcal{F}_{\mathrm{learn}}] E [ d m ∣ F learn ] T c T^{c} T c δ \delta δ q m ( δ ) q_m(\delta) q m ( δ ) t − δ t-\delta t − δ t + δ t+\delta t + δ
sup t ∈ R ∣ P ( T θ ^ ≤ t ∣ F l e a r n ) − Φ ( t ) ∣ ≤ E [ d m ∣ F l e a r n ] + q m ( δ ) + δ 2 π , \sup_{t\in\mathbb{R}}\left|\mathbb{P}(T_{\hat{\theta}}\leq t\mid\mathcal{F}_{\mathrm{learn}})-\Phi(t)\right|
\leq\mathbb{E}[d_m\mid\mathcal{F}_{\mathrm{learn}}]+q_m(\delta)+\frac{\delta}{\sqrt{2\pi}}, t ∈ R sup P ( T θ ^ ≤ t ∣ F learn ) − Φ ( t ) ≤ E [ d m ∣ F learn ] + q m ( δ ) + 2 π δ , where the last term follows because the standard normal density is bounded by 1 / 2 π 1/\sqrt{2\pi} 1/ 2 π m → ∞ m\to\infty m → ∞ δ → 0 \delta\to0 δ → 0 α \alpha α θ ^ = θ \hat{\theta}=\theta θ ^ = θ F l e a r n \mathcal{F}_{\mathrm{learn}} F learn
Uniform Control of the Learning Criterion
We prove Theorem 4.4 by separating sampling error from nuisance error. Throughout this subsection, expectations, variances and norms involving fitted functions are conditional on F f i t \mathcal{F}_{\mathrm{fit}} F fit θ \theta θ n n n m m m P 0 \mathbb{P}_0 P 0 F f i t \mathcal{F}_{\mathrm{fit}} F fit H \mathcal{H} H ∥ f ∥ L 2 ( P 0 ; H ) 2 : = E 0 ∥ f ( O ) ∥ H 2 \lVert f\rVert_{L^2(\mathbb{P}_0;\mathcal{H})}^2 := \mathbb{E}_0\lVert f(O)\rVert_{\mathcal{H}}^2 ∥ f ∥ L 2 ( P 0 ; H ) 2 := E 0 ∥ f ( O ) ∥ H 2
Write O : = X × A × Y \mathcal{O} := \mathcal{X} \times\mathcal{A} \times\mathcal{Y} O := X × A × Y Θ = Ξ M \Theta= \Xi^M Θ = Ξ M Ξ ⊆ R d 0 \Xi\subseteq\mathbb{R}^{d_0} Ξ ⊆ R d 0 d Θ = M d 0 d_\Theta= Md_0 d Θ = M d 0 M = 1 M=1 M = 1 Θ = Ξ \Theta=\Xi Θ = Ξ d Θ = d 0 d_\Theta=d_0 d Θ = d 0 ϑ ∈ Ξ \vartheta\in\Xi ϑ ∈ Ξ k ϑ d e e p k^{\mathrm{deep}}_\vartheta k ϑ deep w ϑ w_\vartheta w ϑ w ϑ b = w b w_{\vartheta_b}=w_b w ϑ b = w b
Let g ~ θ : = g θ ( ⋅ ; π ^ 0 , μ ^ θ ) \widetilde{g}_\theta:= g_\theta(\mathord{\cdot};\widehat{\pi}_0,\widehat{\mu}_\theta) g θ := g θ ( ⋅ ; π 0 , μ θ )
Δ ~ θ : = E 0 [ g ~ θ ( O ) ] , D ~ θ : = ∥ Δ ~ θ ∥ H θ 2 , \widetilde{\Delta}_\theta:= \mathbb{E}_0[\widetilde{g}_\theta(O)], \qquad\widetilde{D}_\theta:= \lVert\widetilde{\Delta}_\theta\rVert_{\mathcal{H}_\theta}^2, Δ θ := E 0 [ g θ ( O )] , D θ := ∥ Δ θ ∥ H θ 2 , s ~ θ 2 : = Var P 0 ( ⟨ Δ ~ θ , g ~ θ ( O ) ⟩ H θ ) . \widetilde{s}_\theta^2 := \operatorname{Var}_{\mathbb{P}_0}\left(\langle\widetilde{\Delta}_\theta,\widetilde{g}_\theta(O)\rangle_{\mathcal{H}_\theta}\right). s θ 2 := Var P 0 ( ⟨ Δ θ , g θ ( O ) ⟩ H θ ) . A subscript o r \mathrm{or} or
Measurable versions. We use the common response kernel from Proposition B.1 :
μ θ ( a , x ) = ∫ Y φ θ ( y ) Q ( d y ∣ a , x ) . (10) \mu_\theta(a,x) = \int_{\mathcal{Y}} \varphi_\theta(y)Q(\mathrm{d}y\mid a,x). \tag*{(10)} μ θ ( a , x ) = ∫ Y φ θ ( y ) Q ( d y ∣ a , x ) . (10) The measurable null-set convention in Section B.1 applies whenever only first moments are assumed. Under the uniform diagonal bound below, the integral exists for every ( θ , a , x ) (\theta,a,x) ( θ , a , x ) ∥ μ θ ( a , x ) ∥ H θ ≤ κ \lVert\mu_\theta(a,x)\rVert_{\mathcal{H}_\theta} \le\sqrt{\kappa} ∥ μ θ ( a , x ) ∥ H θ ≤ κ
Write the estimators from § 3 in terms of the fitted observed-data kernel h ~ θ ( o , o ′ ) : = ⟨ g ~ θ ( o ) , g ~ θ ( o ′ ) ⟩ H θ \widetilde{h}_\theta(o,o') := \langle\widetilde{g}_\theta(o),\widetilde{g}_\theta(o')\rangle_{\mathcal{H}_\theta} h θ ( o , o ′ ) := ⟨ g θ ( o ) , g θ ( o ′ ) ⟩ H θ U θ , j : = m − 1 ∑ i ∈ I 1 h ~ θ ( O i , O j ) U_{\theta,j} := m^{-1}\sum_{i\in I_1}\widetilde{h}_\theta(O_i,O_j) U θ , j := m − 1 ∑ i ∈ I 1 h θ ( O i , O j )
D ~ ^ θ c r o s s = 1 m ∑ j ∈ I 2 U θ , j . (11) \widehat{\widetilde{D}}_\theta^{\mathrm{cross}} = \frac{1}{m}\sum_{j\in I_2} U_{\theta,j}. \tag*{(11)} D θ cross = m 1 j ∈ I 2 ∑ U θ , j . (11) and
s ^ θ 2 = 1 m ∑ j ∈ I 2 U θ , j 2 − ( D ~ ^ θ c r o s s ) 2 . (12) \widehat{s}_\theta^2 = \frac{1}{m}\sum_{j\in I_2} U_{\theta,j}^2 - \left(\widehat{\widetilde{D}}_\theta^{\mathrm{cross}}\right)^2. \tag*{(12)} s θ 2 = m 1 j ∈ I 2 ∑ U θ , j 2 − ( D θ cross ) 2 . (12) Assumption B.5 (Selection regularity). The set Θ ⊂ R d Θ \Theta\subset\mathbb{R}^{d_\Theta} Θ ⊂ R d Θ is compact and lies in a Euclidean ball of radius R Θ R_\Theta R Θ There are fixed κ , W , A , B μ , L < ∞ \kappa, W, A, B_\mu, L < \infty κ , W , A , B μ , L < ∞ such that, almost surely over the nuisance-fitting sample, k θ ( y , y ) ≤ κ k_\theta(y,y) \le\kappa k θ ( y , y ) ≤ κ for every ( θ , y ) (\theta,y) ( θ , y ) and, outside one common ( P X ⊗ ρ ) (\mathbb{P}_X \otimes\rho) ( P X ⊗ ρ ) null set, whenever ∣ r ( a , x ) ∣ > 0 \lvert r(a,x)\rvert> 0 ∣ r ( a , x )∣ > 0 π 0 ( a ∣ x ) , π ^ 0 ( a ∣ x ) > 0 \pi_0(a\mid x), \widehat{\pi}_0(a\mid x) > 0 π 0 ( a ∣ x ) , π 0 ( a ∣ x ) > 0 and, for every θ ∈ Θ \theta\in\Theta θ ∈ Θ
∣ r ( a , x ) ∣ π 0 ( a ∣ x ) ≤ W , ∣ π 0 ( a ∣ x ) π ^ 0 ( a ∣ x ) − 1 ∣ ≤ A , ∥ μ ^ θ ( a , x ) ∥ H θ ≤ B μ . (13) \frac{\lvert r(a,x)\rvert}{\pi_0(a\mid x)} \le W,\qquad\left\lvert\frac{\pi_0(a\mid x)}{\widehat{\pi}_0(a\mid x)} - 1\right\rvert\le A,\qquad\lVert\widehat{\mu}_\theta(a,x)\rVert_{\mathcal{H}_\theta} \le B_\mu. \tag*{(13)} π 0 ( a ∣ x ) ∣ r ( a , x )∣ ≤ W , π 0 ( a ∣ x ) π 0 ( a ∣ x ) − 1 ≤ A , ∥ μ θ ( a , x ) ∥ H θ ≤ B μ . (13) For every θ \theta θ μ ^ θ \widehat{\mu}_\theta μ θ and g ~ θ \widetilde{g}_\theta g θ are strongly measurable. Moreover, h ~ θ ( o , o ′ ) : = ⟨ g ~ θ ( o ) , g ~ θ ( o ′ ) ⟩ H θ \widetilde{h}_\theta(o,o') := \langle\widetilde{g}_\theta(o),\widetilde{g}_\theta(o')\rangle_{\mathcal{H}_\theta} h θ ( o , o ′ ) := ⟨ g θ ( o ) , g θ ( o ′ ) ⟩ H θ is jointly measurable and, almost surely over the nuisance-fitting sample, on a common P 0 2 \mathbb{P}_0^2 P 0 2 full set,
∣ h ~ θ ( o , o ′ ) − h ~ θ ′ ( o , o ′ ) ∣ ≤ L ∥ θ − θ ′ ∥ 2 . (14) \left\lvert\widetilde{h}_\theta(o,o')-\widetilde{h}_{\theta'}(o,o')\right\rvert\le L\lVert\theta-\theta'\rVert_2. \tag*{(14)} h θ ( o , o ′ ) − h θ ′ ( o , o ′ ) ≤ L ∥ θ − θ ′ ∥ 2 . (14) Conditional on F f i t \mathcal{F}_{\mathrm{fit}} F fit
ϵ π , n 2 : = E 0 [ ∫ A ∣ r ( a , X ) ∣ { π 0 ( a ∣ X ) π ^ 0 ( a ∣ X ) − 1 } 2 ρ ( d a ) ] , ϵ μ , n 2 : = sup E 0 [ ∫ A ∣ r ( a , X ) ∣ ∥ μ ^ θ ( a , X ) − μ θ ( a , X ) ∥ H θ 2 ρ ( d a ) ] . (15) \begin{aligned}
\epsilon_{\pi,n}^{2} &:= \mathbb{E}_{0}\left[\int_{\mathcal{A}} |r(a,X)|\left\{\frac{\pi_{0}(a\mid X)}{\widehat{\pi}_{0}(a\mid X)}-1\right\}^{2}\rho(\mathrm{d}a)\right],\\
\epsilon_{\mu,n}^{2} &:= \sup\mathbb{E}_{0}\left[\int_{\mathcal{A}} |r(a,X)|\left\|\widehat{\mu}_{\theta}(a,X)-\mu_{\theta}(a,X)\right\|_{\mathcal{H}_{\theta}}^{2}\rho(\mathrm{d}a)\right]. \tag*{(15)}
\end{aligned} ϵ π , n 2 ϵ μ , n 2 := E 0 [ ∫ A ∣ r ( a , X ) ∣ { π 0 ( a ∣ X ) π 0 ( a ∣ X ) − 1 } 2 ρ ( d a ) ] , := sup E 0 [ ∫ A ∣ r ( a , X ) ∣ ∥ μ θ ( a , X ) − μ θ ( a , X ) ∥ H θ 2 ρ ( d a ) ] . (15) We assume that ϵ μ , n 2 \epsilon_{\mu,n}^{2} ϵ μ , n 2 F f i t \mathcal{F}_{\mathrm{fit}} F fit
For λ > 0 \lambda>0 λ > 0 Θ \Theta Θ h ~ θ \widetilde{h}_{\theta} h θ Θ \Theta Θ J λ , θ J_{\lambda,\theta} J λ , θ f ↦ sup θ ∈ Θ ∣ f ( θ ) − J λ , θ ∣ f\mapsto\sup_{\theta\in\Theta}|f(\theta)-J_{\lambda,\theta}| f ↦ sup θ ∈ Θ ∣ f ( θ ) − J λ , θ ∣ C ( Θ ) C(\Theta) C ( Θ ) J λ , θ J_{\lambda,\theta} J λ , θ
For a failure probability η ∈ ( 0 , 1 ) \eta\in(0,1) η ∈ ( 0 , 1 )
c m ( η ) : = d Θ log ( 1 + 2 R Θ m ) + log ( 4 / η ) + 1 m . (16) c_m(\eta):=\frac{\sqrt{d_{\Theta}\log(1+2R_{\Theta}\sqrt{m})+\log(4/\eta)+1}}{\sqrt{m}}. \tag*{(16)} c m ( η ) := m d Θ log ( 1 + 2 R Θ m ) + log ( 4/ η ) + 1 . (16) Kernel-Class Regularity
We give sufficient conditions for regularity of k θ k_{\theta} k θ h θ h_{\theta} h θ h ~ θ \widetilde{h}_{\theta} h θ
For o , o ′ ∈ O o,o'\in\mathcal{O} o , o ′ ∈ O
h θ ( o , o ′ ) : = ⟨ g θ π , π ′ ( o ) , g θ π , π ′ ( o ′ ) ⟩ H θ . (17) h_{\theta}(o,o'):=\left\langle g_{\theta}^{\pi,\pi'}(o),g_{\theta}^{\pi,\pi'}(o')\right\rangle_{\mathcal{H}_{\theta}}. \tag*{(17)} h θ ( o , o ′ ) := ⟨ g θ π , π ′ ( o ) , g θ π , π ′ ( o ′ ) ⟩ H θ . (17) Assumption B.6 (Oracle induced-kernel class). The parameter set Θ \Theta Θ R Θ R_{\Theta} R Θ O 0 ⊆ O \mathcal{O}_{0}\subseteq\mathcal{O} O 0 ⊆ O P 0 ( O 0 ) = 1 \mathbb{P}_{0}(\mathcal{O}_{0})=1 P 0 ( O 0 ) = 1 B g , L h < ∞ B_{g},L_{h}<\infty B g , L h < ∞ o , o ′ ∈ O 0 o,o'\in\mathcal{O}_{0} o , o ′ ∈ O 0 θ , θ ′ ∈ Θ \theta,\theta'\in\Theta θ , θ ′ ∈ Θ
∥ g θ π , π ′ ( o ) ∥ H θ ≤ B g , ∣ h θ ( o , o ′ ) − h θ ′ ( o , o ′ ) ∣ ≤ L h ∥ θ − θ ′ ∥ 2 . \left\|g_{\theta}^{\pi,\pi'}(o)\right\|_{\mathcal{H}_{\theta}}\le B_{g},\qquad
\left|h_{\theta}(o,o')-h_{\theta'}(o,o')\right|\le L_{h}\left\|\theta-\theta'\right\|_{2}. g θ π , π ′ ( o ) H θ ≤ B g , ∣ h θ ( o , o ′ ) − h θ ′ ( o , o ′ ) ∣ ≤ L h ∥ θ − θ ′ ∥ 2 . The scalar map ( θ , o , o ′ ) ↦ h θ ( o , o ′ ) (\theta,o,o')\mapsto h_{\theta}(o,o') ( θ , o , o ′ ) ↦ h θ ( o , o ′ ) Θ × O 0 2 \Theta\times\mathcal{O}_{0}^{2} Θ × O 0 2 ∣ h θ ( o , o ′ ) ∣ ≤ B g 2 \left|h_{\theta}(o,o')\right|\le B_{g}^{2} ∣ h θ ( o , o ′ ) ∣ ≤ B g 2 B h : = B g 2 B_{h}:=B_{g}^{2} B h := B g 2
Lemma B.8 (Regularity of the deep-kernel ensemble). Suppose there exist finite constants
K d e e p , K c h a r , L d e e p , L c h a r , L w < ∞ K_{\mathrm{deep}},\ K_{\mathrm{char}},\ L_{\mathrm{deep}},\ L_{\mathrm{char}},\ L_{w}<\infty K deep , K char , L deep , L char , L w < ∞ such that, for all ϑ , ϑ ′ ∈ Ξ \vartheta,\vartheta'\in\Xi ϑ , ϑ ′ ∈ Ξ y , y ′ ∈ Y y,y'\in\mathcal{Y} y , y ′ ∈ Y
∣ k ϑ d e e p ( y , y ′ ) ∣ ≤ K d e e p , ∣ k ϑ c h a r ( y , y ′ ) ∣ ≤ K c h a r , \left|k_{\vartheta}^{\mathrm{deep}}(y,y')\right|\le K_{\mathrm{deep}},\qquad
\left|k_{\vartheta}^{\mathrm{char}}(y,y')\right|\le K_{\mathrm{char}}, k ϑ deep ( y , y ′ ) ≤ K deep , k ϑ char ( y , y ′ ) ≤ K char , ∣ k ϑ d e e p ( y , y ′ ) − k ϑ ′ d e e p ( y , y ′ ) ∣ ≤ L d e e p ∥ ϑ − ϑ ′ ∥ 2 , ∣ k ϑ c h a r ( y , y ′ ) − k ϑ ′ c h a r ( y , y ′ ) ∣ ≤ L c h a r ∥ ϑ − ϑ ′ ∥ 2 , \left|k_{\vartheta}^{\mathrm{deep}}(y,y')-k_{\vartheta'}^{\mathrm{deep}}(y,y')\right|
\le L_{\mathrm{deep}}\left\|\vartheta-\vartheta'\right\|_{2},\qquad
\left|k_{\vartheta}^{\mathrm{char}}(y,y')-k_{\vartheta'}^{\mathrm{char}}(y,y')\right|
\le L_{\mathrm{char}}\left\|\vartheta-\vartheta'\right\|_{2}, k ϑ deep ( y , y ′ ) − k ϑ ′ deep ( y , y ′ ) ≤ L deep ∥ ϑ − ϑ ′ ∥ 2 , k ϑ char ( y , y ′ ) − k ϑ ′ char ( y , y ′ ) ≤ L char ∥ ϑ − ϑ ′ ∥ 2 , and
∣ w ϑ − w ϑ ′ ∣ ≤ L w ∥ ϑ − ϑ ′ ∥ 2 . |w_{\vartheta}-w_{\vartheta'}| \le L_w\|\vartheta-\vartheta'\|_2. ∣ w ϑ − w ϑ ′ ∣ ≤ L w ∥ ϑ − ϑ ′ ∥ 2 . Writing L 0 : = L d e e p + L c h a r + L w ( K d e e p + K c h a r ) L_0 := L_{\mathrm{deep}} + L_{\mathrm{char}} + L_w(K_{\mathrm{deep}} + K_{\mathrm{char}}) L 0 := L deep + L char + L w ( K deep + K char ) θ = ( ϑ 1 , … , ϑ M ) \theta= (\vartheta_1,\ldots,\vartheta_M) θ = ( ϑ 1 , … , ϑ M ) θ ′ = ( ϑ 1 ′ , … , ϑ M ′ ) \theta' = (\vartheta'_1,\ldots,\vartheta'_M) θ ′ = ( ϑ 1 ′ , … , ϑ M ′ )
∣ k θ ( y , y ′ ) − k θ ′ ( y , y ′ ) ∣ ≤ L 0 M ∥ θ − θ ′ ∥ 2 . |k_\theta(y,y')-k_{\theta'}(y,y')| \le\frac{L_0}{\sqrt{M}}\|\theta-\theta'\|_2. ∣ k θ ( y , y ′ ) − k θ ′ ( y , y ′ ) ∣ ≤ M L 0 ∥ θ − θ ′ ∥ 2 . Moreover,
sup θ , y k θ ( y , y ) ≤ max { K d e e p , K c h a r } . \sup_{\theta,y} k_\theta(y,y) \le\max\{K_{\mathrm{deep}},K_{\mathrm{char}}\}. θ , y sup k θ ( y , y ) ≤ max { K deep , K char } . Proof. For one member, add and subtract ( 1 − w ϑ ′ ) k ϑ ′ d e e p + w ϑ ′ k ϑ ′ c h a r (1-w_{\vartheta'})k^{\mathrm{deep}}_{\vartheta'}+w_{\vartheta'}k^{\mathrm{char}}_{\vartheta'} ( 1 − w ϑ ′ ) k ϑ ′ deep + w ϑ ′ k ϑ ′ char 0 ≤ w ϑ ≤ 1 0\le w_{\vartheta}\le1 0 ≤ w ϑ ≤ 1
∣ ( 1 − w ϑ ) k ϑ d e e p + w ϑ k ϑ c h a r − ( 1 − w ϑ ′ ) k ϑ ′ d e e p − w ϑ ′ k ϑ ′ c h a r ∣ ≤ ( 1 − w ϑ ) ∣ k ϑ d e e p − k ϑ ′ d e e p ∣ + w ϑ ∣ k ϑ c h a r − k ϑ ′ c h a r ∣ + ∣ w ϑ − w ϑ ′ ∣ ∣ k ϑ ′ c h a r − k ϑ ′ d e e p ∣ ≤ L 0 ∥ ϑ − ϑ ′ ∥ 2 . \begin{aligned}
\big|(1-w_{\vartheta})k^{\mathrm{deep}}_{\vartheta}+w_{\vartheta}k^{\mathrm{char}}_{\vartheta}-(1-w_{\vartheta'})k^{\mathrm{deep}}_{\vartheta'}-w_{\vartheta'}k^{\mathrm{char}}_{\vartheta'}\big|
&\le(1-w_{\vartheta})|k^{\mathrm{deep}}_{\vartheta}-k^{\mathrm{deep}}_{\vartheta'}|+w_{\vartheta}|k^{\mathrm{char}}_{\vartheta}-k^{\mathrm{char}}_{\vartheta'}| \\
&\quad+|w_{\vartheta}-w_{\vartheta'}||k^{\mathrm{char}}_{\vartheta'}-k^{\mathrm{deep}}_{\vartheta'}| \\
&\le L_0\|\vartheta-\vartheta'\|_2.
\end{aligned} ( 1 − w ϑ ) k ϑ deep + w ϑ k ϑ char − ( 1 − w ϑ ′ ) k ϑ ′ deep − w ϑ ′ k ϑ ′ char ≤ ( 1 − w ϑ ) ∣ k ϑ deep − k ϑ ′ deep ∣ + w ϑ ∣ k ϑ char − k ϑ ′ char ∣ + ∣ w ϑ − w ϑ ′ ∣∣ k ϑ ′ char − k ϑ ′ deep ∣ ≤ L 0 ∥ ϑ − ϑ ′ ∥ 2 . Average this bound over members and apply Cauchy-Schwarz:
∣ k θ − k θ ′ ∣ ≤ L 0 M ∑ b = 1 M ∥ ϑ b − ϑ b ′ ∥ 2 ≤ L 0 M ∥ θ − θ ′ ∥ 2 . |k_\theta-k_{\theta'}|\le\frac{L_0}{M}\sum_{b=1}^{M}\|\vartheta_b-\vartheta'_b\|_2\le\frac{L_0}{\sqrt{M}}\|\theta-\theta'\|_2. ∣ k θ − k θ ′ ∣ ≤ M L 0 b = 1 ∑ M ∥ ϑ b − ϑ b ′ ∥ 2 ≤ M L 0 ∥ θ − θ ′ ∥ 2 . The diagonal bound follows first from convexity within each member and then from averaging the members. □ \square □
Proposition B.9 (Primitive conditions for the induced kernel). Suppose Θ \Theta Θ ( θ , y , y ′ ) ↦ k θ ( y , y ′ ) (\theta,y,y')\mapsto k_\theta(y,y') ( θ , y , y ′ ) ↦ k θ ( y , y ′ ) κ , L k \kappa,L_k κ , L k
sup θ ∈ Θ sup y ∈ Y k θ ( y , y ) ≤ κ , sup y , y ′ ∈ Y ∣ k θ ( y , y ′ ) − k θ ′ ( y , y ′ ) ∣ ≤ L k ∥ θ − θ ′ ∥ 2 . \sup_{\theta\in\Theta}\sup_{y\in\mathcal{Y}} k_\theta(y,y)\le\kappa,\qquad\sup_{y,y'\in\mathcal{Y}}|k_\theta(y,y')-k_{\theta'}(y,y')|\le L_k\|\theta-\theta'\|_2. θ ∈ Θ sup y ∈ Y sup k θ ( y , y ) ≤ κ , y , y ′ ∈ Y sup ∣ k θ ( y , y ′ ) − k θ ′ ( y , y ′ ) ∣ ≤ L k ∥ θ − θ ′ ∥ 2 . Let
W : = ess sup P X ⊗ ρ ∣ r ( a , x ) ∣ π 0 ( a ∣ x ) . W:=\operatorname*{ess\,sup}_{P_X\otimes\rho}\frac{|r(a,x)|}{\pi_0(a\mid x)}. W := P X ⊗ ρ ess sup π 0 ( a ∣ x ) ∣ r ( a , x ) ∣ . If W < ∞ W<\infty W < ∞ O 0 ⊆ O O_0\subseteq\mathcal{O} O 0 ⊆ O P 0 ( O 0 ) = 1 P_0(O_0)=1 P 0 ( O 0 ) = 1
sup θ ∈ Θ ∥ g θ π , π ′ ( o ) ∥ H θ ≤ B g : = 2 ( W + 1 ) κ for every o ∈ O 0 , \sup_{\theta\in\Theta}\|g^{\pi,\pi'}_\theta(o)\|_{\mathcal{H}_\theta}\le B_g:=2(W+1)\sqrt{\kappa}\qquad\text{for every }o\in O_0, θ ∈ Θ sup ∥ g θ π , π ′ ( o ) ∥ H θ ≤ B g := 2 ( W + 1 ) κ for every o ∈ O 0 , and the induced kernels in (17) satisfy Assumption B.6 with
B h = B g 2 = 4 ( W + 1 ) 2 κ , L h = 4 ( W + 1 ) 2 L k . B_h=B_g^2=4(W+1)^2\kappa,\qquad L_h=4(W+1)^2L_k. B h = B g 2 = 4 ( W + 1 ) 2 κ , L h = 4 ( W + 1 ) 2 L k . Proof. Choose a common full-measure set on which the contrast-overlap bound defining W W W x x x ρ \rho ρ O 0 O_0 O 0
∣ k θ ( y , y ′ ) ∣ ≤ k θ ( y , y ) k θ ( y ′ , y ′ ) ≤ κ . |k_\theta(y,y')|\le\sqrt{k_\theta(y,y)k_\theta(y',y')}\le\kappa. ∣ k θ ( y , y ′ ) ∣ ≤ k θ ( y , y ) k θ ( y ′ , y ′ ) ≤ κ . Hence ∥ φ θ ( y ) ∥ H θ ≤ κ \|\varphi_\theta(y)\|_{\mathcal{H}_\theta}\le\sqrt{\kappa} ∥ φ θ ( y ) ∥ H θ ≤ κ ∥ μ θ ( a , x ) ∥ H θ ≤ κ \|\mu_\theta(a,x)\|_{\mathcal{H}_\theta}\le\sqrt{\kappa} ∥ μ θ ( a , x ) ∥ H θ ≤ κ ∫ ∣ r ( a , x ) ∣ ρ ( d a ) ≤ 2 \int|r(a,x)|\rho(da)\le2 ∫ ∣ r ( a , x ) ∣ ρ ( d a ) ≤ 2
∥ g θ π , π ′ ( o ) ∥ H θ ≤ ∣ r ( a , x ) ∣ π 0 ( a ∣ x ) ∥ φ θ ( y ) − μ θ ( a , x ) ∥ H θ + ∫ ∣ r ( a ~ , x ) ∣ ∥ μ θ ( a ~ , x ) ∥ H θ ρ ( d a ~ ) ≤ 2 W κ + 2 κ = 2 ( W + 1 ) κ . \begin{aligned}
\|g^{\pi,\pi'}_\theta(o)\|_{\mathcal{H}_\theta}
&\le\frac{|r(a,x)|}{\pi_0(a\mid x)}\|\varphi_\theta(y)-\mu_\theta(a,x)\|_{\mathcal{H}_\theta}
+\int|r(\tilde a,x)|\|\mu_\theta(\tilde a,x)\|_{\mathcal{H}_\theta}\rho(d\tilde a)\\
&\le2W\sqrt{\kappa}+2\sqrt{\kappa}=2(W+1)\sqrt{\kappa}.
\end{aligned} ∥ g θ π , π ′ ( o ) ∥ H θ ≤ π 0 ( a ∣ x ) ∣ r ( a , x ) ∣ ∥ φ θ ( y ) − μ θ ( a , x ) ∥ H θ + ∫ ∣ r ( a ~ , x ) ∣∥ μ θ ( a ~ , x ) ∥ H θ ρ ( d a ~ ) ≤ 2 W κ + 2 κ = 2 ( W + 1 ) κ . Cauchy-Schwarz gives ∣ h θ ( o , o ′ ) ∣ ≤ B g 2 = B h |h_\theta(o,o')| \le B_g^2 = B_h ∣ h θ ( o , o ′ ) ∣ ≤ B g 2 = B h
To compare h θ h_\theta h θ θ \theta θ o = ( x , a , y ) o=(x,a,y) o = ( x , a , y )
M o ( B ) : = r ( a , x ) π 0 ( a ∣ x ) { 1 ( y ∈ B ) − Q ( B ∣ a , x ) } + ∫ A r ( a ~ , x ) Q ( B ∣ a ~ , x ) ρ ( d a ~ ) . M_o(B) := \frac{r(a,x)}{\pi_0(a \mid x)}\{1(y \in B)-Q(B \mid a,x)\}+\int_{\mathcal{A}}r(\tilde a,x)Q(B \mid\tilde a,x)\rho(\mathrm{d}\tilde a). M o ( B ) := π 0 ( a ∣ x ) r ( a , x ) { 1 ( y ∈ B ) − Q ( B ∣ a , x )} + ∫ A r ( a ~ , x ) Q ( B ∣ a ~ , x ) ρ ( d a ~ ) . The common kernel Q Q Q M o M_o M o θ \theta θ ∥ M o ∥ T V ≤ 2 ( W + 1 ) \|M_o\|_{\mathrm{TV}} \le2(W+1) ∥ M o ∥ TV ≤ 2 ( W + 1 )
g θ , θ ′ π ( o ) = ∫ Y φ θ ( u ) M o ( d u ) , g_{\theta,\theta'}^\pi(o)=\int_{\mathcal{Y}}\varphi_\theta(u)M_o(\mathrm{d}u), g θ , θ ′ π ( o ) = ∫ Y φ θ ( u ) M o ( d u ) , so
h θ ( o , o ′ ) = ∫ Y ∫ Y k θ ( u , v ) M o ( d u ) M o ′ ( d v ) . h_\theta(o,o')=\int_{\mathcal{Y}}\int_{\mathcal{Y}}k_\theta(u,v)M_o(\mathrm{d}u)M_{o'}(\mathrm{d}v). h θ ( o , o ′ ) = ∫ Y ∫ Y k θ ( u , v ) M o ( d u ) M o ′ ( d v ) . Measurability of Q Q Q ( θ , o , o ′ ) ↦ h θ ( o , o ′ ) (\theta,o,o')\mapsto h_\theta(o,o') ( θ , o , o ′ ) ↦ h θ ( o , o ′ ) Θ × O 0 2 \Theta\times\mathcal{O}_0^2 Θ × O 0 2
∣ h θ ( o , o ′ ) − h θ ′ ( o , o ′ ) ∣ ≤ ∥ M o ∥ T V ∥ M o ′ ∥ T V sup u , v ∣ k θ ( u , v ) − k θ ′ ( u , v ) ∣ |h_\theta(o,o')-h_{\theta'}(o,o')|
\le\|M_o\|_{\mathrm{TV}}\|M_{o'}\|_{\mathrm{TV}}\sup_{u,v}|k_\theta(u,v)-k_{\theta'}(u,v)| ∣ h θ ( o , o ′ ) − h θ ′ ( o , o ′ ) ∣ ≤ ∥ M o ∥ TV ∥ M o ′ ∥ TV u , v sup ∣ k θ ( u , v ) − k θ ′ ( u , v ) ∣ ≤ 4 ( W + 1 ) 2 L k ∥ θ − θ ′ ∥ 2 . \le4(W+1)^2L_k\|\theta-\theta'\|_2. ≤ 4 ( W + 1 ) 2 L k ∥ θ − θ ′ ∥ 2 . Proposition B.10 (Primitive conditions for the fitted CME family). Condition on F f i t \mathcal{F}_{\mathrm{fit}} F fit π ^ 0 \hat\pi_0 π ^ 0 θ \theta θ k θ k_\theta k θ κ \kappa κ L k L_k L k φ θ \varphi_\theta φ θ
μ ^ θ ( a , x ) = ∑ i = 1 n β i ( a , x ) φ θ ( Y i ) , sup a , x ∑ i = 1 n ∣ β i ( a , x ) ∣ ≤ B β , \hat\mu_\theta(a,x)=\sum_{i=1}^{n}\beta_i(a,x)\varphi_\theta(Y_i),\qquad\sup_{a,x}\sum_{i=1}^{n}|\beta_i(a,x)|\le B_\beta, μ ^ θ ( a , x ) = i = 1 ∑ n β i ( a , x ) φ θ ( Y i ) , a , x sup i = 1 ∑ n ∣ β i ( a , x ) ∣ ≤ B β , where the fitting outcomes Y i Y_i Y i β i \beta_i β i F f i t \mathcal{F}_{\mathrm{fit}} F fit θ \theta θ
B μ = B β κ , L = C β 2 L k , C β : = W ( 1 + A ) ( 1 + B β ) + 2 B β , B_\mu=B_\beta\sqrt{\kappa},\qquad L=C_\beta^2L_k,\qquad C_\beta:=W(1+A)(1+B_\beta)+2B_\beta, B μ = B β κ , L = C β 2 L k , C β := W ( 1 + A ) ( 1 + B β ) + 2 B β , and ∣ h ~ θ ( o , o ′ ) ∣ ≤ κ C β 2 |\tilde h_\theta(o,o')|\le\kappa C_\beta^2 ∣ h ~ θ ( o , o ′ ) ∣ ≤ κ C β 2 P 0 2 \mathbb{P}_0^2 P 0 2
Proof. The CME bound follows by the triangle inequality. Define the finite signed measure
M ~ o = r ( a , x ) π ^ 0 ( a ∣ x ) ( δ y − ∑ i β i ( a , x ) δ Y i ) + ∑ i { ∫ r ( t , x ) β i ( t , x ) ρ ( d t ) } δ Y i , \widetilde M_o=\frac{r(a,x)}{\hat\pi_0(a\mid x)}\left(\delta_y-\sum_i\beta_i(a,x)\delta_{Y_i}\right)+\sum_i\left\{\int r(t,x)\beta_i(t,x)\rho(\mathrm{d}t)\right\}\delta_{Y_i}, M o = π ^ 0 ( a ∣ x ) r ( a , x ) ( δ y − i ∑ β i ( a , x ) δ Y i ) + i ∑ { ∫ r ( t , x ) β i ( t , x ) ρ ( d t ) } δ Y i , with the ratio set to zero where r = 0 r=0 r = 0 ∣ r ∣ / π ^ 0 ≤ W ( 1 + A ) |r|/\hat\pi_0\le W(1+A) ∣ r ∣/ π ^ 0 ≤ W ( 1 + A ) ∫ ∣ r ( t , x ) ∣ ρ ( d t ) ≤ 2 \int|r(t,x)|\rho(\mathrm{d}t)\le2 ∫ ∣ r ( t , x ) ∣ ρ ( d t ) ≤ 2 C β C_\beta C β θ \theta θ
g ~ θ ( o ) = ∫ φ θ ( u ) M ~ o ( d u ) , h ~ θ ( o , o ′ ) = ∫ ∫ k θ ( u , v ) M ~ o ( d u ) M ~ o ′ ( d v ) . \tilde g_\theta(o)=\int\varphi_\theta(u)\widetilde M_o(\mathrm{d}u),\qquad\tilde h_\theta(o,o')=\int\int k_\theta(u,v)\widetilde M_o(\mathrm{d}u)\widetilde M_{o'}(\mathrm{d}v). g ~ θ ( o ) = ∫ φ θ ( u ) M o ( d u ) , h ~ θ ( o , o ′ ) = ∫∫ k θ ( u , v ) M o ( d u ) M o ′ ( d v ) . These representations give the envelope and Lipschitz bounds by ∣ k θ ∣ ≤ κ |k_\theta|\le\kappa ∣ k θ ∣ ≤ κ k θ − k θ ′ k_\theta-k_{\theta'} k θ − k θ ′
For kernel ridge regression with a fixed input kernel and ridge, the coefficients β i ( a , x ) \beta_i(a,x) β i ( a , x ) θ \theta θ φ θ \varphi_\theta φ θ K + n λ μ I K+n\lambda_\mu I K + n λ μ I λ μ \lambda_\mu λ μ
Oracle Concentration and Adaptation
Let ( λ m ) m ≥ 1 (\lambda_m)_{m\geq1} ( λ m ) m ≥ 1 m m m (11) , (12) , and (3) by replacing the fitted pseudo-features with g θ π , π ′ g_\theta^{\pi,\pi'} g θ π , π ′
D ^ θ , o r c r o s s , s ^ θ , o r 2 , J ^ λ m , θ o r . \widehat{D}^{\mathrm{cross}}_{\theta,\mathrm{or}},\qquad\widehat{s}^{2}_{\theta,\mathrm{or}},\qquad\widehat{J}^{\mathrm{or}}_{\lambda_m,\theta}. D θ , or cross , s θ , or 2 , J λ m , θ or . For δ ∈ ( 0 , 1 ) \delta\in(0,1) δ ∈ ( 0 , 1 )
r D , m ( δ ) : = C { B h c m ( δ ) + L h m } . (18) r_{D,m}(\delta):=C\left\{B_h c_m(\delta)+\frac{L_h}{\sqrt{m}}\right\}. \tag*{(18)} r D , m ( δ ) := C { B h c m ( δ ) + m L h } . (18) r s , m ( δ ) : = C { B h 2 c m ( δ ) + B h L h m + B h 2 m } . (19) r_{s,m}(\delta):=C\left\{B_h^2 c_m(\delta)+\frac{B_hL_h}{\sqrt{m}}+\frac{B_h^2}{m}\right\}. \tag*{(19)} r s , m ( δ ) := C { B h 2 c m ( δ ) + m B h L h + m B h 2 } . (19) where C > 0 C>0 C > 0 s > 0 s>0 s > 0
Θ s : = { θ ∈ Θ : s θ 2 ≥ s 2 } . \Theta_s:=\{\theta\in\Theta:s_\theta^2\geq s^2\}. Θ s := { θ ∈ Θ : s θ 2 ≥ s 2 } . Membership in Θ s \Theta_s Θ s
Lemma B.11 (Uniform oracle concentration). Under Assumptions 3.1 and 3.2 and Assumption B.6 , with probability at least 1 − δ 1-\delta 1 − δ
sup θ ∈ Θ ∣ D ^ θ , o r c r o s s − D θ ∣ ≤ r D , m ( δ ) , sup θ ∈ Θ ∣ s ^ θ , o r 2 − s θ 2 ∣ ≤ r s , m ( δ ) . \sup_{\theta\in\Theta}\left|\widehat{D}^{\mathrm{cross}}_{\theta,\mathrm{or}}-D_\theta\right|\leq r_{D,m}(\delta),\qquad
\sup_{\theta\in\Theta}\left|\widehat{s}^{2}_{\theta,\mathrm{or}}-s_\theta^2\right|\leq r_{s,m}(\delta). θ ∈ Θ sup D θ , or cross − D θ ≤ r D , m ( δ ) , θ ∈ Θ sup s θ , or 2 − s θ 2 ≤ r s , m ( δ ) . Proof. We adapt the covering and bounded-difference arguments of Liu et al. [33 ] to the cross-statistic. Earlier uniform bounds for kernel mixtures appear in Gretton et al. [20 ] . Let τ m = m − 1 / 2 \tau_m=m^{-1/2} τ m = m − 1/2 Θ \Theta Θ R Θ R_\Theta R Θ τ m \tau_m τ m N τ m \mathcal{N}_{\tau_m} N τ m
∣ N τ m ∣ ≤ { 1 + 2 R Θ m } d Θ . \left|\mathcal{N}_{\tau_m}\right|\leq\left\{1+2R_\Theta\sqrt{m}\right\}^{d_\Theta}. ∣ N τ m ∣ ≤ { 1 + 2 R Θ m } d Θ . For fixed θ \theta θ
D ^ θ = 1 m 2 ∑ i = 1 m ∑ j = 1 m h θ ( O i ( 1 ) , O j ( 2 ) ) , E 0 [ D ^ θ ] = D θ . \widehat{D}_\theta=\frac{1}{m^2}\sum_{i=1}^{m}\sum_{j=1}^{m}h_\theta\left(O_i^{(1)},O_j^{(2)}\right),\qquad\mathbb{E}_0\left[\widehat{D}_\theta\right]=D_\theta. D θ = m 2 1 i = 1 ∑ m j = 1 ∑ m h θ ( O i ( 1 ) , O j ( 2 ) ) , E 0 [ D θ ] = D θ . Replacing one observation in either half changes D ^ θ \widehat{D}_\theta D θ 2 B h / m 2B_h/m 2 B h / m
P { ∣ D ^ θ − D θ ∣ > t } ≤ 2 exp ( − m t 2 4 B h 2 ) . \mathbb{P}\left\{\left|\widehat{D}_\theta-D_\theta\right|>t\right\}\leq2\exp\left(-\frac{mt^2}{4B_h^2}\right). P { D θ − D θ > t } ≤ 2 exp ( − 4 B h 2 m t 2 ) . A union bound over N τ m \mathcal{N}_{\tau_m} N τ m
B h c m ( δ ) . B_hc_m(\delta). B h c m ( δ ) . Since both D ^ θ \widehat{D}_\theta D θ D θ D_\theta D θ L h L_h L h 2 L h / m 2L_h/\sqrt{m} 2 L h / m
G ˉ 1 : = 1 m ∑ i = 1 m g θ ( O i ( 1 ) ) , G ˉ 2 : = 1 m ∑ j = 1 m g θ ( O j ( 2 ) ) , \bar{G}_1 := \frac{1}{m}\sum_{i=1}^{m} g_\theta(O_i^{(1)}), \qquad\bar{G}_2 := \frac{1}{m}\sum_{j=1}^{m} g_\theta(O_j^{(2)}), G ˉ 1 := m 1 i = 1 ∑ m g θ ( O i ( 1 ) ) , G ˉ 2 := m 1 j = 1 ∑ m g θ ( O j ( 2 ) ) , and
Z j : = ⟨ G ˉ 1 , g θ ( O j ( 2 ) ) − G ˉ 2 ⟩ H θ . Z_j := \langle\bar{G}_1, g_\theta(O_j^{(2)})-\bar{G}_2\rangle_{H_\theta}. Z j := ⟨ G ˉ 1 , g θ ( O j ( 2 ) ) − G ˉ 2 ⟩ H θ . Then s ^ θ 2 = m − 1 ∑ j Z j 2 \widehat{s}_\theta^2=m^{-1}\sum_j Z_j^2 s θ 2 = m − 1 ∑ j Z j 2 ∣ Z j ∣ ≤ 2 B h |Z_j|\le2B_h ∣ Z j ∣ ≤ 2 B h s ^ θ 2 \widehat{s}_\theta^2 s θ 2 C B h 2 / m CB_h^2/m C B h 2 / m C C C
P { ∣ s ^ θ 2 − E 0 s ^ θ 2 ∣ > t } ≤ 2 exp ( − c m t 2 B h 4 ) \mathbb{P}\{|\widehat{s}_\theta^2-\mathbb{E}_0\widehat{s}_\theta^2|>t\}\le2\exp\left(-c\frac{mt^2}{B_h^4}\right) P { ∣ s θ 2 − E 0 s θ 2 ∣ > t } ≤ 2 exp ( − c B h 4 m t 2 ) for a universal c > 0 c>0 c > 0
We now control the bias term. Let H = g θ ( O ) H=g_\theta(O) H = g θ ( O ) Δ = E 0 H \Delta=\mathbb{E}_0H Δ = E 0 H ξ = H − Δ \xi=H-\Delta ξ = H − Δ Σ : = E 0 [ ξ ⊗ ξ ] \Sigma:=\mathbb{E}_0[\xi\otimes\xi] Σ := E 0 [ ξ ⊗ ξ ] G ˉ 1 \bar{G}_1 G ˉ 1 s ^ θ 2 \widehat{s}_\theta^2 s θ 2 m m m ⟨ G ˉ 1 , H j ⟩ \langle\bar{G}_1,H_j\rangle ⟨ G ˉ 1 , H j ⟩
E 0 [ s ^ θ 2 ∣ G ˉ 1 ] = m − 1 m ⟨ G ˉ 1 , Σ G ˉ 1 ⟩ . \mathbb{E}_0[\widehat{s}_\theta^2\mid\bar{G}_1]=\frac{m-1}{m}\langle\bar{G}_1,\Sigma\bar{G}_1\rangle. E 0 [ s θ 2 ∣ G ˉ 1 ] = m m − 1 ⟨ G ˉ 1 , Σ G ˉ 1 ⟩ . Since G ˉ 1 = Δ + m − 1 ∑ i ξ i \bar{G}_1=\Delta+m^{-1}\sum_i\xi_i G ˉ 1 = Δ + m − 1 ∑ i ξ i
E 0 ⟨ G ˉ 1 , Σ G ˉ 1 ⟩ = s θ 2 + 1 m E 0 ⟨ ξ , Σ ξ ⟩ . \mathbb{E}_0\langle\bar{G}_1,\Sigma\bar{G}_1\rangle=s_\theta^2+\frac{1}{m}\mathbb{E}_0\langle\xi,\Sigma\xi\rangle. E 0 ⟨ G ˉ 1 , Σ G ˉ 1 ⟩ = s θ 2 + m 1 E 0 ⟨ ξ , Σ ξ ⟩ . Thus
E 0 s ^ θ 2 − s θ 2 = − s θ 2 m + m − 1 m 2 E 0 ⟨ ξ , Σ ξ ⟩ , \mathbb{E}_0\widehat{s}_\theta^2-s_\theta^2=-\frac{s_\theta^2}{m}+\frac{m-1}{m^2}\mathbb{E}_0\langle\xi,\Sigma\xi\rangle, E 0 s θ 2 − s θ 2 = − m s θ 2 + m 2 m − 1 E 0 ⟨ ξ , Σ ξ ⟩ , and boundedness of h θ h_\theta h θ
∣ E 0 s ^ θ 2 − s θ 2 ∣ ≤ C B h 2 m . |\mathbb{E}_0\widehat{s}_\theta^2-s_\theta^2|\le C\frac{B_h^2}{m}. ∣ E 0 s θ 2 − s θ 2 ∣ ≤ C m B h 2 . Finally, θ ↦ s ^ θ 2 \theta\mapsto\widehat{s}_\theta^2 θ ↦ s θ 2 8 B h L h 8B_hL_h 8 B h L h θ ↦ s θ 2 \theta\mapsto s_\theta^2 θ ↦ s θ 2 4 B h L h 4B_hL_h 4 B h L h
Lemma B.12 (Stable ratio bound). Let D , D ^ ∈ R D,\widehat{D}\in\mathbb{R} D , D ∈ R v , v ^ ∈ R + v,\widehat{v}\in\mathbb{R}_+ v , v ∈ R + s , B > 0 s,B>0 s , B > 0
0 ≤ D ≤ B , v ≥ s 2 , ∣ v ^ − v ∣ ≤ s 2 / 2. 0\le D\le B,\qquad v\ge s^2,\qquad|\widehat{v}-v|\le s^2/2. 0 ≤ D ≤ B , v ≥ s 2 , ∣ v − v ∣ ≤ s 2 /2. Then, for every λ ≥ 0 \lambda\ge0 λ ≥ 0
∣ D ^ v ^ + λ − D v ∣ ≤ 2 s ∣ D ^ − D ∣ + 2 B s 3 { ∣ v ^ − v ∣ + λ } . (20) \left|\frac{\widehat{D}}{\sqrt{\widehat{v}+\lambda}}-\frac{D}{\sqrt{v}}\right|
\le\frac{\sqrt{2}}{s}|\widehat{D}-D|+\frac{\sqrt{2B}}{s^3}\{|\widehat{v}-v|+\lambda\}. \tag*{(20)} v + λ D − v D ≤ s 2 ∣ D − D ∣ + s 3 2 B { ∣ v − v ∣ + λ } . (20) Proof. The variance condition gives v ^ ≥ s 2 / 2 \widehat{v}\ge s^2/2 v ≥ s 2 /2
∣ D ^ v ^ + λ − D v ∣ ≤ ∣ D ^ − D ∣ v ^ + λ + D ∣ ( v ^ + λ ) − 1 / 2 − v − 1 / 2 ∣ ≤ 2 s ∣ D ^ − D ∣ + 2 D s 3 ∣ v ^ + λ − v ∣ , \begin{aligned}
\left|\frac{\widehat{D}}{\sqrt{\widehat{v}+\lambda}}-\frac{D}{\sqrt{v}}\right|
\le\frac{|\widehat{D}-D|}{\sqrt{\widehat{v}+\lambda}}+D\left|(\widehat{v}+\lambda)^{-1/2}-v^{-1/2}\right| \\
&\le\frac{\sqrt{2}}{s}|\widehat{D}-D|+\frac{\sqrt{2D}}{s^3}|\widehat{v}+\lambda-v|,
\end{aligned} v + λ D − v D ≤ v + λ ∣ D − D ∣ + D ( v + λ ) − 1/2 − v − 1/2 ≤ s 2 ∣ D − D ∣ + s 3 2 D ∣ v + λ − v ∣ , where the second line uses the mean-value theorem on [ s 2 / 2 , ∞ ) [s^2/2,\infty) [ s 2 /2 , ∞ ) D ≤ B D\le B D ≤ B
Theorem B.13 (Variance-stable oracle adaptation). Let s > 0 s > 0 s > 0 Θ s ≠ ∅ \Theta_s \ne\varnothing Θ s = ∅ r s , m ( δ ) ≤ s 2 / 2 r_{s,m}(\delta) \le s^2/2 r s , m ( δ ) ≤ s 2 /2 1 − δ 1-\delta 1 − δ
sup θ ∈ Θ s ∣ J ^ λ m , θ o r − J θ ∣ ≤ ϵ m ( δ ) , (21) \sup_{\theta\in\Theta_s} \left|\hat{J}^{\mathrm{or}}_{\lambda_m,\theta} - J_\theta\right| \le\epsilon_m(\delta), \tag*{(21)} θ ∈ Θ s sup J ^ λ m , θ or − J θ ≤ ϵ m ( δ ) , (21) where
ϵ m ( δ ) : = 2 s r D , m ( δ ) + 2 B h s 3 { r s , m ( δ ) + λ m } . (22) \epsilon_m(\delta) := \frac{\sqrt{2}}{s}r_{D,m}(\delta) + \frac{\sqrt{2B_h}}{s^3}\{r_{s,m}(\delta)+\lambda_m\}. \tag*{(22)} ϵ m ( δ ) := s 2 r D , m ( δ ) + s 3 2 B h { r s , m ( δ ) + λ m } . (22) If B h B_h B h L h L_h L h s s s
d Θ log ( 1 + 2 R Θ m ) + 1 m ⟶ 0 , λ m = O ( d Θ log ( 1 + 2 R Θ m ) + 1 m ) , \frac{d_\Theta\log(1+2R_\Theta\sqrt{m})+1}{m}\longrightarrow0,\qquad\lambda_m=O\left(\sqrt{\frac{d_\Theta\log(1+2R_\Theta\sqrt{m})+1}{m}}\right), m d Θ log ( 1 + 2 R Θ m ) + 1 ⟶ 0 , λ m = O ( m d Θ log ( 1 + 2 R Θ m ) + 1 ) , then
sup θ ∈ Θ s ∣ J ^ λ m , θ o r − J θ ∣ = O P ( d Θ log ( 1 + 2 R Θ m ) + 1 m ) . \sup_{\theta\in\Theta_s}\left|\hat{J}^{\mathrm{or}}_{\lambda_m,\theta}-J_\theta\right|=O_{\mathbb{P}}\left(\sqrt{\frac{d_\Theta\log(1+2R_\Theta\sqrt{m})+1}{m}}\right). θ ∈ Θ s sup J ^ λ m , θ or − J θ = O P ( m d Θ log ( 1 + 2 R Θ m ) + 1 ) . Proof of Theorem B.13. On the event in Lemma B.11, every θ ∈ Θ s \theta\in\Theta_s θ ∈ Θ s
s ^ θ , o r 2 ≥ s θ 2 − r s , m ( δ ) ≥ s 2 / 2. \hat{s}_{\theta,\mathrm{or}}^2 \ge s_\theta^2-r_{s,m}(\delta)\ge s^2/2. s ^ θ , or 2 ≥ s θ 2 − r s , m ( δ ) ≥ s 2 /2. Moreover, 0 ≤ D θ = E 0 h θ ( O 1 , O 2 ) ≤ B h 0\le D_\theta=\mathbb{E}_0h_\theta(O_1,O_2)\le B_h 0 ≤ D θ = E 0 h θ ( O 1 , O 2 ) ≤ B h Lemma B.12 pointwise with
D = D θ , D ^ = D ^ θ , o r c r o s s , v = s θ 2 , v ^ = s ^ θ , o r 2 , B = B h , D=D_\theta,\qquad\widehat{D}=\widehat{D}^{\mathrm{cross}}_{\theta,\mathrm{or}},\qquad v=s_\theta^2,\qquad\widehat{v}=\widehat{s}_{\theta,\mathrm{or}}^2,\qquad B=B_h, D = D θ , D = D θ , or cross , v = s θ 2 , v = s θ , or 2 , B = B h , and take the supremum over Θ s \Theta_s Θ s ϵ m ( δ ) \epsilon_m(\delta) ϵ m ( δ ) r D , m r_{D,m} r D , m r s , m r_{s,m} r s , m
Theorem 1 of Gretton et al. [20 ] and Theorem 11 of Liu et al. [33 ] choose a ridge of order t − 1 / 3 t^{-1/3} t − 1/3 t t t t t t λ t − 1 / 2 \lambda_t^{-1/2} λ t − 1/2 O ( λ t ) O(\lambda_t) O ( λ t ) Θ s \Theta_s Θ s Lemma B.12 avoids this amplification. The same refinement applies on the variance-bounded subclass of Liu et al. using their Propositions 15-16 and λ t = O ( log t / t ) \lambda_t=O(\sqrt{\log t/t}) λ t = O ( log t / t )
Nuisance Perturbation
We now transfer the oracle concentration bounds to fitted pseudo-features.
Lemma B.14 (Score envelopes from selection regularity). Under Assumption B.5, on a common full-measure set,
sup θ ∈ Θ { ∥ g θ π , π ′ ( o ) ∥ H θ ∨ ∥ g ~ θ ( o ) ∥ H θ } ≤ M g , \sup_{\theta\in\Theta}\left\{\left\|g_{\theta}^{\pi,\pi'}(o)\right\|_{\mathcal{H}_\theta}\vee\left\|\widetilde{g}_\theta(o)\right\|_{\mathcal{H}_\theta}\right\}\le M_g, θ ∈ Θ sup { g θ π , π ′ ( o ) H θ ∨ ∥ g θ ( o ) ∥ H θ } ≤ M g , where
M g : = max { 2 ( W + 1 ) κ , W ( 1 + A ) ( κ + B μ ) + 2 B μ } . M_g:=\max\left\{2(W+1)\sqrt{\kappa},W(1+A)(\sqrt{\kappa}+B_\mu)+2B_\mu\right\}. M g := max { 2 ( W + 1 ) κ , W ( 1 + A ) ( κ + B μ ) + 2 B μ } . Consequently ∣ h ~ θ ( o , o ′ ) ∣ ≤ M g 2 \left|\widetilde{h}_\theta(o,o')\right|\le M_g^2 h θ ( o , o ′ ) ≤ M g 2
Proof. The diagonal kernel bound and Jensen’s inequality give ∥ ϕ θ ( y ) ∥ ≤ κ \lVert\phi_\theta(y)\rVert\le\sqrt{\kappa} ∥ ϕ θ ( y )∥ ≤ κ ∥ μ θ ( a , x ) ∥ ≤ κ \lVert\mu_\theta(a,x)\rVert\le\sqrt{\kappa} ∥ μ θ ( a , x )∥ ≤ κ ∫ ∣ r ( a , x ) ∣ ρ ( d a ) ≤ 2 \int|r(a,x)|\rho(\mathrm{d}a) \le2 ∫ ∣ r ( a , x ) ∣ ρ ( d a ) ≤ 2
∥ g θ ( o ) ∥ ≤ ∣ r ( a , x ) ∣ π 0 ( a ∣ x ) ∥ ϕ θ ( y ) − μ θ ( a , x ) ∥ + ∫ ∣ r ( a ~ , x ) ∣ ∥ μ θ ( a ~ , x ) ∥ ρ ( d a ~ ) . \lVert g_\theta(o)\rVert\le\frac{|r(a,x)|}{\pi_0(a\mid x)}\lVert\phi_\theta(y)-\mu_\theta(a,x)\rVert+\int|r(\tilde a,x)|\lVert\mu_\theta(\tilde a,x)\rVert\rho(\mathrm{d}\tilde a). ∥ g θ ( o )∥ ≤ π 0 ( a ∣ x ) ∣ r ( a , x ) ∣ ∥ ϕ θ ( y ) − μ θ ( a , x )∥ + ∫ ∣ r ( a ~ , x ) ∣ ∥ μ θ ( a ~ , x )∥ ρ ( d a ~ ) . On the support of ∣ r ∣ |r| ∣ r ∣
∣ r ∣ π ^ 0 = ∣ r ∣ π 0 π 0 π ^ 0 ≤ W ( 1 + A ) , \frac{|r|}{\widehat{\pi}_0}=\frac{|r|\pi_0}{\pi_0\widehat{\pi}_0}\le W(1+A), π 0 ∣ r ∣ = π 0 π 0 ∣ r ∣ π 0 ≤ W ( 1 + A ) , which gives the fitted bound in the same way. Cauchy-Schwarz gives the final claim.
Write B ~ h : = M q 2 \widetilde B_h:=M_q^2 B h := M q 2 r ~ D , m ( δ ) \widetilde r_{D,m}(\delta) r D , m ( δ ) r ~ s , m ( δ ) \widetilde r_{s,m}(\delta) r s , m ( δ ) ( B h , L h ) (B_h,L_h) ( B h , L h ) ( B ~ h , L ) (\widetilde B_h,L) ( B h , L )
Lemma B.15 (Exact nuisance decomposition). Under Assumptions B.5 and 3.1, define
α ( a , x ) : = π 0 ( a ∣ x ) π ^ 0 ( a ∣ x ) − 1 , d θ ( a , x ) : = μ ^ θ ( a , x ) − μ θ ( a , x ) , q ( a , x ) : = r ( a , x ) π 0 ( a ∣ x ) , \alpha(a,x):=\frac{\pi_0(a\mid x)}{\widehat{\pi}_0(a\mid x)}-1,\qquad d_\theta(a,x):=\widehat{\mu}_\theta(a,x)-\mu_\theta(a,x),\qquad q(a,x):=\frac{r(a,x)}{\pi_0(a\mid x)}, α ( a , x ) := π 0 ( a ∣ x ) π 0 ( a ∣ x ) − 1 , d θ ( a , x ) := μ θ ( a , x ) − μ θ ( a , x ) , q ( a , x ) := π 0 ( a ∣ x ) r ( a , x ) , where q : = 0 q:=0 q := 0 { ∣ r ∣ > 0 } \{|r|>0\} { ∣ r ∣ > 0 } ξ θ ( O ) : = ϕ θ ( Y ) − μ θ ( A , X ) \xi_\theta(O):=\phi_\theta(Y)-\mu_\theta(A,X) ξ θ ( O ) := ϕ θ ( Y ) − μ θ ( A , X ) t θ , d ( X ) : = ∫ r ( a , X ) d θ ( a , X ) ρ ( d a ) t_{\theta,d}(X):=\int r(a,X)d_\theta(a,X)\rho(\mathrm{d}a) t θ , d ( X ) := ∫ r ( a , X ) d θ ( a , X ) ρ ( d a )
g ~ θ − g θ = q α ξ θ + { t θ , d − q d θ } − q α d θ . (23) \widetilde g_\theta-g_\theta=q\alpha\xi_\theta+\{t_{\theta,d}-qd_\theta\}-q\alpha d_\theta. \tag*{(23)} g θ − g θ = q α ξ θ + { t θ , d − q d θ } − q α d θ . (23) The first term has conditional mean zero given ( A , X ) (A,X) ( A , X ) X X X B θ : = Δ ~ θ − Δ θ B_\theta:=\widetilde\Delta_\theta-\Delta_\theta B θ := Δ θ − Δ θ
B θ = − E 0 [ ∫ A r ( a , X ) α ( a , X ) d θ ( a , X ) ρ ( d a ) ] , sup θ ∈ Θ ∥ B θ ∥ ≤ ϵ π , n ϵ μ , n . (24) B_\theta=-\mathbb{E}_0\left[\int_{\mathcal A}r(a,X)\alpha(a,X)d_\theta(a,X)\rho(\mathrm{d}a)\right],\qquad\sup_{\theta\in\Theta}\lVert B_\theta\rVert\le\epsilon_{\pi,n}\epsilon_{\mu,n}. \tag*{(24)} B θ = − E 0 [ ∫ A r ( a , X ) α ( a , X ) d θ ( a , X ) ρ ( d a ) ] , θ ∈ Θ sup ∥ B θ ∥ ≤ ϵ π , n ϵ μ , n . (24) Moreover, the squared L 2 ( P 0 ; H θ ) L^2(\mathbb{P}_0;\mathcal H_\theta) L 2 ( P 0 ; H θ )
4 κ W ϵ π , n 2 , W ϵ μ , n 2 , W ( B μ + κ ) 2 ϵ π , n 2 . (25) 4\kappa W\epsilon_{\pi,n}^2,\qquad W\epsilon_{\mu,n}^2,\qquad W(B_\mu+\sqrt{\kappa})^2\epsilon_{\pi,n}^2. \tag*{(25)} 4 κW ϵ π , n 2 , W ϵ μ , n 2 , W ( B μ + κ ) 2 ϵ π , n 2 . (25) Proof. Because r / π ^ 0 = q ( 1 + α ) r/\widehat{\pi}_0=q(1+\alpha) r / π 0 = q ( 1 + α ) μ ^ θ = μ θ + d θ \widehat{\mu}_\theta=\mu_\theta+d_\theta μ θ = μ θ + d θ
g ~ θ − g θ = q ( 1 + α ) ( ξ θ − d θ ) + ∫ r ( μ θ + d θ ) ρ ( d a ) − q ξ θ − ∫ r μ θ ρ ( d a ) = q α ξ θ + t θ , d − q d θ − q α d θ . \begin{aligned}
\widetilde g_\theta-g_\theta=q(1+\alpha)(\xi_\theta-d_\theta)+\int r(\mu_\theta+d_\theta)\rho(\mathrm{d}a)-q\xi_\theta-\int r\mu_\theta\rho(\mathrm{d}a) \\
&=q\alpha\xi_\theta+t_{\theta,d}-qd_\theta-q\alpha d_\theta.
\end{aligned} g θ − g θ = q ( 1 + α ) ( ξ θ − d θ ) + ∫ r ( μ θ + d θ ) ρ ( d a ) − q ξ θ − ∫ r μ θ ρ ( d a ) = q α ξ θ + t θ , d − q d θ − q α d θ . where ∫ f d ρ \int f\,\mathrm{d}\rho ∫ f d ρ A \mathcal A A E 0 [ ξ θ ∣ A , X ] = 0 \mathbb{E}_0[\xi_\theta\mid A,X]=0 E 0 [ ξ θ ∣ A , X ] = 0
E 0 [ q d θ ( A , X ) ∣ X ] = ∫ r ( a , X ) d θ ( a , X ) ρ ( d a ) = t θ , d ( X ) , \mathbb{E}_0[qd_\theta(A,X)\mid X]=\int r(a,X)d_\theta(a,X)\rho(\mathrm{d}a)=t_{\theta,d}(X), E 0 [ q d θ ( A , X ) ∣ X ] = ∫ r ( a , X ) d θ ( a , X ) ρ ( d a ) = t θ , d ( X ) , which proves the second. Only the last term contributes to the unconditional mean, and conditioning on X X X B θ B_\theta B θ ∣ r ( a , x ) ∣ P X ( d x ) ρ ( d a ) |r(a,x)|\mathbb{P}_X(\mathrm{d}x)\rho(\mathrm{d}a) ∣ r ( a , x ) ∣ P X ( d x ) ρ ( d a )
For the component bounds, ∥ ξ θ ∥ ≤ 2 κ \lVert\xi_\theta\rVert\le2\sqrt{\kappa} ∥ ξ θ ∥ ≤ 2 κ r 2 / π 0 ≤ W ∣ r ∣ r^2/\pi_0\le W|r| r 2 / π 0 ≤ W ∣ r ∣
E 0 ∥ q α ξ θ ∥ 2 ≤ 4 κ W ϵ π , n 2 . \mathbb{E}_0\lVert q\alpha\xi_\theta\rVert^2\le4\kappa W\epsilon_{\pi,n}^2. E 0 ∥ q α ξ θ ∥ 2 ≤ 4 κW ϵ π , n 2 . The middle term is the negative of q d θ qd_\theta q d θ X X X
E 0 ∥ t θ , d − q d θ ∥ 2 ≤ E 0 ∥ q d θ ∥ 2 ≤ W ϵ μ , n 2 . \mathbb{E}_0\lVert t_{\theta,d}-qd_\theta\rVert^2\le\mathbb{E}_0\lVert qd_\theta\rVert^2\le W\epsilon_{\mu,n}^2. E 0 ∥ t θ , d − q d θ ∥ 2 ≤ E 0 ∥ q d θ ∥ 2 ≤ W ϵ μ , n 2 . Finally, ∥ d θ ∥ ≤ B μ + κ \lVert d_\theta\rVert\le B_\mu+\sqrt{\kappa} ∥ d θ ∥ ≤ B μ + κ
Lemma B.16 (Sharp population nuisance perturbation). Under Assumptions B.5 and 3.1, for every θ ∈ Θ \theta\in\Theta θ ∈ Θ
D ~ θ − D θ = 2 ⟨ Δ θ , B θ ⟩ + ∥ B θ ∥ 2 , ∣ D ~ θ − D θ ∣ ≤ 4 κ ϵ π , n ϵ μ , n + ϵ π , n 2 ϵ μ , n 2 . (26) \widetilde{D}_{\theta} - D_{\theta} = 2\langle\Delta_{\theta},B_{\theta}\rangle+ \lVert B_{\theta}\rVert^{2},
\qquad
\left|\widetilde{D}_{\theta} - D_{\theta}\right| \le4\sqrt{\kappa}\epsilon_{\pi,n}\epsilon_{\mu,n}+\epsilon_{\pi,n}^{2}\epsilon_{\mu,n}^{2}. \tag*{(26)} D θ − D θ = 2 ⟨ Δ θ , B θ ⟩ + ∥ B θ ∥ 2 , D θ − D θ ≤ 4 κ ϵ π , n ϵ μ , n + ϵ π , n 2 ϵ μ , n 2 . (26) There is a finite C C C ( κ , W , A , B μ ) (\kappa,W,A,B_{\mu}) ( κ , W , A , B μ )
sup θ ∈ Θ ∣ s ~ θ 2 − s θ 2 ∣ ≤ C { ϵ π , n + ϵ π , n 2 + ϵ π , n ϵ μ , n + ϵ μ , n 2 } ≤ C { ϵ π , n + ϵ μ , n 2 } . (27) \sup_{\theta\in\Theta}\left|\widetilde{s}_{\theta}^{2}-s_{\theta}^{2}\right|
\le C\left\{\epsilon_{\pi,n}+\epsilon_{\pi,n}^{2}+\epsilon_{\pi,n}\epsilon_{\mu,n}+\epsilon_{\mu,n}^{2}\right\}
\le C\left\{\epsilon_{\pi,n}+\epsilon_{\mu,n}^{2}\right\}. \tag*{(27)} θ ∈ Θ sup s θ 2 − s θ 2 ≤ C { ϵ π , n + ϵ π , n 2 + ϵ π , n ϵ μ , n + ϵ μ , n 2 } ≤ C { ϵ π , n + ϵ μ , n 2 } . (27) If π ^ 0 = π 0 \widehat{\pi}_{0}=\pi_{0} π 0 = π 0 ∣ r ∣ |r| ∣ r ∣ B θ = 0 B_{\theta}=0 B θ = 0
D ~ θ = D θ , s ~ θ 2 = s θ 2 + E 0 [ ⟨ Δ θ , g ~ θ ( O ) − g θ ( O ) ⟩ 2 ] , (28) \widetilde{D}_{\theta}=D_{\theta},
\qquad
\widetilde{s}_{\theta}^{2}=s_{\theta}^{2}+\mathbb{E}_{0}\left[\langle\Delta_{\theta},\widetilde{g}_{\theta}(O)-g_{\theta}(O)\rangle^{2}\right], \tag*{(28)} D θ = D θ , s θ 2 = s θ 2 + E 0 [ ⟨ Δ θ , g θ ( O ) − g θ ( O ) ⟩ 2 ] , (28) so 0 ≤ s ~ θ 2 − s θ 2 ≤ 4 κ W ϵ μ , n 2 0\le\widetilde{s}_{\theta}^{2}-s_{\theta}^{2}\le4\kappa W\epsilon_{\mu,n}^{2} 0 ≤ s θ 2 − s θ 2 ≤ 4 κW ϵ μ , n 2
Proof. The numerator identity follows from Δ ~ θ = Δ θ + B θ \widetilde{\Delta}_{\theta}=\Delta_{\theta}+B_{\theta} Δ θ = Δ θ + B θ ∥ Δ θ ∥ ≤ 2 κ \lVert\Delta_{\theta}\rVert\le2\sqrt{\kappa} ∥ Δ θ ∥ ≤ 2 κ (24) gives the stated bound.
To bound the variance error, we treat the propensity and outcome components separately. Suppress θ \theta θ
G : = g − Δ , e : = g ~ − g , e ˉ : = e − B , Z : = ⟨ Δ , G ⟩ . G:=g-\Delta,\qquad e:=\widetilde{g}-g,\qquad\bar{e}:=e-B,\qquad Z:=\langle\Delta,G\rangle. G := g − Δ , e := g − g , e ˉ := e − B , Z := ⟨ Δ , G ⟩ . The centred fitted projection is
Z ~ : = ⟨ Δ + B , G + e ˉ ⟩ = Z + R , R : = ⟨ Δ , e ˉ ⟩ + ⟨ B , G + e ˉ ⟩ . \widetilde{Z}:=\langle\Delta+B,G+\bar{e}\rangle=Z+R,\qquad R:=\langle\Delta,\bar{e}\rangle+\langle B,G+\bar{e}\rangle. Z := ⟨ Δ + B , G + e ˉ ⟩ = Z + R , R := ⟨ Δ , e ˉ ⟩ + ⟨ B , G + e ˉ ⟩ . Both Z Z Z R R R
s ~ 2 − s 2 = 2 E 0 [ Z R ] + E 0 [ R 2 ] . (29) \widetilde{s}^{2}-s^{2}=2\mathbb{E}_{0}[ZR]+\mathbb{E}_{0}[R^{2}]. \tag*{(29)} s 2 − s 2 = 2 E 0 [ Z R ] + E 0 [ R 2 ] . (29) Let p : = q α ξ p:=q\alpha\xi p := q α ξ u : = t d − q d u:=t d-q d u := t d − q d v : = − q α d v:=-q\alpha d v := − q α d (23) . Also set t ( X ) : = ∫ r ( a , X ) μ ( a , X ) ρ ( d a ) t(X):=\int r(a,X)\mu(a,X)\rho(\mathrm{d}a) t ( X ) := ∫ r ( a , X ) μ ( a , X ) ρ ( d a )
G = q ξ + t ( X ) − Δ . G=q\xi+t(X)-\Delta. G = q ξ + t ( X ) − Δ. The conditional-embedding term has the exact cancellation
E 0 [ Z ⟨ Δ , u ⟩ ] = 0. (30) \mathbb{E}_{0}[Z\langle\Delta,u\rangle]=0. \tag*{(30)} E 0 [ Z ⟨ Δ , u ⟩] = 0. (30) Indeed, u u u ( A , X ) (A,X) ( A , X ) E 0 [ ξ ∣ A , X ] = 0 \mathbb{E}_{0}[\xi\mid A,X]=0 E 0 [ ξ ∣ A , X ] = 0 q ξ q\xi q ξ t ( X ) − Δ t(X)-\Delta t ( X ) − Δ X X X E 0 [ u ∣ X ] = 0 \mathbb{E}_{0}[u\mid X]=0 E 0 [ u ∣ X ] = 0 ϵ μ , n \epsilon_{\mu,n} ϵ μ , n
The same calculation does not cancel the propensity contribution. Instead,
E 0 [ Z ⟨ Δ , p ⟩ ] = E 0 [ q ( A , X ) 2 α ( A , X ) ⟨ Δ , ξ ( O ) ⟩ 2 ] , (31) \mathbb{E}_{0}[Z\langle\Delta,p\rangle]
=\mathbb{E}_{0}\left[q(A,X)^{2}\alpha(A,X)\langle\Delta,\xi(O)\rangle^{2}\right], \tag*{(31)} E 0 [ Z ⟨ Δ , p ⟩] = E 0 [ q ( A , X ) 2 α ( A , X ) ⟨ Δ , ξ ( O ) ⟩ 2 ] , (31) which is not necessarily zero. Since r 2 / π 0 ≤ W ∣ r ∣ r^{2}/\pi_{0}\le W|r| r 2 / π 0 ≤ W ∣ r ∣
∣ E 0 [ Z ⟨ Δ , p ⟩ ] ∣ ≤ C E 0 ∫ ∣ r ( a , X ) ∣ ∣ α ( a , X ) ∣ ρ ( d a ) ≤ C ϵ π , n , \left|\mathbb{E}_{0}[Z\langle\Delta,p\rangle]\right|
\le C\mathbb{E}_{0}\int|r(a,X)||\alpha(a,X)|\rho(\mathrm{d}a)
\le C\epsilon_{\pi,n}, ∣ E 0 [ Z ⟨ Δ , p ⟩] ∣ ≤ C E 0 ∫ ∣ r ( a , X ) ∣∣ α ( a , X ) ∣ ρ ( d a ) ≤ C ϵ π , n , where E 0 ∫ ∣ r ∣ d ρ ≤ 2 \mathbb{E}_{0}\int|r|\mathrm{d}\rho\le2 E 0 ∫ ∣ r ∣ d ρ ≤ 2
For the mixed term,
E 0 [ v ∣ X ] = − ∫ r ( a , X ) α ( a , X ) d ( a , X ) ρ ( d a ) . \mathbb{E}_{0}[v\mid X]=-\int r(a,X)\alpha(a,X)d(a,X)\rho(\mathrm{d}a). E 0 [ v ∣ X ] = − ∫ r ( a , X ) α ( a , X ) d ( a , X ) ρ ( d a ) . The residual part q ξ q\xi q ξ v v v
E 0 ∥ ∫ r α d d ρ ∥ ≤ ϵ π , n ϵ μ , n . \mathbb{E}_{0}\left\lVert\int r\alpha d\,\mathrm{d}\rho\right\rVert
\le\epsilon_{\pi,n}\epsilon_{\mu,n}. E 0 ∫ r α d d ρ ≤ ϵ π , n ϵ μ , n . Since t ( X ) t(X) t ( X ) Δ \Delta Δ
∣ E 0 [ Z ⟨ Δ , v − B ⟩ ] ∣ + ∣ E 0 [ Z ⟨ B , G + e ˉ ⟩ ] ∣ ≤ C ϵ π , n ϵ μ , n . \left|\mathbb{E}_0[Z\langle\Delta,v-B\rangle]\right|+\left|\mathbb{E}_0[Z\langle B,G+\bar e\rangle]\right|\le C\epsilon_{\pi,n}\epsilon_{\mu,n}. ∣ E 0 [ Z ⟨ Δ , v − B ⟩] ∣ + ∣ E 0 [ Z ⟨ B , G + e ˉ ⟩] ∣ ≤ C ϵ π , n ϵ μ , n . We now control the quadratic term in (29) . From (25) ,
E 0 ∥ e ˉ ∥ 2 ≤ E 0 ∥ e ∥ 2 ≤ C ( ϵ π , n 2 + ϵ μ , n 2 ) . \mathbb{E}_0\|\bar e\|^2\le\mathbb{E}_0\|e\|^2\le C(\epsilon_{\pi,n}^2+\epsilon_{\mu,n}^2). E 0 ∥ e ˉ ∥ 2 ≤ E 0 ∥ e ∥ 2 ≤ C ( ϵ π , n 2 + ϵ μ , n 2 ) . Using ∥ B ∥ ≤ ϵ π , n ϵ μ , n \|B\|\le\epsilon_{\pi,n}\epsilon_{\mu,n} ∥ B ∥ ≤ ϵ π , n ϵ μ , n
E 0 [ R 2 ] ≤ C { ϵ π , n 2 + ϵ μ , n 2 + ϵ π , n 2 ϵ μ , n 2 } . \mathbb{E}_0[R^2]\le C\{\epsilon_{\pi,n}^2+\epsilon_{\mu,n}^2+\epsilon_{\pi,n}^2\epsilon_{\mu,n}^2\}. E 0 [ R 2 ] ≤ C { ϵ π , n 2 + ϵ μ , n 2 + ϵ π , n 2 ϵ μ , n 2 } . Substituting into (29) proves the first inequality in (27) . Under Assumption B.5, both nuisance errors are bounded by constants: ∣ α ∣ ≤ A |\alpha|\le A ∣ α ∣ ≤ A ∥ d ∥ ≤ B μ + κ \|d\|\le B_\mu+\sqrt{\kappa} ∥ d ∥ ≤ B μ + κ ∣ r ∣ |r| ∣ r ∣ C ( ϵ π , n + ϵ μ , n ) C(\epsilon_{\pi,n}+\epsilon_{\mu,n}) C ( ϵ π , n + ϵ μ , n )
If π ^ 0 = π 0 \hat\pi_0=\pi_0 π ^ 0 = π 0 p = v = B = 0 p=v=B=0 p = v = B = 0 e = u e=u e = u (30) yields
s ^ 2 = E 0 { Z + ⟨ Δ , u ⟩ } 2 = s 2 + E 0 ⟨ Δ , u ⟩ 2 . \hat s^2=\mathbb{E}_0\{Z+\langle\Delta,u\rangle\}^2=s^2+\mathbb{E}_0\langle\Delta,u\rangle^2. s ^ 2 = E 0 { Z + ⟨ Δ , u ⟩ } 2 = s 2 + E 0 ⟨ Δ , u ⟩ 2 . The middle component bound in (25) and ∥ Δ ∥ 2 ≤ 4 κ \|\Delta\|^2\le4\kappa ∥Δ ∥ 2 ≤ 4 κ
Uniform Criterion Bound and Consequences
Proof of Theorem 4.4. Conditional on F f i t \mathcal{F}_{\mathrm{fit}} F fit 1 − η 1-\eta 1 − η
sup θ ∈ Θ ∣ D ^ θ c r o s s − D ~ θ ∣ ≤ r ~ D , m ( η ) , sup θ ∈ Θ ∣ s ^ θ 2 − s ~ θ 2 ∣ ≤ r ~ s , m ( η ) . (32) \sup_{\theta\in\Theta}\left|\hat D^{\mathrm{cross}}_\theta-\widetilde D_\theta\right|\le\widetilde r_{D,m}(\eta),\qquad
\sup_{\theta\in\Theta}\left|\hat s_\theta^2-\widetilde s_\theta^2\right|\le\widetilde r_{s,m}(\eta). \tag*{(32)} θ ∈ Θ sup D ^ θ cross − D θ ≤ r D , m ( η ) , θ ∈ Θ sup s ^ θ 2 − s θ 2 ≤ r s , m ( η ) . (32) The definitions of these two radii and of c m c_m c m r ~ D , m ( η ) + r ~ s , m ( η ) ≤ C c m ( η ) \widetilde r_{D,m}(\eta)+\widetilde r_{s,m}(\eta)\le Cc_m(\eta) r D , m ( η ) + r s , m ( η ) ≤ C c m ( η ) (32) with (26) and (27) gives the componentwise bounds
sup θ ∈ Θ ∣ D ^ θ c r o s s − D θ ∣ ≤ C { c m ( η ) + ϵ π , n ϵ μ , n } , (33) \sup_{\theta\in\Theta}\left|\hat D^{\mathrm{cross}}_\theta-D_\theta\right|\le C\{c_m(\eta)+\epsilon_{\pi,n}\epsilon_{\mu,n}\}, \tag*{(33)} θ ∈ Θ sup D ^ θ cross − D θ ≤ C { c m ( η ) + ϵ π , n ϵ μ , n } , (33) sup θ ∈ Θ ∣ s ^ θ 2 − s θ 2 ∣ ≤ C { c m ( η ) + ϵ π , n + ϵ μ , n 2 } . (34) \sup_{\theta\in\Theta}\left|\hat s_\theta^2-s_\theta^2\right|\le C\{c_m(\eta)+\epsilon_{\pi,n}+\epsilon_{\mu,n}^2\}. \tag*{(34)} θ ∈ Θ sup s ^ θ 2 − s θ 2 ≤ C { c m ( η ) + ϵ π , n + ϵ μ , n 2 } . (34) Here the bounded nuisance envelopes absorb ( ϵ π , n ϵ μ , n ) 2 (\epsilon_{\pi,n}\epsilon_{\mu,n})^2 ( ϵ π , n ϵ μ , n ) 2
For the ratio, decompose
∣ J ^ λ , θ D R − J λ , θ ∣ ≤ ∣ D ^ θ c r o s s − D θ ∣ λ + D θ ∣ ( s ^ θ 2 + λ ) − 1 / 2 − ( s θ 2 + λ ) − 1 / 2 ∣ ≤ ∣ D ^ θ c r o s s − D θ ∣ λ 0 + 2 κ λ 0 3 / 2 ∣ s ^ θ 2 − s θ 2 ∣ . \begin{aligned}
\left|\hat J^{\mathrm{DR}}_{\lambda,\theta}-J_{\lambda,\theta}\right|
\le\frac{\left|\hat D^{\mathrm{cross}}_\theta-D_\theta\right|}{\sqrt{\lambda}}
+D_\theta\left|(\hat s_\theta^2+\lambda)^{-1/2}-(s_\theta^2+\lambda)^{-1/2}\right| \\
&\le\frac{\left|\hat D^{\mathrm{cross}}_\theta-D_\theta\right|}{\sqrt{\lambda_0}}
+\frac{2\kappa}{\lambda_0^{3/2}}\left|\hat s_\theta^2-s_\theta^2\right|.
\end{aligned} J ^ λ , θ DR − J λ , θ ≤ λ D ^ θ cross − D θ + D θ ( s ^ θ 2 + λ ) − 1/2 − ( s θ 2 + λ ) − 1/2 ≤ λ 0 D ^ θ cross − D θ + λ 0 3/2 2 κ s ^ θ 2 − s θ 2 . where D θ ≤ 4 κ D_\theta\le4\kappa D θ ≤ 4 κ v ↦ ( v + λ ) − 1 / 2 v\mapsto(v+\lambda)^{-1/2} v ↦ ( v + λ ) − 1/2 ( 2 λ 0 3 / 2 ) − 1 (2\lambda_0^{3/2})^{-1} ( 2 λ 0 3/2 ) − 1 (33) to (34) proves (8) .
For a cyclic rotation c c c ϵ π , n ( c ) \epsilon_{\pi,n}^{(c)} ϵ π , n ( c ) ϵ μ , n ( c ) \epsilon_{\mu,n}^{(c)} ϵ μ , n ( c ) 1 − 3 η 1-3\eta 1 − 3 η
sup θ ∈ Θ ∣ 1 3 ∑ c = 1 3 J ^ λ , θ D R , ( c ) − J λ , θ ∣ ≤ C [ c m ( η ) + 1 3 ∑ c = 1 3 { ϵ π , n ( c ) ϵ μ , n ( c ) + ϵ π , n ( c ) + ( ϵ μ , n ( c ) ) 2 } ] . \sup_{\theta\in\Theta}\left|\frac{1}{3}\sum_{c=1}^{3}\hat J_{\lambda,\theta}^{\mathrm{DR},(c)}-J_{\lambda,\theta}\right|
\le C\left[c_m(\eta)+\frac{1}{3}\sum_{c=1}^{3}\left\{\epsilon_{\pi,n}^{(c)}\epsilon_{\mu,n}^{(c)}+\epsilon_{\pi,n}^{(c)}+\left(\epsilon_{\mu,n}^{(c)}\right)^2\right\}\right]. θ ∈ Θ sup 3 1 c = 1 ∑ 3 J ^ λ , θ DR , ( c ) − J λ , θ ≤ C [ c m ( η ) + 3 1 c = 1 ∑ 3 { ϵ π , n ( c ) ϵ μ , n ( c ) + ϵ π , n ( c ) + ( ϵ μ , n ( c ) ) 2 } ] . No independence among the three rotation events is required.
Joint product-class complexity. If Ξ ⊆ { v : ∥ v ∥ 2 ≤ R 0 } \Xi\subseteq\{v : \lVert v\rVert_2 \le R_0\} Ξ ⊆ { v : ∥ v ∥ 2 ≤ R 0 } d Θ = M d 0 d_{\Theta}=Md_0 d Θ = M d 0 R Θ ≤ M R 0 R_{\Theta}\le\sqrt{M}R_0 R Θ ≤ M R 0
c m ( η ) = O ( M d 0 log ( 1 + 2 R 0 M m ) + log ( 1 / η ) + 1 m ) . c_m(\eta)=O\left(\sqrt{\frac{Md_0\log(1+2R_0\sqrt{Mm})+\log(1/\eta)+1}{m}}\right). c m ( η ) = O m M d 0 log ( 1 + 2 R 0 M m ) + log ( 1/ η ) + 1 . For fixed class constants and nuisance rates ϵ π , n = O P ( a π , n ) \epsilon_{\pi,n}=O_{\mathbb{P}}(a_{\pi,n}) ϵ π , n = O P ( a π , n ) ϵ μ , n = O P ( a μ , n ) \epsilon_{\mu,n}=O_{\mathbb{P}}(a_{\mu,n}) ϵ μ , n = O P ( a μ , n ) Theorem 4.4 gives criterion error
O P ( M d 0 log ( 1 + 2 R 0 M m ) + 1 m + a π , n + a μ , n 2 ) . O_{\mathbb{P}}\left(\sqrt{\frac{Md_0\log(1+2R_0\sqrt{Mm})+1}{m}}+a_{\pi,n}+a_{\mu,n}^{2}\right). O P m M d 0 log ( 1 + 2 R 0 M m ) + 1 + a π , n + a μ , n 2 . If M M M M d 0 log ( 1 + 2 R 0 M m ) = o ( m ) Md_0\log(1+2R_0\sqrt{Mm})=o(m) M d 0 log ( 1 + 2 R 0 M m ) = o ( m ) M = 1 M=1 M = 1
Theorem B.17 (Approximation of unregularised efficacy on a variance-stable class). Let Θ s : = { θ ∈ Θ : s θ 2 ≥ s 2 } \Theta_s:=\{\theta\in\Theta:\ s_\theta^2\ge s^2\} Θ s := { θ ∈ Θ : s θ 2 ≥ s 2 } s > 0 s>0 s > 0 C v , C s C_v,C_s C v , C s C v C_v C v C s C_s C s s s s F f i t \mathcal{F}_{\mathrm{fit}} F fit
C v { c m ( η ) + ϵ π , n + ϵ μ , n 2 } ≤ s 2 / 2 , C_v\{c_m(\eta)+\epsilon_{\pi,n}+\epsilon_{\mu,n}^2\}\le s^2/2, C v { c m ( η ) + ϵ π , n + ϵ μ , n 2 } ≤ s 2 /2 , for every λ m > 0 \lambda_m>0 λ m > 0 1 − η 1-\eta 1 − η
sup θ ∈ Θ s ∣ J ^ λ m , θ D R − J θ ∣ ≤ C s { c m ( η ) + ϵ π , n ϵ μ , n + ϵ π , n + ϵ μ , n 2 + λ m } . (35) \sup_{\theta\in\Theta_s}\left|\widehat{J}^{\mathrm{DR}}_{\lambda_m,\theta}-J_\theta\right|\le C_s\{c_m(\eta)+\epsilon_{\pi,n}\epsilon_{\mu,n}+\epsilon_{\pi,n}+\epsilon_{\mu,n}^2+\lambda_m\}. \tag*{(35)} θ ∈ Θ s sup J λ m , θ DR − J θ ≤ C s { c m ( η ) + ϵ π , n ϵ μ , n + ϵ π , n + ϵ μ , n 2 + λ m } . (35) Suppose also that deterministic sequences a π , n , a μ , n → 0 a_{\pi,n},a_{\mu,n}\to0 a π , n , a μ , n → 0 ϵ π , n = O P ( a π , n ) \epsilon_{\pi,n}=O_{\mathbb{P}}(a_{\pi,n}) ϵ π , n = O P ( a π , n ) ϵ μ , n = O P ( a μ , n ) \epsilon_{\mu,n}=O_{\mathbb{P}}(a_{\mu,n}) ϵ μ , n = O P ( a μ , n )
d Θ log ( 1 + 2 R Θ m ) + 1 m ⟶ 0 , λ m = O ( d Θ log ( 1 + 2 R Θ m ) + 1 m ) . \frac{d_{\Theta}\log(1+2R_{\Theta}\sqrt{m})+1}{m}\longrightarrow0,\qquad
\lambda_m=O\left(\sqrt{\frac{d_{\Theta}\log(1+2R_{\Theta}\sqrt{m})+1}{m}}\right). m d Θ log ( 1 + 2 R Θ m ) + 1 ⟶ 0 , λ m = O ( m d Θ log ( 1 + 2 R Θ m ) + 1 ) . Then the margin holds with probability tending to one for fixed s s s η \eta η (35) is
O P ( d Θ log ( 1 + 2 R Θ m ) + 1 m + a π , n + a μ , n 2 ) . O_{\mathbb{P}}\left(\sqrt{\frac{d_{\Theta}\log(1+2R_{\Theta}\sqrt{m})+1}{m}}+a_{\pi,n}+a_{\mu,n}^2\right). O P ( m d Θ log ( 1 + 2 R Θ m ) + 1 + a π , n + a μ , n 2 ) . Proof. The proof of (34) supplies the constant C v C_v C v s ^ θ 2 ≥ s 2 / 2 \widehat{s}_\theta^2\ge s^2/2 s θ 2 ≥ s 2 /2 Θ s \Theta_s Θ s Lemma B.12 with D = D θ D=D_\theta D = D θ D ^ = D ^ θ c r o s s \widehat{D}=\widehat{D}^{\mathrm{cross}}_\theta D = D θ cross v = s θ 2 v=s_\theta^2 v = s θ 2 v ^ = s ^ θ 2 \widehat{v}=\widehat{s}_\theta^2 v = s θ 2 B = 4 κ B=4\kappa B = 4 κ (33) to (34) then give (35) . The rate statement follows from (16) .
Under the preceding ridge choice, for n ≍ m n\asymp m n ≍ m O P ( log m / m ) O_{\mathbb{P}}(\sqrt{\log m/m}) O P ( log m / m ) J θ J_\theta J θ Θ s \Theta_s Θ s ϵ π , n = O P ( log m / m ) \epsilon_{\pi,n}=O_{\mathbb{P}}(\sqrt{\log m/m}) ϵ π , n = O P ( log m / m ) ϵ μ , n = O P ( ( log m / m ) 1 / 4 ) \epsilon_{\mu,n}=O_{\mathbb{P}}((\log m/m)^{1/4}) ϵ μ , n = O P (( log m / m ) 1/4 ) θ \theta θ
Corollary B.18 (Known logging policy). Under the conditions of Theorem 4.4 , if π ^ 0 = π 0 \widehat{\pi}_0=\pi_0 π 0 = π 0 ∣ r ∣ |r| ∣ r ∣ λ ≥ λ 0 \lambda\ge\lambda_0 λ ≥ λ 0 1 − η 1-\eta 1 − η
sup θ ∈ Θ ∣ J ^ λ , θ D R − J λ , θ ∣ ≤ C { c m ( η ) + ϵ μ , n 2 } . \sup_{\theta\in\Theta}\left|\widehat{J}^{\mathrm{DR}}_{\lambda,\theta}-J_{\lambda,\theta}\right|\le C\{c_m(\eta)+\epsilon_{\mu,n}^2\}. θ ∈ Θ sup J λ , θ DR − J λ , θ ≤ C { c m ( η ) + ϵ μ , n 2 } . If, in addition, Θ s \Theta_s Θ s
C v { c m ( η ) + ϵ μ , n 2 } ≤ s 2 / 2 , C_v\{c_m(\eta)+\epsilon_{\mu,n}^2\}\le s^2/2, C v { c m ( η ) + ϵ μ , n 2 } ≤ s 2 /2 , then, for every λ m > 0 \lambda_m>0 λ m > 0 1 − η 1-\eta 1 − η
sup θ ∈ Θ s ∣ J ^ λ m , θ D R − J θ ∣ ≤ C s { c m ( η ) + ϵ μ , n 2 + λ m } . \sup_{\theta\in\Theta_s}\left|\widehat{J}^{\mathrm{DR}}_{\lambda_m,\theta}-J_\theta\right|\le C_s\{c_m(\eta)+\epsilon_{\mu,n}^2+\lambda_m\}. θ ∈ Θ s sup J λ m , θ DR − J θ ≤ C s { c m ( η ) + ϵ μ , n 2 + λ m } . Proof. With known logging, ϵ π , n = 0 \epsilon_{\pi,n}=0 ϵ π , n = 0 4 κ W ϵ μ , n 2 4\kappa_W\epsilon_{\mu,n}^{2} 4 κ W ϵ μ , n 2 Theorem B.17 proves the second result.
Null Second-Order Structure
We use the conditional notation and nuisance errors from Section B.7.
Proposition B.20 (Null second-order structure). Fix θ \theta θ t ↦ P t t\mapsto P_t t ↦ P t P 0 P_0 P 0 Δ θ , t = Δ θ , 0 + t Δ ˙ θ + o ( t ) \Delta_{\theta,t}=\Delta_{\theta,0}+t\dot{\Delta}_{\theta}+o(t) Δ θ , t = Δ θ , 0 + t Δ ˙ θ + o ( t ) H θ \mathcal{H}_\theta H θ H 0 H_0 H 0
D θ , t = t 2 ∥ Δ θ ∥ H θ 2 + o ( t 2 ) . (36) D_{\theta,t}=t^2\lVert\Delta_\theta\rVert_{\mathcal{H}_\theta}^{2}+o(t^2). \tag*{(36)} D θ , t = t 2 ∥ Δ θ ∥ H θ 2 + o ( t 2 ) . (36) If the perturbation has score ℓ ∈ L 0 2 ( P 0 ) \ell\in L_0^2(P_0) ℓ ∈ L 0 2 ( P 0 ) Δ ˙ θ = E 0 [ g θ ( O ) ℓ ( O ) ] \dot{\Delta}_\theta=\mathbb{E}_0[g_\theta(O)\ell(O)] Δ ˙ θ = E 0 [ g θ ( O ) ℓ ( O )] (36) is induced by
h θ , 0 ( o 1 , o 2 ) ≔ ⟨ g θ ( o 1 ) , g θ ( o 2 ) ⟩ H θ . h_{\theta,0}(o_1,o_2)\coloneqq\langle g_\theta(o_1),g_\theta(o_2)\rangle_{\mathcal{H}_\theta}. h θ , 0 ( o 1 , o 2 ) : = ⟨ g θ ( o 1 ) , g θ ( o 2 ) ⟩ H θ . Equivalently, D θ , 0 ′ ′ [ ℓ , ℓ ] = 2 E 0 ⊗ 2 [ h θ , 0 ( O 1 , O 2 ) ℓ ( O 1 ) ℓ ( O 2 ) ] D_{\theta,0}^{\prime\prime}[\ell,\ell]=2\mathbb{E}_0^{\otimes2}[h_{\theta,0}(O_1,O_2)\ell(O_1)\ell(O_2)] D θ , 0 ′′ [ ℓ , ℓ ] = 2 E 0 ⊗ 2 [ h θ , 0 ( O 1 , O 2 ) ℓ ( O 1 ) ℓ ( O 2 )] t 2 / 2 t^2/2 t 2 /2 2 h θ , 0 2h_{\theta,0} 2 h θ , 0 h θ , 0 h_{\theta,0} h θ , 0 U U U
E 0 [ h θ , 0 ( O 1 , O 2 ) ∣ O 1 ] = E 0 [ h θ , 0 ( O 1 , O 2 ) ∣ O 2 ] = 0 almost surely , \mathbb{E}_0[h_{\theta,0}(O_1,O_2)\mid O_1]=\mathbb{E}_0[h_{\theta,0}(O_1,O_2)\mid O_2]=0\quad\text{almost surely}, E 0 [ h θ , 0 ( O 1 , O 2 ) ∣ O 1 ] = E 0 [ h θ , 0 ( O 1 , O 2 ) ∣ O 2 ] = 0 almost surely , so h θ , 0 h_{\theta,0} h θ , 0
For the fitted cross-product, suppose Assumptions B.5 and 3.1 hold, let ξ ~ θ , i : = g ~ θ ( O i ) − Δ ~ θ \widetilde{\xi}_{\theta,i}:=\widetilde{g}_\theta(O_i)-\widetilde{\Delta}_\theta ξ θ , i := g θ ( O i ) − Δ θ ξ ~ ‾ θ , ℓ \overline{\widetilde{\xi}}_{\theta,\ell} ξ θ , ℓ ℓ \ell ℓ H 0 H_0 H 0 Δ ~ θ = B θ \widetilde{\Delta}_\theta=B_\theta Δ θ = B θ
D ^ θ c r o s s = ∥ B θ ∥ 2 + ⟨ B θ , ξ ~ ‾ θ , 1 + ξ ~ ‾ θ , 2 ⟩ + ⟨ ξ ~ ‾ θ , 1 , ξ ~ ‾ θ , 2 ⟩ . (37) \widehat{D}^{\mathrm{cross}}_\theta=\lVert B_\theta\rVert^2+\langle B_\theta,\overline{\widetilde{\xi}}_{\theta,1}+\overline{\widetilde{\xi}}_{\theta,2}\rangle+\langle\overline{\widetilde{\xi}}_{\theta,1},\overline{\widetilde{\xi}}_{\theta,2}\rangle. \tag*{(37)} D θ cross = ∥ B θ ∥ 2 + ⟨ B θ , ξ θ , 1 + ξ θ , 2 ⟩ + ⟨ ξ θ , 1 , ξ θ , 2 ⟩ . (37) s ^ θ 2 = 1 m ∑ j ∈ I 2 ⟨ B θ + ξ ~ ‾ θ , 1 , ξ ~ θ , j − ξ ~ ‾ θ , 2 ⟩ 2 . (38) \widehat{s}_\theta^2=\frac{1}{m}\sum_{j\in\mathcal{I}_2}\left\langle B_\theta+\overline{\widetilde{\xi}}_{\theta,1},\widetilde{\xi}_{\theta,j}-\overline{\widetilde{\xi}}_{\theta,2}\right\rangle^2. \tag*{(38)} s θ 2 = m 1 j ∈ I 2 ∑ ⟨ B θ + ξ θ , 1 , ξ θ , j − ξ θ , 2 ⟩ 2 . (38) Consequently,
E 0 [ D ^ θ c r o s s ∣ F f i t ] = ∥ B θ ∥ 2 ≤ ϵ π , n 2 ϵ μ , n 2 , \mathbb{E}_0[\widehat{D}^{\mathrm{cross}}_\theta\mid\mathcal{F}_{\mathrm{fit}}]=\lVert B_\theta\rVert^2\le\epsilon_{\pi,n}^2\epsilon_{\mu,n}^2, E 0 [ D θ cross ∣ F fit ] = ∥ B θ ∥ 2 ≤ ϵ π , n 2 ϵ μ , n 2 , and, for this fixed θ \theta θ
D ^ θ c r o s s = O P ( { ϵ π , n ϵ μ , n + m − 1 / 2 } 2 | F f i t ) , s ^ θ 2 = O P ( { ϵ π , n ϵ μ , n + m − 1 / 2 } 2 | F f i t ) . (39) \widehat{D}^{\mathrm{cross}}_\theta=O_{\mathbb{P}}\left(\left\{\epsilon_{\pi,n}\epsilon_{\mu,n}+m^{-1/2}\right\}^2\middle|\mathcal{F}_{\mathrm{fit}}\right),\qquad
\widehat{s}_\theta^2=O_{\mathbb{P}}\left(\left\{\epsilon_{\pi,n}\epsilon_{\mu,n}+m^{-1/2}\right\}^2\middle|\mathcal{F}_{\mathrm{fit}}\right). \tag*{(39)} D θ cross = O P ( { ϵ π , n ϵ μ , n + m − 1/2 } 2 F fit ) , s θ 2 = O P ( { ϵ π , n ϵ μ , n + m − 1/2 } 2 F fit ) . (39) Proof. Under H 0 H_0 H 0 Δ θ , 0 = 0 \Delta_{\theta,0}=0 Δ θ , 0 = 0
∥ t Δ ˙ θ + o ( t ) ∥ 2 = t 2 ∥ Δ ˙ θ ∥ 2 + o ( t 2 ) , \left\|t\dot{\Delta}_{\theta}+o(t)\right\|^2=t^2\left\|\dot{\Delta}_{\theta}\right\|^2+o(t^2), t Δ ˙ θ + o ( t ) 2 = t 2 Δ ˙ θ 2 + o ( t 2 ) , which proves (36) . If Δ ˙ θ = E 0 [ g θ ℓ ] \dot{\Delta}_{\theta}=E_0[g_{\theta}\ell] Δ ˙ θ = E 0 [ g θ ℓ ]
∥ Δ ˙ θ ∥ 2 = E 0 ⊗ 2 [ h θ , 0 ( O 1 , O 2 ) ℓ ( O 1 ) ℓ ( O 2 ) ] . \left\|\dot{\Delta}_{\theta}\right\|^2=E_0^{\otimes2}\left[h_{\theta,0}(O_1,O_2)\ell(O_1)\ell(O_2)\right]. Δ ˙ θ 2 = E 0 ⊗ 2 [ h θ , 0 ( O 1 , O 2 ) ℓ ( O 1 ) ℓ ( O 2 ) ] . Multiplying this identity by two gives the asserted second-derivative convention. Since E 0 g θ = Δ θ , 0 = 0 E_0g_{\theta}=\Delta_{\theta,0}=0 E 0 g θ = Δ θ , 0 = 0
E 0 [ h θ , 0 ( O 1 , O 2 ) ∣ O 1 ] = ⟨ g θ ( O 1 ) , E 0 g θ ( O 2 ) ⟩ = 0 E_0\left[h_{\theta,0}(O_1,O_2)\mid O_1\right]=\langle g_{\theta}(O_1),E_0g_{\theta}(O_2)\rangle=0 E 0 [ h θ , 0 ( O 1 , O 2 ) ∣ O 1 ] = ⟨ g θ ( O 1 ) , E 0 g θ ( O 2 )⟩ = 0 almost surely, and similarly conditioning on O 2 O_2 O 2
For the empirical identities, write the two fitted half-means as B θ + ξ ˉ θ , 1 B_{\theta}+\bar{\xi}_{\theta,1} B θ + ξ ˉ θ , 1 B θ + ξ ˉ θ , 2 B_{\theta}+\bar{\xi}_{\theta,2} B θ + ξ ˉ θ , 2 (38) . Independence and centring of the two halves then give the conditional expectation. Finally, the Hilbert-space second-moment bound yields ξ ˉ θ , ℓ = O P ( m − 1 / 2 ∣ F f i t ) \bar{\xi}_{\theta,\ell}=O_P(m^{-1/2}\mid\mathcal{F}_{\mathrm{fit}}) ξ ˉ θ , ℓ = O P ( m − 1/2 ∣ F fit ) m − 1 ∑ j ∈ I 2 ∥ ξ θ , j − ξ ˉ θ , 2 ∥ 2 = O P ( 1 ∣ F f i t ) m^{-1}\sum_{j\in I_2}\left\|\xi_{\theta,j}-\bar{\xi}_{\theta,2}\right\|^2=O_P(1\mid\mathcal{F}_{\mathrm{fit}}) m − 1 ∑ j ∈ I 2 ξ θ , j − ξ ˉ θ , 2 2 = O P ( 1 ∣ F fit ) (24) proves (39) .
This expansion parallels Luedtke et al. [37 ] , whose U-statistic uses all off-diagonal pairs. Here the cross-product uses independent halves, with conditional studentisation as in Kim & Ramdas [29 ] ; Shekhar et al. [52 ] . With fitted nuisances, only the centred term in (37) is degenerate; the B θ B_{\theta} B θ
Held-Out Fixed-Alternative Scaling
Proposition 4.1 concerns a fixed kernel and true nuisance functions. We now extend its scaling limit to the selected kernel with fitted nuisances, conditional on the learning procedure. Write m t e s t m_{\mathrm{test}} m test
Theorem B.21 (Held-out fixed-alternative scaling). Suppose Assumptions 3.1 and 3.2 hold. Consider a joint sequence with m t e s t → ∞ m_{\mathrm{test}}\to\infty m test → ∞ under a fixed alternative. Assume that the final fitted score is F l e a r n \mathcal{F}_{\mathrm{learn}} F learn measurable and conditionally Bochner integrable, and that the two test halves consist of i.i.d. observations from P 0 P_0 P 0 are mutually independent, and are independent of F l e a r n \mathcal{F}_{\mathrm{learn}} F learn For a random fitted score f : O → H θ ^ f:\mathcal{O}\to\mathcal{H}_{\hat{\theta}} f : O → H θ ^ write
∥ f ∥ 2 , l e a r n 2 : = E 0 [ ∥ f ( O ) ∥ H θ ^ 2 ∣ F l e a r n ] . \|f\|^2_{2,\mathrm{learn}}:=E_0\left[\|f(O)\|^2_{\mathcal{H}_{\hat{\theta}}}\mid\mathcal{F}_{\mathrm{learn}}\right]. ∥ f ∥ 2 , learn 2 := E 0 [ ∥ f ( O ) ∥ H θ ^ 2 ∣ F learn ] . For some fixed s > 0 s>0 s > 0 suppose P ( θ ^ ∈ Θ s ) → 1 P(\hat{\theta}\in\Theta_s)\to1 P ( θ ^ ∈ Θ s ) → 1 define J θ ^ : = 0 J_{\hat{\theta}}:=0 J θ ^ := 0 on { θ ^ ∉ Θ s } \{\hat{\theta}\notin\Theta_s\} { θ ^ ∈ / Θ s } and assume
∥ g ^ θ ^ − g θ ^ π , π ′ ∥ 2 , l e a r n = o P ( 1 ) , E 0 [ ∥ g ^ θ ^ ( O ) ∥ H θ ^ 4 ∣ F l e a r n ] = O P ( 1 ) . \left\|\hat{g}_{\hat{\theta}}-g_{\hat{\theta}}^{\pi,\pi'}\right\|_{2,\mathrm{learn}}=o_P(1),\qquad E_0\left[\|\hat{g}_{\hat{\theta}}(O)\|^4_{\mathcal{H}_{\hat{\theta}}}\mid\mathcal{F}_{\mathrm{learn}}\right]=O_P(1). g ^ θ ^ − g θ ^ π , π ′ 2 , learn = o P ( 1 ) , E 0 [ ∥ g ^ θ ^ ( O ) ∥ H θ ^ 4 ∣ F learn ] = O P ( 1 ) . Then
T θ ^ m t e s t − J θ ^ → P 0. (40) \frac{T_{\hat{\theta}}}{\sqrt{m_{\mathrm{test}}}}-J_{\hat{\theta}}\xrightarrow{\mathbb{P}}0. \tag*{(40)} m test T θ ^ − J θ ^ P 0. (40) Proof of Theorem B.21. Condition on F l e a r n \mathcal{F}_{\mathrm{learn}} F learn
G : = g ^ θ ^ , g : = g θ ^ π , π ′ , μ G : = E 0 [ G ( O ) ∣ F l e a r n ] , Δ : = Δ θ ^ . G:=\hat{g}_{\hat{\theta}},\qquad g:=g_{\hat{\theta}}^{\pi,\pi'},\qquad\mu_G:=E_0[G(O)\mid\mathcal{F}_{\mathrm{learn}}],\qquad\Delta:=\Delta_{\hat{\theta}}. G := g ^ θ ^ , g := g θ ^ π , π ′ , μ G := E 0 [ G ( O ) ∣ F learn ] , Δ := Δ θ ^ . The conditional fourth-moment bound and conditional laws of large numbers on the two independent test halves give
D ^ θ ^ c r o s s − ∥ μ G ∥ H θ ^ 2 = o P ( 1 ) , \widehat{D}^{\mathrm{cross}}_{\hat{\theta}}-\|\mu_G\|^2_{\mathcal{H}_{\hat{\theta}}}=o_P(1), D θ ^ cross − ∥ μ G ∥ H θ ^ 2 = o P ( 1 ) , s ^ θ ^ 2 − Var P 0 ( ⟨ μ G , G ( O ) ⟩ H θ ^ ∣ F l e a r n ) = o P ( 1 ) . \hat{s}^2_{\hat{\theta}}-\operatorname{Var}_{P_0}\left(\langle\mu_G,G(O)\rangle_{\mathcal{H}_{\hat{\theta}}}\mid\mathcal{F}_{\mathrm{learn}}\right)=o_P(1). s ^ θ ^ 2 − Var P 0 ( ⟨ μ G , G ( O ) ⟩ H θ ^ ∣ F learn ) = o P ( 1 ) . For the second line, first condition additionally on the first test half; the second-half studentiser is a scalar sample variance. The conditional fourth-moment bound makes its law of large numbers uniform along the joint sequence, and replacing the first-half mean by μ G \mu_G μ G o P ( 1 ) o_P(1) o P ( 1 )
Since E 0 [ g ( O ) ∣ F l e a r n ] = Δ \mathbb{E}_0[g(O)\mid\mathcal{F}_{\mathrm{learn}}]=\Delta E 0 [ g ( O ) ∣ F learn ] = Δ
∥ μ G − Δ ∥ H θ ^ ≤ ∥ G − g ∥ 2 , l e a r n = o P ( 1 ) . \lVert\mu_G-\Delta\rVert_{\mathcal{H}_{\hat{\theta}}}\leq\lVert G-g\rVert_{2,\mathrm{learn}}=o_P(1). ∥ μ G − Δ ∥ H θ ^ ≤ ∥ G − g ∥ 2 , learn = o P ( 1 ) . Moreover,
∥ ⟨ μ G , G ( O ) ⟩ H θ ^ − ⟨ Δ , g ( O ) ⟩ H θ ^ ∥ L 2 ( P 0 ∣ F l e a r n ) ≤ ∥ μ G − Δ ∥ H θ ^ ∥ G ∥ 2 , l e a r n + ∥ Δ ∥ H θ ^ ∥ G − g ∥ 2 , l e a r n = o P ( 1 ) . \begin{aligned}
\left\lVert\langle\mu_G,G(O)\rangle_{\mathcal{H}_{\hat{\theta}}}-\langle\Delta,g(O)\rangle_{\mathcal{H}_{\hat{\theta}}}\right\rVert_{L^2(P_0\mid\mathcal{F}_{\mathrm{learn}})}
&\leq\lVert\mu_G-\Delta\rVert_{\mathcal{H}_{\hat{\theta}}}\lVert G\rVert_{2,\mathrm{learn}}\\
&\quad+\lVert\Delta\rVert_{\mathcal{H}_{\hat{\theta}}}\lVert G-g\rVert_{2,\mathrm{learn}}=o_P(1).
\end{aligned} ⟨ μ G , G ( O ) ⟩ H θ ^ − ⟨ Δ , g ( O ) ⟩ H θ ^ L 2 ( P 0 ∣ F learn ) ≤ ∥ μ G − Δ ∥ H θ ^ ∥ G ∥ 2 , learn + ∥ Δ ∥ H θ ^ ∥ G − g ∥ 2 , learn = o P ( 1 ) . Here ∥ G ∥ 2 , l e a r n = O P ( 1 ) \lVert G\rVert_{2,\mathrm{learn}}=O_P(1) ∥ G ∥ 2 , learn = O P ( 1 ) ∥ Δ ∥ = O P ( 1 ) \lVert\Delta\rVert=O_P(1) ∥ Δ ∥ = O P ( 1 ) ∥ μ G ∥ ≤ ∥ G ∥ 2 , l e a r n \lVert\mu_G\rVert\leq\lVert G\rVert_{2,\mathrm{learn}} ∥ μ G ∥ ≤ ∥ G ∥ 2 , learn D θ ^ D_{\hat{\theta}} D θ ^ s θ ^ 2 s_{\hat{\theta}}^2 s θ ^ 2 o P ( 1 ) o_P(1) o P ( 1 ) { θ ^ ∈ Θ s } \{\hat{\theta}\in\Theta_s\} { θ ^ ∈ Θ s } s θ ^ ≥ s s_{\hat{\theta}}\geq s s θ ^ ≥ s P ( s ^ θ ^ = 0 ) → 0 \mathbb{P}(\hat{s}_{\hat{\theta}}=0)\to0 P ( s ^ θ ^ = 0 ) → 0
T θ ^ m t e s t = J θ ^ + o P ( 1 ) , \frac{T_{\hat{\theta}}}{\sqrt{m_{\mathrm{test}}}}=J_{\hat{\theta}}+o_P(1), m test T θ ^ = J θ ^ + o P ( 1 ) , which proves (40).
Experimental Details
We now detail our experimental setting. We first describe the training and testing procedure, then specify the construction of each experiment. Finally, we detail the neural network architectures and hyperparameters used for the optimisation.
Training and Testing Procedure
We describe the training and testing procedure in Algorithm 1. Each ensemble member is trained separately by maximising its own learning criterion, using the same learning sample D l e a r n \mathcal{D}_{\mathrm{learn}} D learn ϑ b \vartheta_b ϑ b D v a l \mathcal{D}_{\mathrm{val}} D val D l e a r n \mathcal{D}_{\mathrm{learn}} D learn D l e a r n ∪ D v a l \mathcal{D}_{\mathrm{learn}}\cup\mathcal{D}_{\mathrm{val}} D learn ∪ D val
We keep the selected kernels and fitted nuisance estimators fixed and evaluate all methods on the same 200 fresh test datasets D t e s t \mathcal{D}_{\mathrm{test}} D test α = 0.05 \alpha=0.05 α = 0.05 ± \pm ± N : = ∣ D l e a r n ∣ + ∣ D v a l ∣ + ∣ D t e s t ∣ N:=|\mathcal{D}_{\mathrm{learn}}|+|\mathcal{D}_{\mathrm{val}}|+|\mathcal{D}_{\mathrm{test}}| N := ∣ D learn ∣ + ∣ D val ∣ + ∣ D test ∣ N / 3 N/3 N /3 ∣ D t e s t ∣ |\mathcal{D}_{\mathrm{test}}| ∣ D test ∣
For experiments with binary treatments, we compare two deterministic interventions π ( 1 ∣ x ) = 1 \pi(1\mid x)=1 π ( 1 ∣ x ) = 1 π ′ ( 0 ∣ x ) = 1 \pi'(0\mid x)=1 π ′ ( 0 ∣ x ) = 1
For DR-KTE, we use the Gaussian outcome kernel k ( y , y ′ ) = exp { − ∥ ϕ ( y ) − ϕ ( y ′ ) ∥ 2 / σ 2 } k(y,y')=\exp\{-\lVert\phi(y)-\phi(y')\rVert^2/\sigma^2\} k ( y , y ′ ) = exp { − ∥ ϕ ( y ) − ϕ ( y ′ ) ∥ 2 / σ 2 } ϕ ( y ) \phi(y) ϕ ( y ) σ \sigma σ
Algorithm 1 Training and testing procedure of DR-KTE-Deep
Require: Logged data D \mathcal{D} D π , π ′ \pi,\pi' π , π ′ M M M { k ϑ : ϑ ∈ Ξ } \{k_{\vartheta}:\vartheta\in\Xi\} { k ϑ : ϑ ∈ Ξ } λ > 0 \lambda> 0 λ > 0 T max T_{\max} T m a x γ \gamma γ
Split D \mathcal{D} D D l e a r n \mathcal{D}_{\mathrm{learn}} D learn D v a l \mathcal{D}_{\mathrm{val}} D val D t e s t \mathcal{D}_{\mathrm{test}} D test
Split D l e a r n \mathcal{D}_{\mathrm{learn}} D learn D ( 1 ) \mathcal{D}^{(1)} D ( 1 ) D ( 2 ) \mathcal{D}^{(2)} D ( 2 ) D ( 3 ) \mathcal{D}^{(3)} D ( 3 )
for b = 1 , … , M b=1,\ldots,M b = 1 , … , M
Initialise ϑ b ∈ Ξ \vartheta_b \in\Xi ϑ b ∈ Ξ ϑ ~ b ← ϑ b \widetilde{\vartheta}_b \leftarrow\vartheta_b ϑ b ← ϑ b
for t = 1 , … , T max t=1,\ldots,T_{\max} t = 1 , … , T m a x
for r = 1 , 2 , 3 r=1,2,3 r = 1 , 2 , 3
Assign the folds cyclically to D f i t ( r ) \mathcal{D}_{\mathrm{fit}}^{(r)} D fit ( r ) D s e l , 1 ( r ) \mathcal{D}_{\mathrm{sel},1}^{(r)} D sel , 1 ( r ) D s e l , 2 ( r ) \mathcal{D}_{\mathrm{sel},2}^{(r)} D sel , 2 ( r )
Fit π ^ ( r ) \widehat{\pi}^{(r)} π ( r ) μ ^ ϑ ~ b ( r ) \widehat{\mu}_{\widetilde{\vartheta}_b}^{(r)} μ ϑ b ( r ) D f i t ( r ) \mathcal{D}_{\mathrm{fit}}^{(r)} D fit ( r )
Compute pseudo-features on D s e l , 1 ( r ) \mathcal{D}_{\mathrm{sel},1}^{(r)} D sel , 1 ( r ) D s e l , 2 ( r ) \mathcal{D}_{\mathrm{sel},2}^{(r)} D sel , 2 ( r )
Compute J λ , ϑ b D R , ( r ) J_{\lambda,\vartheta_b}^{\mathrm{DR},(r)} J λ , ϑ b DR , ( r )
Update ϑ b \vartheta_b ϑ b 1 3 ∑ r = 1 3 J λ , ϑ b D R , ( r ) \frac{1}{3}\sum_{r=1}^{3}J_{\lambda,\vartheta_b}^{\mathrm{DR},(r)} 3 1 ∑ r = 1 3 J λ , ϑ b DR , ( r )
Update the EMA parameters ϑ ~ b \widetilde{\vartheta}_b ϑ b
Fit nuisances on D l e a r n \mathcal{D}_{\mathrm{learn}} D learn T ϑ ~ b T_{\widetilde{\vartheta}_b} T ϑ b D v a l \mathcal{D}_{\mathrm{val}} D val
Set ϑ ^ b \widehat{\vartheta}_b ϑ b D v a l \mathcal{D}_{\mathrm{val}} D val
Set θ ^ = ( ϑ ^ 1 , … , ϑ ^ M ) \widehat{\theta}=(\widehat{\vartheta}_1,\ldots,\widehat{\vartheta}_M) θ = ( ϑ 1 , … , ϑ M ) k θ ^ k_{\widehat{\theta}} k θ
Refit the nuisance estimators on the combined learning and validation samples D l e a r n ∪ D v a l \mathcal{D}_{\mathrm{learn}}\cup\mathcal{D}_{\mathrm{val}} D learn ∪ D val
Split D t e s t \mathcal{D}_{\mathrm{test}} D test I 1 , I 2 \mathcal{I}_1,\mathcal{I}_2 I 1 , I 2 m t e s t m_{\mathrm{test}} m test
Compute the test pseudo-features and form T θ ^ T_{\widehat{\theta}} T θ
return T θ ^ T_{\widehat{\theta}} T θ p p p 1 − Φ ( T θ ^ ) 1-\Phi(T_{\widehat{\theta}}) 1 − Φ ( T θ )
Image Outcomes
We detail the construction of experiments on image outcomes from § 5 . We extend the OCTMNIST experiments of Zenati & Gretton (2026). We apply semi-synthetic perturbations to baseline images sampled from MedMNIST v2 ([66 ] ), released under CC BY-NC 4.0 for DermaMNIST and CC BY 4.0 for the remaining datasets.
For each unit, we sample a synthetic covariate
X i ∈ R d X , X i ∼ N ( 0 , I d X ) , (41) X_i \in\mathbb{R}^{d_X}, \qquad X_i \sim\mathcal{N}(0,I_{d_X}), \tag*{(41)} X i ∈ R d X , X i ∼ N ( 0 , I d X ) , (41) and assign treatment as
A i ∣ X i ∼ Bernoulli { π 0 ( 1 ∣ X i ) } , π 0 ( 1 ∣ X i ) = clip { σ ( α ⊤ X i ) , 0.07 , 0.93 } . (42) A_i \mid X_i \sim\operatorname{Bernoulli}\{\pi_0(1\mid X_i)\}, \qquad\pi_0(1\mid X_i)=\operatorname{clip}\{\sigma(\alpha^\top X_i),0.07,0.93\}. \tag*{(42)} A i ∣ X i ∼ Bernoulli { π 0 ( 1 ∣ X i )} , π 0 ( 1 ∣ X i ) = clip { σ ( α ⊤ X i ) , 0.07 , 0.93 } . (42) The potential outcomes are generated as
Y i a = B i + R i ( a ) , a ∈ { 0 , 1 } , Y_i^a = B_i + R_i(a), \qquad a \in\{0,1\}, Y i a = B i + R i ( a ) , a ∈ { 0 , 1 } , where B i B_i B i R i ( a ) R_i(a) R i ( a ) Y i a Y_i^a Y i a a a a Y i = Y i A i Y_i=Y_i^{A_i} Y i = Y i A i
q i = q min + ( q max − q min ) σ ( β ⊤ X i ) q_i = q_{\min} + (q_{\max} - q_{\min})\sigma(\beta^{\top}X_i) q i = q m i n + ( q m a x − q m i n ) σ ( β ⊤ X i ) where q i ∈ ( q min , q max ) q_i \in(q_{\min},q_{\max}) q i ∈ ( q m i n , q m a x )
S i ∣ X i ∼ Bernoulli ( q i ) , S_i \mid X_i \sim\operatorname{Bernoulli}(q_i), S i ∣ X i ∼ Bernoulli ( q i ) , define mean-one multiplicative jitter factors
J i a = exp ( τ U i a − τ 2 / 2 ) , U i a ∼ N ( 0 , 1 ) , J_{ia} = \exp(\tau U_{ia} - \tau^2/2), \qquad U_{ia} \sim\mathcal{N}(0,1), J ia = exp ( τ U ia − τ 2 /2 ) , U ia ∼ N ( 0 , 1 ) , and independent pixel noise ϵ i a ∼ N ( 0 , σ ϵ 2 I ) \epsilon_{ia} \sim\mathcal{N}(0,\sigma_{\epsilon}^{2}I) ϵ ia ∼ N ( 0 , σ ϵ 2 I )
R i ( 0 ) = q i δ J i 0 M i + ϵ i 0 , R i ( 1 ) = S i δ J i 1 M i + ϵ i 1 , R_i(0) = q_i\delta J_{i0}M_i + \epsilon_{i0}, \qquad R_i(1) = S_i\delta J_{i1}M_i + \epsilon_{i1}, R i ( 0 ) = q i δ J i 0 M i + ϵ i 0 , R i ( 1 ) = S i δ J i 1 M i + ϵ i 1 , where M i M_i M i
We use d X = 6 d_X = 6 d X = 6 α = ( 0.90 , − 0.75 , 0.55 , − 0.40 , 0.30 , − 0.20 ) ⊤ \alpha= (0.90,-0.75,0.55,-0.40,0.30,-0.20)^{\top} α = ( 0.90 , − 0.75 , 0.55 , − 0.40 , 0.30 , − 0.20 ) ⊤ β = ( 0.60 , − 0.50 , 0.40 , − 0.30 , 0.20 , − 0.10 ) ⊤ \beta= (0.60,-0.50,0.40,-0.30,0.20,-0.10)^{\top} β = ( 0.60 , − 0.50 , 0.40 , − 0.30 , 0.20 , − 0.10 ) ⊤ q min = 0.10 q_{\min} = 0.10 q m i n = 0.10 q max = 0.40 q_{\max} = 0.40 q m a x = 0.40 τ = 0.20 \tau= 0.20 τ = 0.20 σ ϵ = 0.025 \sigma_{\epsilon} = 0.025 σ ϵ = 0.025
For Table 1 under H 1 H_1 H 1 δ = 0.7 \delta= 0.7 δ = 0.7 δ = 0 \delta= 0 δ = 0 H 0 H_0 H 0
Table 1. Power (H 1 H_1 H 1 H 0 H_0 H 0 N = 1500 N = 1500 N = 1500 α = 0.05 \alpha= 0.05 α = 0.05 H 1 / H 0 H_1/H_0 H 1 / H 0 ± \pm ±
Text Outcomes
We now detail the generation of text outcomes in Section 5. Given some covariate X i X_i X i A i A_i A i (42) and generate pseudo-outcomes
Y ~ i 0 = β ⊤ X i + ϵ i 0 , Y ~ i 1 = β ⊤ X i + b i ( δ ) + ϵ i 1 , ϵ i 0 , ϵ i 1 ∼ i . i . d . N ( 0 , σ ϵ 2 ) . (43) \widetilde{Y}_i^0 = \beta^{\top}X_i + \epsilon_{i0}, \qquad\widetilde{Y}_i^1 = \beta^{\top}X_i + b_i(\delta) + \epsilon_{i1}, \qquad\epsilon_{i0},\epsilon_{i1} \mathrel{\overset{\mathrm{i.i.d.}}{\sim}} \mathcal{N}(0,\sigma_{\epsilon}^{2}). \tag*{(43)} Y i 0 = β ⊤ X i + ϵ i 0 , Y i 1 = β ⊤ X i + b i ( δ ) + ϵ i 1 , ϵ i 0 , ϵ i 1 ∼ i.i.d. N ( 0 , σ ϵ 2 ) . (43) where b i ( δ ) = δ ( 2 Z i − 1 ) b_i(\delta) = \delta(2Z_i - 1) b i ( δ ) = δ ( 2 Z i − 1 ) Z i ∼ i . i . d . Bernoulli ( 0.5 ) Z_i \mathrel{\overset{\mathrm{i.i.d.}}{\sim}} \operatorname{Bernoulli}(0.5) Z i ∼ i.i.d. Bernoulli ( 0.5 ) Y a Y^a Y a a a a Y i = Y i A i Y_i = Y_i^{A_i} Y i = Y i A i
Oncology Notes. We sample a five-dimensional covariate X i ∼ N ( 0 , I 5 ) X_i \sim\mathcal{N}(0,I_5) X i ∼ N ( 0 , I 5 )
Movie Reviews. For each unit, we first sample a movie review from the Rotten Tomatoes dataset [44 ] and prompt the language model to rewrite the review as positive, negative and neutral. We construct the covariate X i ∈ R 5 X_i \in\mathbb{R}^5 X i ∈ R 5
Clinical Notes. For each unit, we first sample a clinical note from the MTSamples dataset [42 ] . We then append to the note two language model generated sentences describing a positive, negative or neutral clinical outcome. As with movie reviews, we construct X i ∈ R 5 X_i \in\mathbb{R}^{5} X i ∈ R 5
We use α = ( 0.90 , − 0.75 , 0.55 , − 0.40 , 0.30 ) T \boldsymbol{\alpha} = (0.90, -0.75, 0.55, -0.40, 0.30)^{\mathsf{T}} α = ( 0.90 , − 0.75 , 0.55 , − 0.40 , 0.30 ) T β = ( 0.60 , − 0.50 , 0.40 , − 0.30 , 0.20 ) T \boldsymbol{\beta} = (0.60, -0.50, 0.40, -0.30, 0.20)^{\mathsf{T}} β = ( 0.60 , − 0.50 , 0.40 , − 0.30 , 0.20 ) T σ ε = 0.1 \sigma_{\varepsilon} = 0.1 σ ε = 0.1 − 1 -1 − 1 [ − 1 , 1 ) [-1, 1) [ − 1 , 1 ) 1 1 1 H 1 H_1 H 1 δ = 1.0 \delta= 1.0 δ = 1.0 δ = 2.6 \delta= 2.6 δ = 2.6 H 0 H_0 H 0 δ = 0 \delta= 0 δ = 0
We generate the text variants with Qwen3-8B, with thinking disabled, temperature 0.7 0.7 0.7 p = 0.9 p = 0.9 p = 0.9
Graph Outcomes
Finally, we detail the graph outcome constructions of § 5 . For a pre-treatment graph G i G_i G i c i c_i c i G i G_i G i X i X_i X i (42) with α = 1 \alpha= 1 α = 1 G i G_i G i Y i Y_i Y i
Contact Networks. Each unit’s pre-treatment graph G i G_i G i [24 ] . Each node is an individual, and each edge is weighted by the number of 20-second intervals during which two individuals were in close proximity. We use mean excess degree as a proxy for epidemic risk, motivated by its role in random-network epidemic thresholds [43 ] :
c i = ∑ v d v 2 ∑ v d v − 1 , c_i = \frac{\sum_v d_v^2}{\sum_v d_v} - 1, c i = ∑ v d v ∑ v d v 2 − 1 , where d v d_v d v v v v
We sample an infection source with probability proportional to each node’s weighted degree. Under A i = 0 A_i = 0 A i = 0 A i = 1 A_i = 1 A i = 1 H 0 H_0 H 0
LLM Code Repairs. Each unit’s pre-treatment graph G i G_i G i [63 ] , represented as its abstract syntax tree (AST).
Under A i = 0 A_i = 0 A i = 0 Y i 0 = G i Y_i^0 = G_i Y i 0 = G i A i = 1 A_i = 1 A i = 1 Y i 1 Y_i^1 Y i 1 [14 ] . We choose c i c_i c i
c i = log ( 1 + ∣ V i ∣ ) , c_i = \log(1 + \lvert\mathcal{V}_i \rvert), c i = log ( 1 + ∣ V i ∣) , where ∣ V i ∣ \lvert\mathcal{V}_i \rvert ∣ V i ∣
Under H 0 H_0 H 0 Y i 0 = Y i 1 = G i Y_i^0 = Y_i^1 = G_i Y i 0 = Y i 1 = G i H 1 H_1 H 1
Each outcome tree is encoded as a padded sequence of node labels and parent indices. This representation is injective for the labelled tree, although constructing the tree may discard information from the original program or contact network. The WL subtree kernel is not generally characteristic, and therefore does not provide the same omnibus guarantees as the Gaussian kernel tests.
Architectures and Hyperparameters
We describe the neural network architectures and hyperparameters used for DR-KTE-Deep. For scalar outcomes, Causal-Blob, and text outcomes when represented as an embedding, ζ θ \zeta_\theta ζ θ Figure 5 ). Text outcomes, when represented as a sequence of tokens, use a two-layer transformer (Figure 7 ). For image outcomes, we use a four-layer convolutional network with LeakyReLU activations (Figure 6 ). Finally for graph outcomes, we use a three-layer message-passing graph neural network (GNN) (Figure 8 ).
Figure 5. Four-layer fully connected feature extractor for scalar outcomes, Causal-Blob, and text outcomes under an embedded representation.
Figure 6. Four-layer convolutional feature extractor for image outcomes.
Figure 7. Two-layer transformer feature extractor for text outcomes represented as a sequence of tokens.
Figure 8. Three-layer message-passing feature extractor for graph outcomes.
We use an Adam optimiser (Kingma & Ba, [31 ] ), with separate learning rates for the neural features and the remaining parameters. Hyperparameters are summarised in Table 4 and Table 5 .
Table 4. Shared hyperparameters for our experiments.
Table 5. Experiment-specific hyperparameters.
DR-KTE-Opt shares the same optimiser, regularisation parameters and number of epochs, but only optimises the lengthscale of its kernel, using the same learning rate as the non-neural parameters of DR-KTE-Deep.
Additional Experiments
Additional Results on Image Outcomes
Medical-Image Outcomes. We run additional simulations on medical-image outcomes, varying both sample size and effect strength. On OCT-DME, deep-kernel power is already near one throughout the sample-size sweep, while the Gaussian baselines have low power (Figure 9 ). On Derma-Melanoma, increasing N N N Figure 10 ).
Figure 9. Results for OCT-DME. From left to right: (a) Type-I error under H 0 H_0 H 0 δ = 0 \delta= 0 δ = 0 N N N δ = 0.4 \delta= 0.4 δ = 0.4 δ \delta δ N = 1500 N = 1500 N = 1500 ± \pm ± α = 0.05 \alpha= 0.05 α = 0.05
Figure 10. Results for Derma-Melanoma. From left to right: (a) Type-I error under H 0 H_0 H 0 δ = 0 \delta= 0 δ = 0 N N N δ = 1.4 \delta= 1.4 δ = 1.4 δ \delta δ N = 1500 N = 1500 N = 1500 ± \pm ± α = 0.05 \alpha= 0.05 α = 0.05
MNIST, RetinaMNIST, BUSI. We next study a construction where treatment changes the sampling frequency of image labels of the MNIST ([32 ] ), RetinaMNIST ([65 , 66 , 34 ] ), and BUSI ([2 ] ) datasets.
Covariates and treatments are sampled as in Equation (41) and Equation (42) . We define the scalar pseudo-outcomes
Y ~ i 0 = β ⊤ X i + ε i 0 , Y ~ i 1 = β ⊤ X i + b i ( δ ) + ε i 1 , ε i 0 , ε i 1 ∼ i . i . d . N ( 0 , σ ε 2 ) , (44) \widetilde{Y}^{0}_{i} = \beta^{\top}X_i + \varepsilon_{i0}, \qquad\widetilde{Y}^{1}_{i} = \beta^{\top}X_i + b_i(\delta) + \varepsilon_{i1}, \qquad\varepsilon_{i0},\varepsilon_{i1} \overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}(0,\sigma_{\varepsilon}^{2}), \tag*{(44)} Y i 0 = β ⊤ X i + ε i 0 , Y i 1 = β ⊤ X i + b i ( δ ) + ε i 1 , ε i 0 , ε i 1 ∼ i.i.d. N ( 0 , σ ε 2 ) , (44) where d X = 5 d_X = 5 d X = 5 α = ( 0.90 , − 0.75 , 0.55 , − 0.40 , 0.30 ) ⊤ \alpha= (0.90,-0.75,0.55,-0.40,0.30)^{\top} α = ( 0.90 , − 0.75 , 0.55 , − 0.40 , 0.30 ) ⊤ β = ( 0.60 , − 0.50 , 0.40 , − 0.30 , 0.20 ) ⊤ \beta= (0.60,-0.50,0.40,-0.30,0.20)^{\top} β = ( 0.60 , − 0.50 , 0.40 , − 0.30 , 0.20 ) ⊤ σ ε = 0.1 \sigma_{\varepsilon} = 0.1 σ ε = 0.1
We consider the three scenarios of Muandet et al. [40 ] . For Scenario I, b i ( δ ) = 0 b_i(\delta) = 0 b i ( δ ) = 0 b i ( δ ) = δ b_i(\delta) = \delta b i ( δ ) = δ b i ( δ ) = δ ( 2 Z i − 1 ) b_i(\delta) = \delta(2Z_i - 1) b i ( δ ) = δ ( 2 Z i − 1 ) Z i ∼ i . i . d . Bernoulli ( 0.5 ) Z_i \overset{\mathrm{i.i.d.}}{\sim} \operatorname{Bernoulli}(0.5) Z i ∼ i.i.d. Bernoulli ( 0.5 ) H 0 H_0 H 0 H 1 H_1 H 1 ℓ i a \ell_i^a ℓ i a Y i a Y_i^a Y i a ℓ i a \ell_i^a ℓ i a Y i = Y i A i Y_i = Y_i^{A_i} Y i = Y i A i Figure 11 , and Figure 12 illustrates the construction.
Figure 11. Data-generating process for MNIST, RetinaMNIST and BUSI. The covariate X X X A A A Y ~ \widetilde{Y} Y Y Y Y
Figure 12. MNIST (Scenario II, left) and RetinaMNIST (Scenario III, right). Panels (a) and (c) show label distributions under each treatment. Panels (b) and (d) show sampled image outcomes.
We report average Type-I error for Scenario I and average power for Scenarios II and III in Table 6 for the three datasets at N = 1500 N = 1500 N = 1500 N = 3000 N = 3000 N = 3000 α = 0.05 \alpha= 0.05 α = 0.05
Table 6. Average rejection rates for MNIST, RetinaMNIST, and BUSI image outcomes at α = 0.05 \alpha= 0.05 α = 0.05 δ = 1.0 \delta= 1.0 δ = 1.0 1.5 1.5 1.5 δ = 2.0 \delta= 2.0 δ = 2.0 4.0 4.0 4.0 δ = 4.0 \delta= 4.0 δ = 4.0 ± \pm ±
Fake-MNIST. We now consider a treatment that changes the source dataset of the sampled image, adapting the comparison of Liu et al. [33 ] between real MNIST images and images generated by a generative adversarial network (GAN). We use a single digit covariate X i ∈ { 0 , … , 9 } X_i \in\{0,\ldots,9\} X i ∈ { 0 , … , 9 }
A i ∣ X i ∼ Bernoulli { π 0 ( 1 ∣ X i ) } , π 0 ( 1 ∣ X i ) = clip { σ ( s π X i − 4.5 8.25 ) , 0.07 , 0.93 } , A_i \mid X_i \sim\operatorname{Bernoulli}\{\pi_0(1 \mid X_i)\}, \qquad\pi_0(1 \mid X_i) = \operatorname{clip}\left\{\sigma\left(s_\pi\frac{X_i-4.5}{\sqrt{8.25}}\right),0.07,0.93\right\}, A i ∣ X i ∼ Bernoulli { π 0 ( 1 ∣ X i )} , π 0 ( 1 ∣ X i ) = clip { σ ( s π 8.25 X i − 4.5 ) , 0.07 , 0.93 } , where 4.5 and 8.25 \sqrt{8.25} 8.25 s π s_\pi s π
Under H 0 H_0 H 0 X i X_i X i H 1 H_1 H 1 X i X_i X i
r ( A i , X i ) = A i X i + 1 s r , r(A_i,X_i)=A_i\frac{X_i+1}{s_r}, r ( A i , X i ) = A i s r X i + 1 , and a real MNIST image otherwise, where s r s_r s r A i = 0 A_i=0 A i = 0 A i = 1 A_i=1 A i = 1 A i = 1 A_i=1 A i = 1 A i = 1 A_i=1 A i = 1
Average Type-I error and power at N = 1500 N=1500 N = 1500 N = 3000 N=3000 N = 3000 Table 7 . Under H 1 H_1 H 1 H 0 H_0 H 0
Table 7. Type-I error (H 0 H_0 H 0 H 1 H_1 H 1 α = 0.05 \alpha=0.05 α = 0.05 ± \pm ±
Causal-Blob
We evaluate our method on a Gaussian mixture dataset adapted from the Blob dataset of Liu et al. [33 ] to a causal setting with covariates. For each unit, we sample a mode index M i ∈ { 1 , … , 9 } M_i\in\{1,\ldots,9\} M i ∈ { 1 , … , 9 } { μ m } m = 1 9 = { ( j , k ) : j , k ∈ { 0 , 1 , 2 } } \{\mu_m\}_{m=1}^{9}=\{(j,k):j,k\in\{0,1,2\}\} { μ m } m = 1 9 = {( j , k ) : j , k ∈ { 0 , 1 , 2 }}
X i = μ M i + ε X , ε X ∼ N ( 0 , σ X 2 I 2 ) . X_i=\mu_{M_i}+\varepsilon_X,\qquad\varepsilon_X\sim\mathcal{N}(0,\sigma_X^2I_2). X i = μ M i + ε X , ε X ∼ N ( 0 , σ X 2 I 2 ) . Treatment is sampled as in Equation (42) . The potential outcomes follow the mode-centred Gaussians
Y i 0 ∣ M i = m ∼ N ( μ m , Σ P ) , Y i 1 ∣ M i = m ∼ N ( μ m , Σ Q , m ) , Y_i^0\mid M_i=m\sim\mathcal{N}(\mu_m,\Sigma_P),\qquad Y_i^1\mid M_i=m\sim\mathcal{N}(\mu_m,\Sigma_{Q,m}), Y i 0 ∣ M i = m ∼ N ( μ m , Σ P ) , Y i 1 ∣ M i = m ∼ N ( μ m , Σ Q , m ) , where Σ P = 0.03 I 2 \Sigma_P=0.03I_2 Σ P = 0.03 I 2 Σ Q , m \Sigma_{Q,m} Σ Q , m Σ P \Sigma_P Σ P ρ m \rho_m ρ m σ X = 0.3 \sigma_X=0.3 σ X = 0.3 α = ( 1.2 , − 0.8 ) ⊤ \alpha=(1.2,-0.8)^\top α = ( 1.2 , − 0.8 ) ⊤ H 1 H_1 H 1
Table 8 shows that DR-KTE-Deep achieves the highest power on Causal-Blob at both sample sizes, with the largest gain over DR-KTE-Opt at N = 3000 N=3000 N = 3000
Table 8. Power and Type-I error for Causal-Blob at α = 0.05 \alpha= 0.05 α = 0.05 H 1 / H 0 H_1/H_0 H 1 / H 0 ± \pm ±
Scalar Outcomes
We also evaluate our method on scalar outcomes. We adapt the scalar-outcome scenarios of Muandet et al. [40 ] with binary treatments. Covariates and treatments are sampled as in Equations (41) and (42) , and we use the scalar outcomes of Equation (44) directly. Scenarios I, II and III cover the null hypothesis, a mean shift and a mean-preserving distributional shift respectively. We additionally consider Scenario IV of Zenati et al. [69 ] , with b i ( δ ) = 4 δ Z i b_i(\delta) = 4\delta Z_i b i ( δ ) = 4 δ Z i Z i ∼ Bernoulli ( 0.5 ) Z_i \sim\operatorname{Bernoulli}(0.5) Z i ∼ Bernoulli ( 0.5 )
Table 9 reports average Type-I error for Scenario I and average power for Scenarios II-IV. Type-I error is close to the nominal level for the kernel tests at both sample sizes. Power is close to one in Scenarios II and IV, with little improvement from learning a deep kernel. Scenario III is more challenging, although power increases substantially with sample size. DR-KTE-Opt achieves the highest estimated power in Scenario III at both sample sizes.
Table 9. Power and Type-I error for scalar outcomes at α = 0.05 \alpha= 0.05 α = 0.05 δ = 0.8 \delta= 0.8 δ = 0.8 M = 5 M = 5 M = 5 H 1 / H 0 H_1/H_0 H 1 / H 0 ± \pm ±
Estimator Comparison
We study how the choice of estimator affects test power and Type-I error, using the same learned outcome kernel for all methods. We consider the continuous-treatment construction of Zenati et al. [69 ] . We compare doubly robust estimation with inverse-propensity weighting (IPW) and a kernel policy test. For each unit, we sample the covariate as
X i ∼ N ( 0 , I 5 ) , X_i \sim\mathcal{N}(0, I_5), X i ∼ N ( 0 , I 5 ) , and treatment is assigned as
A i ∣ X i ∼ N ( X i ⊤ w , 1 ) . A_i \mid X_i \sim\mathcal{N}(X_i^\top w, 1). A i ∣ X i ∼ N ( X i ⊤ w , 1 ) . The outcome is then
Y i = X i ⊤ β + A i + 0.1 ϵ i , ϵ i ∼ N ( 0 , 1 ) , Y_i = X_i^\top\beta+ A_i + 0.1\epsilon_i,\qquad\epsilon_i \sim\mathcal{N}(0,1), Y i = X i ⊤ β + A i + 0.1 ϵ i , ϵ i ∼ N ( 0 , 1 ) , where w , β ∈ R 5 w, \beta\in\mathbb{R}^{5} w , β ∈ R 5
We compare two interventions π \pi π π ′ \pi^{\prime} π ′ π : = π 0 \pi:= \pi_0 π := π 0 [69 ] with continuous treatments. Scenario I tests the null hypothesis with π ′ ( ⋅ ∣ x ) = N ( x ⊤ w I , 1 ) \pi^{\prime}(\cdot\mid x) = \mathcal{N}(x^\top w_I,1) π ′ ( ⋅ ∣ x ) = N ( x ⊤ w I , 1 ) w I ≠ w w_I \ne w w I = w ∥ β + w I ∥ = ∥ β + w ∥ \lVert\beta+ w_I\rVert= \lVert\beta+ w\rVert ∥ β + w I ∥ = ∥ β + w ∥ π ′ ( ⋅ ∣ x ) = N ( x ⊤ ( w + 2 15 ) , 1 ) \pi^{\prime}(\cdot\mid x) = \mathcal{N}(x^\top(w + 2^{15}),1) π ′ ( ⋅ ∣ x ) = N ( x ⊤ ( w + 2 15 ) , 1 ) π ′ ( ⋅ ∣ x ) \pi^{\prime}(\cdot\mid x) π ′ ( ⋅ ∣ x ) N ( x ⊤ ( w + 15 ) , 1 ) \mathcal{N}(x^\top(w + 15),1) N ( x ⊤ ( w + 15 ) , 1 ) N ( x ⊤ ( w − 15 ) , 1 ) \mathcal{N}(x^\top(w - 15),1) N ( x ⊤ ( w − 15 ) , 1 ) π ′ ( ⋅ ∣ x ) \pi^{\prime}(\cdot\mid x) π ′ ( ⋅ ∣ x ) N ( x ⊤ ( w + 2 15 ) , 1 ) \mathcal{N}(x^\top(w + 2^{15}),1) N ( x ⊤ ( w + 2 15 ) , 1 ) N ( x ⊤ w , 1 ) \mathcal{N}(x^\top w,1) N ( x ⊤ w , 1 )
We compare our doubly robust estimator, DR-KTE-Deep, to a cross-U IPW estimator, IPW-Cross, obtained by setting μ ^ θ ( a , x ) : = 0 \hat{\mu}_{\theta}(a,x) := 0 μ ^ θ ( a , x ) := 0 (1) . We additionally compare our method to a kernel policy test, KPT, following Zenati et al. [69 ] . All methods use the same deep outcome kernel, learned with the doubly robust criterion. We divide the data into four equal samples for learning, validation, nuisance fitting and testing. For this comparison, checkpoint selection uses variance regularisation 10 − 4 10^{-4} 1 0 − 4
Table 10 shows that DR-KTE-Deep achieves higher estimated power than IPW-Cross and KPT in all three alternative scenarios, with the largest gains in Scenario III. Under the null hypothesis, its rejection rate is close to the nominal level of α = 0.05 \alpha= 0.05 α = 0.05
Table 10. Power and Type-I error at α = 0.05 \alpha= 0.05 α = 0.05 N = 1600 N = 1600 N = 1600 M = 1 M = 1 M = 1 H 1 / H 0 H_1/H_0 H 1 / H 0 ± \pm ±
Robustness to Outcome Model Misspecification
We analyse the contribution of fitting the conditional outcome embedding with the logging policy known. We compare the fitted embedding with μ ^ θ ( a , x ) = 0 \hat{\mu}_{\theta}(a,x) = 0 μ ^ θ ( a , x ) = 0 π ^ 0 ( a ∣ x ) ≡ π 0 ( a ∣ x ) \hat{\pi}_0(a \mid x) \equiv\pi_0(a \mid x) π ^ 0 ( a ∣ x ) ≡ π 0 ( a ∣ x )
Table 11 shows similar Type-I error for the fitted and zero outcome embeddings, with rejection rates close to the nominal level. Using the fitted outcome embedding improves power on Causal-Blob and in Scenarios II and III, with the largest gain in Scenario III. Both estimators achieve power close to one in Scenario IV.
Table 11. Power/Type-I error (H 1 / H 0 H_1/H_0 H 1 / H 0 α = 0.05 \alpha= 0.05 α = 0.05 N = 1500 N = 1500 N = 1500 M = 1 M = 1 M = 1 H 1 / H 0 H_1/H_0 H 1 / H 0 ± \pm ±
Training Objective Comparison
We test whether our power learning objective improves detection compared to a reconstruction objective. On OCT-Drusen and Fake-MNIST, we compare DR-KTE-Deep with a frozen randomly initialised CNN and a pixel autoencoder. All variants share the same encoder architecture, initial encoder weights, learning and validation observations, and test datasets.
DR-KTE-Deep has the highest power on both constructions under H 1 H_1 H 1 Table 12 ), while Type-I error remains close to the nominal level for all methods.
Table 12. Power under H 1 H_1 H 1 H 0 H_0 H 0 N = 1500 N = 1500 N = 1500
Text Representation Ablation
We consider an alternative representation for the text outcomes described in § 5 , mapping each outcome to a fixed-length vector using the Qwen3-Embedding-0.6B ((Zhang et al., 2025)) embedding model. Since a language-model embedding can capture semantic information, a simple Gaussian kernel may be well suited to detecting differences in outcome distributions.
Table 13 shows that all three methods achieve high, similar power using this representation. Type-I error remains close to the nominal level for Oncology Notes and Clinical Notes, while all three methods are conservative on Movie Reviews.
However, this embedding is not guaranteed to be injective: distinct texts can collapse to the same embedding. A Gaussian kernel, even if characteristic on the embedding space, might therefore not detect every difference between the original text outcome distributions. Additionally, this representation can be more costly than a sequence of tokens, as it requires inference over an embedding model.
Table 13. Power (H 1 H_1 H 1 H 0 H_0 H 0 N = 1500 N = 1500 N = 1500 α = 0.05 \alpha= 0.05 α = 0.05
Effect of Ensemble Size
We evaluate the effect of memberwise averaging by varying the number of separately trained kernels included in the ensemble. We perform five independent batches, each consisting of 20 independent outer repetitions. In each repetition, we train five members and form ensembles with M ∈ { 1 , 2 , 3 , 5 } M \in\{1, 2, 3, 5\} M ∈ { 1 , 2 , 3 , 5 } M M M N = 1500 N = 1500 N = 1500 δ = 1.0 \delta= 1.0 δ = 1.0
For each outer repetition and ensemble size, we estimate power as the rejection rate over 200 fresh test datasets under H 1 H_1 H 1 α = 0.05 \alpha= 0.05 α = 0.05
On OCT-Drusen, power increases slightly with ensemble size, and its standard deviation is lower at M = 5 M = 5 M = 5 M = 1 M = 1 M = 1 Table 14 and Figure 13 ).
Table 14. Mean and standard deviation (SD) of empirical power at N = 1500 N = 1500 N = 1500 δ = 1.0 \delta= 1.0 δ = 1.0 ± \pm ±
Figure 13. Average standard deviation of empirical power against ensemble size at N = 1500 N = 1500 N = 1500 δ = 1.0 \delta= 1.0 δ = 1.0 ± \pm ±
While Proposition B.3 establishes variance reduction for the population discrepancy with the training data fixed, it does not imply that empirical power becomes less variable across repetitions.
Weisfeiler-Lehman Outcome Kernel
We consider an alternative base kernel for the graph-outcome experiments. We replace the Gaussian kernel used as k c h a r k^{\mathrm{char}} k char [53 ] ) and denote the resulting tests by DR-KTE-WL and DR-KTE-WL-Deep. Since the WL kernel is not generally characteristic, these methods do not retain the same omnibus guarantees as the Gaussian-based ones.
Table 15 reports average power under H 1 H_1 H 1 H 0 H_0 H 0
Table 15. Power (H 1 H_1 H 1 H 0 H_0 H 0 N = 1500 N = 1500 N = 1500 H 1 / H 0 H_1/H_0 H 1 / H 0 ± \pm ± Table 3 .
Linear Outcome Kernel
We compare DR-KTE-Deep with DR-ATE, using a linear outcome kernel k ( y , y ′ ) = ϕ ( y ) T ϕ ( y ′ ) k(y,y') = \phi(y)^{\mathsf{T}}\phi(y') k ( y , y ′ ) = ϕ ( y ) T ϕ ( y ′ ) ϕ ( y ) \phi(y) ϕ ( y )
For the medical-image experiments, Causal-Blob and the scalar outcomes Scenario III, which construct a mean-matched but distributional effect, DR-KTE-Deep has high power, while DR-ATE rejects around the nominal rate (Table 16 , Table 20 and Table 21 ). In the remaining experiments, where the construction introduces a mean-shift in interventional outcomes, DR-ATE often has high power, but DR-KTE-Deep always achieves higher rejection rates (Table 17 to Table 19 ).
Table 16. Comparison with DR-ATE for medical-image outcomes at N = 1500 N=1500 N = 1500 H 1 / H 0 H_1/H_0 H 1 / H 0 ± \pm ±
Table 17. Comparison with DR-ATE for MNIST, RetinaMNIST, BUSI and Fake-MNIST at N = 1500 N=1500 N = 1500 H 0 H_0 H 0 H 1 H_1 H 1 ± \pm ±
Table 19. Comparison with DR-ATE for graph outcomes at N = 1500 N=1500 N = 1500 H 1 / H 0 H_1/H_0 H 1 / H 0 ± \pm ±
Table 20. Comparison with DR-ATE on Causal-Blob at N = 1500 N = 1500 N = 1500 H 1 / H 0 H_1/H_0 H 1 / H 0 ± \pm ±
Table 21. Comparison with DR-ATE for scalar outcomes at N = 1500 N = 1500 N = 1500 H 1 / H 0 H_1/H_0 H 1 / H 0 ± \pm ±
Table 18. Comparison with DR-ATE for text outcomes at N = 1500 N=1500 N = 1500 H 1 / H 0 H_1/H_0 H 1 / H 0 ± \pm ±
Scaling with Outcome Dimension
We compare DR-ATE, DR-KTE, DR-KTE-Opt and DR-KTE-Deep as the outcome dimension increases. We use Causal-Blob (Section D.2), where we vary the outcome dimension by appending independent noise coordinates to the two-dimensional outcomes. For d > 2 d > 2 d > 2
Y i ( d ) = ( Y i ( 2 ) Z i ) , Z i ∼ N ( 0 , 0.17 2 I d − 2 ) , Y i ( 2 ) = ( 1 − A i ) Y i 0 + A i Y i 1 , Y_i^{(d)} = \begin{pmatrix} Y_i^{(2)} \\ Z_i \end{pmatrix}, \quad Z_i \sim\mathcal{N}(0,0.17^2 I_{d-2}), Y_i^{(2)} = (1-A_i)Y_i^0 + A_iY_i^1, Y i ( d ) = ( Y i ( 2 ) Z i ) , Z i ∼ N ( 0 , 0.1 7 2 I d − 2 ) , Y i ( 2 ) = ( 1 − A i ) Y i 0 + A i Y i 1 , where Z i Z_i Z i Y i ( 2 ) Y_i^{(2)} Y i ( 2 ) Z i Z_i Z i
Figure 14 shows that DR-KTE-Opt improves power over DR-KTE, while DR-KTE-Deep achieves substantially higher power through d = 8 d = 8 d = 8 d = 16 d = 16 d = 16 d = 32 d = 32 d = 32
Figure 14. Causal-Blob at N = 4000 N = 4000 N = 4000 δ = 0 \delta= 0 δ = 0 H 0 H_0 H 0 δ = 1.0 \delta= 1.0 δ = 1.0 H 1 H_1 H 1 ± \pm ± α = 0.05 \alpha= 0.05 α = 0.05
Effect of Joint Maximisation
We compare DR-KTE-Deep to a variant that jointly maximises the power learning criterion of the averaged ensemble. In DR-KTE-Deep, each kernel member is trained and selected separately. The final ensemble can therefore be composed of members selected from different training epochs. The joint maximisation variant instead selects a single ensemble checkpoint on the validation sample. Joint maximisation therefore removes the flexibility to select each member at its own best validation checkpoint.
Joint maximisation generally yields similar or lower estimated power than DR-KTE-Deep, while improving on DR-KTE (Table 22 ). OCT-Drusen is the exception, with higher power under joint maximisation. Type-I error remains close to the nominal level.
Table 22. Effect of joint maximisation at N = 1500 N = 1500 N = 1500 α = 0.05 \alpha= 0.05 α = 0.05 H 1 / H 0 H_1/H_0 H 1 / H 0 ± \pm ±
