Modulating Merging Strengths via Joint Loss Estimation for LoRA-based Continual Learning

Introduction

Continual learning (CL) aims to enable models to learn continuously from a stream of tasks without retraining from scratch or accessing data from previous tasks [13, 25], which is essential for evolving open-world scenarios. A central challenge in CL is catastrophic forgetting: models must preserve previously learned capabilities during adapting to new tasks [17, 26]. In this work, we consider the challenging rehearsal-free Continual Learning setting, where the model must continually learn new tasks without storing past data and predict test samples over all seen tasks without task identities.

Recently, with the advancement of Pre-Trained Models (PTMs) [4], Parameter-Efficient Fine-Tuning (PEFT) methods have emerged as a promising solution for CL [31, 35, 43]. By updating a small number of parameters, PEFT-based methods enable efficient adaptation to new tasks while preserving general pre-trained knowledge, making it particularly suitable for rehearsal-free CL.

In the regime of PEFT-based CL, Low-Rank Adaptation (LoRA) [11] methods have gained increasing attention due to their efficiency and scalability. To maintain a single inference model, a common strategy is to merge the current task LoRA update into the accumulated model after each task [28], rather than storing a separate LoRA module per task. This controlled merging reduces interference with prior knowledge while enabling adaptation to the current task. Early studies mainly adopt EMA-style interpolation [20] or weight averaging [6], while recent approaches estimate the interpolation coefficient from joint objectives [28] or Bayesian formulations [16]. Despite these different designs, most existing methods still use a shared scalar factor to control the whole LoRA update. This naturally raises a question: should all LoRA parameters share the same merge strength, or should different positions in the LoRA update be treated differently during merging? To answer this question, we conduct a preliminary analysis in Figure 1, where the LoRA update is split into two groups according to different normalized magnitudes [12]. We then assign an independent merge coefficient to each group. The results show that this simple grouped merge already outperforms the best shared factor merge, suggesting that different parts of the LoRA update may prefer different merge strengths.

Building on this observation, we propose a Merging Strength Modulation via Joint Loss Estimation ($M^{2}$LE) framework for LoRA-based Continual Learning, consists of Hessian Information-guided Adaptive Merging (HIM) and Task Simulation Perturbation for Merging (TSP). Instead of searching for an interpolation coefficient, HIM directly formulates the optimization of the target merged update $\Delta\mathbf{W}^{*}_{t}$ under a joint objective that accounts for losses on all seen tasks. Through a local quadratic approximation to the task-wise losses, we obtain an analytic solution, which naturally takes the form of an element-wise modulation of the current task LoRA update. Specifically, each element is adaptively scaled by the associated Hessian information, allowing the merging process to account for the varying sensitivity of individual parameter elements across tasks, thereby achieving a more fine-grained balance between old and new task performance. However, once the current task update is merged into the inference model, it becomes part of the fixed parameters and will be affected by subsequent task updates. If the merged update lies in a sharp local region, later merging steps can easily increase the loss on old tasks, resulting in forgetting. Therefore, besides determining how to merge the update, it is also important to shape the update itself before merging. To this end, we introduce TSP, which simulates cross-task interference during current task training. TSP induces a local parameter change with one subset of the current mini-batch and evaluates the resulting loss variation on another subset, thereby encouraging the learned update to remain stable during subsequent merging.

Our main contributions can be summarized as follows:

• We propose Hessian Information-guided Adaptive Merging (HIM), a parameter-adaptive merging algorithm derived from an approximated joint loss objective, enabling more fine-grained stability-plasticity balancing beyond a shared scalar merging coefficient.

• We propose Task Simulation Perturbation (TSP), a training strategy that simulates future cross-task interference through local parameter perturbations induced from label-disjoint subset, thereby encouraging flat solutions and improving LoRA merging stability.

• Extensive experiments demonstrate the effectiveness of our proposed method, achieving superior performances over state-of-the-art PEFT-based CL methods on multiple benchmarks.

Related Work

PEFT-based and LoRA-based continual learning. PEFT-based CL [50] typically keeps a pre-trained model frozen and introduces lightweight learnable modules for continual learning. L2P [43] retrieves instance specific prompts from a prompt pool, and SECA [8] uses textual semantic priors to support knowledge transfer and prototype refinement in CLIP-based continual learning. EWC [13] preserves previous knowledge by penalizing changes to important parameters, and EKPC [38] mitigates forgetting through elastic knowledge preservation and compensation. Among PEFT-based method [1, 2, 3, 39, 46, 47, 48, 52], LoRA has become a representative choice for continual learning due to its parameter efficiency and compatibility with frozen pre-trained backbones [11]. InfLoRA [18] constrains new LoRA updates to interference-free subspaces through orthogonality, PLAN [41] proactively allocates low-rank subspaces and optimizes them against worst-case perturbations, Gain-LoRA [19] expands a new LoRA for each task and introduces gating modules to adaptively weight the old and new branches, and MACIL [44] mitigates forgetting through semantic drift calibration. LoDA [7] continually decomposes LoRA updates into general and isolated subspaces for better stability-plasticity trade-off.

Model Merging. Model merging aims to integrate weights trained on different tasks into a single network [33]. RobustMerge [49] improves parameter-efficient merging by enhancing the robustness of low-rank update directions, while OPCM [36] enables sequential merging through orthogonal projection and adaptive scaling. This idea has also been used in continual learning, where the currentiyanas model or update is merged into the inference model after each task [12]. ACMap [6] averages the adapters learned from different tasks into a shared adapter, BECAME [16] derives an adaptive coefficient from a Bayesian formulation, and LoRA-P&M [28] combines the previous and current models using a shared merging factor computed in closed form. However, these methods mainly rely on a shared factor to interpolate updates, while our method derives the merged update from the loss changes of old and new tasks under a joint objective.

Preliminary Analysis

Research Question. Should LoRA parameters at different positions be treated differently during merging, rather than being controlled by a shared factor?

Experiment Setting. As a case study, we consider the 10 task setting on ImageNet-R and focus on the first merge step. Specifically, we take the model after learning task 0 and the update learned on task 1 for analysis. Let $W_0$ denote the inference parameter after learning task 0, and let $\Delta W_1$ denote the LoRA update learned on task 1. To this end, we build a simple merging setup. Specifically, we compute the normalized magnitude map of $\Delta W_1$ as $R = \lvert\Delta W_1\rvert/ \max_{i,j} \lvert\Delta W_1^{(i,j)}\rvert$ [12], and divide the update into two groups: elements with normalized values below 0.5, and elements with normalized values at least 0.5. We then assign two independent coefficients, $\alpha_1$ and $\alpha_2$, to the two groups, and define the merged model as: $W(\alpha_1,\alpha_2) = W_0 + \alpha_1(M_1 \odot\Delta W_1) + \alpha_2(M_2 \odot\Delta W_1)$, where $M_1$ and $M_2$ denote the corresponding masks, and $\odot$ denotes element-wise multiplication. By varying $\alpha_1$ and $\alpha_2$, we evaluate the joint loss on task 0 and task 1.

Observation. Figure 1 shows that the low loss region is not centered on the diagonal $\alpha_1 = \alpha_2$, but is shifted toward an off-diagonal region. Consistently, the green diamond marks the point with the best average accuracy on task0 and task1, which is located at (0.40, 0.80), while the purple diamond marks the best scalar merge on the diagonal, located at (0.60, 0.60). Compared with the scalar merge, the green-diamond point improves the average accuracy from 91.04% to 91.84%.

Insight. These results suggest that a shared scalar coefficient is overly restrictive: different parameter groups do not, in general, share the same preferred merge strength, and should therefore be treated differently during merging.

Methodology

Preliminaries

In CL, a model learns from a sequence of tasks $\{D_t\}_{t=1}^{T}$. The dataset of task $t$ is denoted by $D_t = \{(x_i^t,y_i^t)\}_{i=1}^{n_t}$, where $x_i^t \in\mathbb{R}^{d}$ is the input and $y_i^t \in\mathcal{C}_t$ is the corresponding label. Each task introduces a set of new classes, and the label spaces of different tasks are mutually disjoint, i.e., $\mathcal{C}_i \cap\mathcal{C}_j = \varnothing$ for any $i \ne j$. Accordingly, the cumulative label space observed up to task $t$ is $\mathcal{C}_{1:t} = \bigcup_{i=1}^{t} \mathcal{C}_i$. Under the standard rehearsal-free CL setting, only the current task data $D_t$ is accessible when learning task $t$. The goal is to learn tasks sequentially while maintaining strong performance over all tasks observed so far.

Overall framework

Let $W_{t-1} \in\mathbb{R}^{d_o \times d_i}$ denote the inference parameter after learning task $t-1$, where $d_i$ and $d_o$ are the input and output dimensions. At $t$-th task, for each ViT layers, we freeze the backbone weights $W_{t-1}$ and attach a LoRA module. The LoRA update is parameterized by two low-rank matrices:

where $r$ is the LoRA rank. A common LoRA merging strategy directly updates the inference model by scaling the current task update with a shared coefficient $W_t = W_{t-1} + \alpha_t \Delta W_t$, where the scalar $\alpha_t$ controls the merge strength of the whole update. However, this scalar merge applies the same merge strength to all elements in $\Delta W_t$, although different elements may have different effects on previous and current tasks.

To address this, $M^{2}$LE consists of two components: Hessian Information-guided Adaptive Merging (HIM) and Task Simulation Perturbation for Merging (TSP). During training, TSP regularizes the LoRA update by simulating cross task perturbations, encouraging the learned update $\Delta W_t$ to improve the stability during subsequent merging. After training, HIM derives a target merged update $\Delta W_t^{*}$ under a joint loss impact objective over previous and current tasks, instead of searching for a global merge coefficient. The inference parameter is then updated by $W_t = W_{t-1} + \Delta W_t^{*}$. An overview of the framework and the complete training procedure are provided in Appendix A (Figure 5 and Algorithm 1).

Hessian Information-guided Adaptive Merging

Existing LoRA-based merging methods for CL always merge the current task LoRA update into the inference model using a shared scalar factor. This forces all parameter elements to be merged with the same strength, although they may affect previous and current tasks differently. This motivates us to account for the different preferred merge strengths of update parameter during merging.

Specifically, we directly solve for the target merged update $\Delta W_t^{*} \in\mathbb{R}^{d_o \times d_i}$ under a joint objective over previous and current tasks. Intuitively, we want this update to incorporate new knowledge as effectively as possible while reducing catastrophic forgetting on previous tasks. To this end, we first define $\mathcal{J}_{\mathrm{old}}$ and $\mathcal{J}_{\mathrm{new}}$ to measure how an ideal merged update $\widehat{\Delta W}_t \in\mathbb{R}^{d_o \times d_i}$ affects the losses on previous and current tasks:

Here, $\Delta W_t \in\mathbb{R}^{d_o \times d_i}$ denotes the update learned for task $t$, $\mathcal{L}_i$ denotes the loss of task $i$. Therefore, we seek a merged update that jointly minimizes $\mathcal{J}_{\mathrm{old}}$ on previous tasks and $\mathcal{J}_{\mathrm{new}}$ on the current task, with a hyper-parameter $\lambda$ controlling the strength of the current task term. The target merged update $\Delta W_t^{*}$ is obtained by solving:

In the rehearsal-free setting, Eq. (4) cannot be optimized directly, since the old-task term $\mathcal{J}_{\mathrm{old}}$ depends on previous task data that are no longer accessible when learning task $t$. To obtain a tractable approximation, we approximate Eq. (2) and Eq. (3) with local second-order Taylor expansions. For notational simplicity, we vectorize $\widehat{\Delta W}_t$ into $\widehat{\Delta w}_t = \operatorname{vec}(\widehat{\Delta W}_t) \in\mathbb{R}^{d_o d_i \times1}$. For the old task term $\mathcal{J}_{\mathrm{old}}(\widehat{\Delta w}_t)$, expanding around $w_{t-1}$ gives:

where $\mathbf{g}_i \in\mathbb{R}^{d_o d_i \times1}$ and $\mathbf{H}_i \in\mathbb{R}^{d_o d_i \times d_o d_i}$ denote the first and second terms of the $i$-th old task loss around $w_{t-1}$, respectively. Similarly, for the current task, expanding around $w_{t-1} + \Delta w_t$ yields:

where $\mathbf{g}_t$ and $\mathbf{H}_t$ denote the first- and second-order terms of the current task loss around $w_{t-1} + \Delta w_t$, respectively. Since the previous model $w_{t-1}$ is obtained by recursively merging past updates and $w_{t-1} + \Delta w_t$ is the learned current-task solution. Under a local approximation, we treat the first-order terms in Eq. (5) and Eq. (6) as approximately zero. Substituting the remaining quadratic terms into Eq. (4), we obtain:

Taking the derivative with respect to $\Delta\widehat{\mathbf{w}}_{t}$ and setting it to zero gives:

A detailed derivation is provided in Appendix § 9, and Appendix § 10.4 further analyzes the effectiveness of accumulated historical Fisher statistics. Following EWC [13], we approximate each Hessian in Eq. (8) with the diagonal empirical Fisher, reusing their historical running sum for the old task term. For task $t$, the $i$-th diagonal element of $\mathbf{F}_{t}$ at the learned parameters $\mathbf{W}_{t-1}+\Delta\mathbf{W}_{t}$ is defined as

Here, $\mathbf{F}_{t}\in\mathbb{R}^{d_{o}\times d_{i}}$ denotes the diagonal Fisher of the trainable update, reshaped to match $\Delta\mathbf{W}_{t}$. Under this diagonal approximation, Eq. (8) gives the following element-wise merged update:

where $\varepsilon$ is a small damping constant for numerical stability and after merging task $t$, we update the accumulated Fisher by $\mathbf{F}_{1:t}=\mathbf{F}_{1:t-1}+\mathbf{F}_{t}$, which is used in the next merging step.

Task Simulation Perturbation for Merging

According to Eq. (9), HIM obtains $\Delta\mathbf{W}^{*}_{t}$ by element-wise reweighting $\Delta\mathbf{W}_{t}$ and merge it into the inference model, where it becomes part of the fixed parameters for later tasks. If the merged update lies in a sharp region, subsequent new task merging can easily increase its loss and lead to forgetting. This naturally connects to the flat-minimum principle, making SAM [5] a straightforward way to improve the local stability of the current task update. However, SAM computes the perturbation from the overall current task loss and minimizes the perturbed loss on the same task, so its flatness constraint remains limited to the current task. Such a design makes it difficult to approximate the case in CL where the update from one task affects another task. To approximating cross-task interference when old task data are unavailable. We randomly split each mini-batch into two label-disjoint subsets ($S_{1},S_{2}$), using one subset to induce a local parameter change and the other to evaluate the resulting loss change.

For the $S_{1}\to S_{2}$ direction, we first compute the gradient on the source subset $S_{1}$:

We then construct a probing perturbation along the induced descent direction:

where $\rho$ controls the perturbation scale and $\varepsilon$ is a small constant for numerical stability. The target subset $S_{2}$ is evaluated under both the perturbed and original parameters:

The perturbed loss evaluates whether the direction induced by $S_{1}$ remains effective on the disjoint subset $S_{2}$ under a local displacement. To prevent the perturbation from pulling training too far away from the normal update and degrading the clean loss on $S_{2}$, we retain the clean loss as an anchor and define the TSP objective as:

where $\beta$ balances the perturbed and clean terms. We first update the parameters for the $S_{1}\to S_{2}$ step by:

where $\eta$ is the learning rate. We then exchange the roles of the two subsets and perform the reverse update:

where $\mathcal{L}_{\mathrm{TSP}}^{2\to1}$ is defined in the same way, with the roles of $\mathcal{S}_1$ and $\mathcal{S}_2$ exchanged. Thus, within each mini-batch, the optimization proceeds sequentially as $B_t A_t \xrightarrow{\mathcal{S}_1\to\mathcal{S}_2} B_t^{\prime} A_t^{\prime} \xrightarrow{\mathcal{S}_2\to\mathcal{S}_1} B_t^{\prime\prime} A_t^{\prime\prime}$. This dual training framework learns the current task update under task simulating perturbations, so that the subsequent merge better preserves previously learned knowledge.

Experiment

Datasets. We evaluate our method on five standard continual learning benchmarks: ImageNet-R [9], ImageNet-A [10], CIFAR100 [15], CUB200 [37], and DomainNet [27]. ImageNet-R contains 30,000 images from 200 classes with diverse styles, while ImageNet-A consists of 7,500 natural adversarial examples spanning 200 classes. CIFAR100 is a standard benchmark with 60,000 images from 100 classes, and CUB200 is a fine-grained bird classification dataset with 200 categories. For DomainNet, following prior works [7], we use the commonly adopted 200-class evaluation protocol.

Evaluation Metrics. Following prior PEFT-based continual learning works [35, 44], we adopt two standard evaluation metrics: (1) $A_{\mathrm{Last}}$, i.e., the final accuracy over all classes after the last task, and (2) $A_{\mathrm{Avg}}$, i.e., the average accuracy across all incremental tasks.

Implementation Details. We use ViT-B/16 [4] pre-trained on ImageNet-21K [30] as the backbone. All experiments are conducted on a single NVIDIA RTX 3090 GPU. The model is optimized with SGD and a Cosine Annealing scheduler following [44]. We train each task for 20 epochs with a batch size of 48. The initial learning rate is set to 0.02 on ImageNet-R and 0.01 on the other datasets. For the HIM, the hyper-parameter $\lambda$ in Eq. (9) is set to 5 for all datasets. For the TSP, $\rho$ in Eq. (11) is set to 0.3 on ImageNet-A and 0.5 on the other datasets, while $\beta$ in Eq. (13) is set to 0.6 on ImageNet-A and 0.8 on the other datasets.

Variant with classifier alignment. For fair comparison with methods that use feature replay or classifier alignment (CA), we also evaluate an enhanced variant, denoted as $M^{2}$LE+CA. $M^{2}$LE follows the end-to-end continual learning protocol without feature replay or post-hoc classifier adjustment. $M^{2}$LE+CA further applies the widely used classifier alignment strategy after each task, following SLCA [51]. We report these two variants separately to distinguish the effect of our proposed method from the additional classifier alignment stage.

Comparison with the State-of-the-Arts

We compare our method with representative state-of-the-art continual learning approaches from three categories, including prompt-based methods [31, 43], adapter-based methods [21, 34, 35, 40] and LoRA-based methods [18, 28, 41, 44, 45, 53, 57]. For completeness, methods that rely on feature replay or classifier alignment are reported separately from strictly rehearsal-free methods.

As shown in Table 1 and Table 2, our method demonstrates clear advantages over existing PEFT-based continual learning methods. Without classifier alignment, compared with the LoRA-P&M, $M^{2}$LE consistently improves $\mathcal{A}_{Last}$ across all settings, with gains of 1.39%–5.88% on the 10 task benchmarks and 0.43%–2.25% under the additional task settings. With classifier alignment, $M^{2}$LE+CA achieves highly competitive performance against feature-replay-based methods, outperforming MACIL in $\mathcal{A}_{Last}$ by 0.80%–1.82% on the 10 task ImageNet-R/ImageNet-A settings and by 1.69%–4.89% on their 20 task settings. The margins are relatively smaller on CIFAR100 and CUB, where strong pre-trained representations already provide discriminative features and thus reduce the need for further backbone adaptation.

Table 1: Experimental results for 10 incremental tasks on ImageNet-R, ImageNet-A, CIFAR100, and CUB. We report the averaged results over 3 random seeds. Within each group, the highest results are in bold, and the second highest results are _underlined_. "FR" denotes the use of feature replay.

Method	Venue	FR	10S-ImageNet-R $A_{\mathrm{Last}} \uparrow$	10S-ImageNet-R $A_{\mathrm{Avg}} \uparrow$	10S-ImageNet-A $A_{\mathrm{Last}} \uparrow$	10S-ImageNet-A $A_{\mathrm{Avg}} \uparrow$	10S-CIFAR100 $A_{\mathrm{Last}} \uparrow$	10S-CIFAR100 $A_{\mathrm{Avg}} \uparrow$	10S-CUB $A_{\mathrm{Last}} \uparrow$	10S-CUB $A_{\mathrm{Avg}} \uparrow$
L2P [43]	CVPR’22	✗	72.34±0.17	77.36±0.64	44.04±0.93	51.24±2.26	84.06±0.88	88.26±1.34	67.02±1.90	70.62±1.60
CODAPrompt [31]	CVPR’23	✗	73.31±0.50	78.47±0.53	52.08±0.12	63.92±0.12	83.21±3.39	87.71±3.17	77.23±1.12	81.90±0.85
InfLoRA [18]	CVPR’24	✗	74.75±0.64	80.67±0.55	49.20±1.12	60.92±0.61	86.75±0.35	91.72±0.15	70.82±0.23	81.39±0.14
PLAN [41]	ICCV’25	✗	75.25±0.42	80.41±0.56	–	–	87.54±0.31	92.21±0.35	–	–
SEMA [40]	CVPR’25	✗	78.00	83.56	53.32	64.53	86.98	91.37	–	–
SD-LoRA [45]	ICLR’25	✗	77.34±0.35	82.04±0.24	55.96±0.73	64.95±1.63	88.01±0.31	92.54±0.18	77.48±0.20	85.59±0.44
BiLoRA [57]	CVPR’25	✗	77.95±0.14	81.52±0.26	–	–	87.46±0.76	92.50±0.62	–	–
LoRA-P&M [28]	NIPS’25	✗	79.95±0.18	85.29±0.93	56.57±0.78	65.35±1.81	88.45±0.35	92.89±1.13	78.29±0.50	83.39±0.61
EWC-LoRA [53]	ICLR’26	✗	72.86±0.79	78.95±0.86	59.89±0.26	68.33±0.67	87.91±0.57	92.27±0.39	–	–
M²LE (Ours)	–	✗	81.34±0.16	86.53±0.51	62.45±1.34	71.02±1.82	89.97±0.04	93.31±0.95	81.40±0.21	89.23±0.48
SLCA [51]	ICCV’23	✓	79.35±0.28	83.29±0.46	61.05±0.63	68.88±2.31	91.26±0.37	94.29±0.92	84.68±0.09	90.77±0.79
SSIAT [35]	CVPR’24	✓	79.38±0.59	83.63±0.43	62.43±1.63	70.83±1.63	91.35±0.26	94.35±0.60	88.75±0.38	93.00±0.90
MOS [34]	AAAI’25	✓	77.68±0.41	82.06±0.53	54.75±1.09	63.32±2.38	91.53±0.35	94.21±0.91	89.97±0.32	93.43±0.60
MACIL [44]	ICML’25	✓	81.88±0.07	85.95±0.27	64.14±0.58	71.45±1.35	91.94±0.17	94.43±0.79	90.52±0.13	93.93±0.47
SAEF [21]	CVPR’26	✓	80.02	84.54	64.91	73.39	90.60	94.53	–	–
M²LE (Ours) + CA	–	✓	82.68±0.30	87.05±0.51	65.96±0.90	74.02±0.85	91.86±0.21	94.62±0.90	90.35±0.16	93.92±0.38

Table 2: Additional results under different task settings, including 5 and 20 tasks on ImageNet-R, 20 tasks on ImageNet-A, and 10 tasks on DomainNet. Within each group, the highest results are in bold, and the second highest results are _underlined_. “FR” denotes the use of feature replay.

Method	Venue	FR	5S-ImageNet-R $A_{\mathrm{Last}} \uparrow$	5S-ImageNet-R $A_{\mathrm{Avg}} \uparrow$	20S-ImageNet-R $A_{\mathrm{Last}} \uparrow$	20S-ImageNet-R $A_{\mathrm{Avg}} \uparrow$	20S-ImageNet-A $A_{\mathrm{Last}} \uparrow$	20S-ImageNet-A $A_{\mathrm{Avg}} \uparrow$	10S-DomainNet $A_{\mathrm{Last}} \uparrow$	10S-DomainNet $A_{\mathrm{Avg}} \uparrow$
L2P [43]	CVPR’22	✗	70.83±0.58	78.34±0.47	69.64±0.42	75.28±0.57	40.48±1.78	49.62±1.46	81.17±0.83	87.43±0.95
CODAPrompt [31]	CVPR’23	✗	74.91±0.33	79.25±0.53	69.96±0.50	75.34±0.85	46.62±1.92	54.86±0.50	80.04±0.79	86.27±0.82
InfLoRA [18]	CVPR’24	✗	77.53±0.37	82.01±0.12	71.01±0.45	77.28±0.45	46.81±2.30	57.13±0.95	81.45±0.68	87.55±0.57
SEMA [40]	CVPR’25	✗	79.78	84.75	74.53	81.75	–	–	–	–
SD-LoRA [45]	ICLR’25	✗	79.15±0.20	83.01±0.42	75.26±0.37	80.22±0.72	49.12±1.78	59.63±1.57	81.01±0.42	86.85±0.40
BiLoRA [57]	CVPR’25	✗	78.32±0.37	83.31±0.52	72.41±0.65	79.28±0.76	–	–	76.56±0.41	83.20±0.35
LoRA-P&M [28]	NIPS’25	✗	81.47±0.56	85.96±0.52	76.37±0.09	82.77±0.71	52.27±1.84	62.15±1.33	84.35±0.48	89.43±0.27
EWC-LoRA [53]	ICLR’26	✗	76.36±0.21	81.43±0.13	70.18±1.06	77.06±0.54	–	–	–	–
M²LE (Ours)	–	✗	82.67±0.18	87.31±0.35	78.62±0.43	84.62±0.26	53.87±1.45	63.69±1.87	84.78±0.13	89.74±0.44
SLCA [51]	ICCV’23	✓	81.01±0.11	84.18±0.28	74.63±1.55	79.99±1.29	36.69±21.31	56.35±7.09	84.35±0.72	88.56±0.21
SSIAT [35]	CVPR’24	✓	80.52±0.07	84.25±0.31	75.67±0.14	82.30±0.36	59.16±1.03	68.45±1.92	85.11±0.56	89.80±0.34
MOS [34]	AAAI’25	✓	78.76±0.11	82.37±0.26	75.16±0.59	80.53±0.70	–	–	–	–
MACIL [44]	ICML’25	✓	83.37±0.26	87.00±0.35	79.43±0.34	84.34±0.32	59.36±1.27	68.03±2.00	86.62±0.35	90.88±0.20
M²LE (Ours) + CA	–	✓	82.74±0.29	87.25±0.28	81.12±0.14	86.12±0.33	64.25±0.56	72.54±0.92	87.61±0.13	91.51±0.18

Ablation Studies

We conduct ablation studies on the main components of our training pipeline, including the sequential fine-tuning baseline, the proposed HIM, and TSP. As shown in Table 3, HIM (Idx 2) brings the main improvements over the baseline (Idx 1), increasing $\mathcal{A}_{Last}$ by 6.74%–11.90% across all four settings. This shows that reweighting the current task update under the joint loss impact objective in Eq. (4) is more effective than directly adding it to the previous inference model. TSP (Idx 3) also consistently improves the baseline, with $\mathcal{A}_{Last}$ gains of 1.15%–4.01%. The improvement is particularly clear on the more challenging ImageNet-A benchmark, suggesting that TSP helps the current task update remain stable under task simulating perturbations, making it less sensitive to subsequent merge-induced changes. Notably, combining HIM and TSP (Idx 4) further improves $\mathcal{A}_{Last}$ by 0.58%–1.10% over HIM alone and achieves the best overall performance across all settings, indicating that the two components play complementary roles.

Idx	HIM	TSP	10S-ImageNet-R		10S-ImageNet-A		20S-ImageNet-R		20S-ImageNet-A
			$\mathcal{A}_{Last} \uparrow$	$\mathcal{A}_{Avg} \uparrow$	$\mathcal{A}_{Last} \uparrow$	$\mathcal{A}_{Avg} \uparrow$	$\mathcal{A}_{Last} \uparrow$	$\mathcal{A}_{Avg} \uparrow$	$\mathcal{A}_{Last} \uparrow$	$\mathcal{A}_{Avg} \uparrow$
1	–	–	73.05±0.35	80.40±0.60	52.56±2.08	63.36±1.32	66.14±0.20	73.72±0.25	46.06±2.14	59.43±1.77
2	✓	–	80.36±0.40	85.78±0.44	61.35±1.06	70.23±2.04	78.04±0.35	84.31±0.54	52.80±1.46	62.64±2.14
3	–	✓	74.20±0.91	80.67±0.47	56.57±2.31	66.26±1.46	69.17±0.10	77.16±0.39	49.50±0.72	60.45±1.52
4	✓	✓	81.34±0.16	86.53±0.51	62.45±1.34	71.02±1.82	78.62±0.43	84.62±0.26	53.87±1.45	63.69±1.87

In-Depth Analysis

Comparison of different merge strategies. We further compare different merge strategies on 10S-ImageNet-R and 10S-ImageNet-A to analyze how the choice of merge rule affects continual learning performance. Seq-FT is used as a without merge baseline that updates the inference model by directly adding the current task LoRA update without any merge rule. As shown in Table 4, EMA-style and LoRA-P&M both outperform Seq-FT, confirming the benefit of merging in rehearsal-free CL.

Strategy	10S-ImageNet-R		10S-ImageNet-A
	$\mathcal{A}_{Last} \uparrow$	$\mathcal{A}_{Avg} \uparrow$	$\mathcal{A}_{Last} \uparrow$	$\mathcal{A}_{Avg} \uparrow$
Seq-FT	73.05±0.35	80.40±0.60	52.56±2.08	63.36±1.32
EMA-style ($\theta= \frac{t-1}{t}\theta_{t-1} + \frac{1}{t}\theta_t$) [20]	79.14±0.10	85.08±0.41	59.64±1.22	69.52±1.82
LoRA-P&M [28]	78.89±0.44	84.99±0.45	59.93±1.57	69.87±2.05
HIM-last ($\mathcal{J}_{old} = \mathcal{L}_{t-1}(\mathbf{W}_{t-1} + \Delta\hat{\mathbf{W}}_t) - \mathcal{L}_{t-1}(\mathbf{W}_{t-1})$)	76.23±0.76	83.81±0.61	58.17±2.38	68.84±1.93
HIM (Ours)	80.36±0.40	85.78±0.44	61.35±1.06	70.23±2.04

However, these methods still rely on a scalar factor shared by all parameters, matching the limitation discussed in § 3. In contrast, HIM directly solves the target merged update under Eq. (4) and obtains the element-wise solution in Eq. (9). It achieves the best results, improving $A_{Last}$ over EMA-style averaging by 1.22%/1.71% and over LoRA-P&M by 1.47%/1.42% on ImageNet-R/ImageNet-A, showing the assigning merge strengths at the element level is more effective than using a shared scalar factor. We further compare HIM with a HIM variant (HIM-last) that only uses task $t = 1$ in Eq. (2); without explicit constraints from earlier tasks, this variant suffers from more severe forgetting, indicating that accumulated old task information in $\mathbf{F}_{1:t-1}$ is important for merging. More HIM variants are reported in Appendix C.3.

Analysis of perturbation training under cross-task merging. We further compare TSP with SAM [5] and without TSP to examine the effect of different perturbation training strategies during subsequent merging. As shown in Table 5, SAM performs worse than HIM alone. Compared with HIM alone, TSP improves $A_{Last}$ by 0.98%/1.10% on ImageNet-R/ImageNet-A. To better understand this result, we visualize its behavior on both the current task and the old tasks after merging. Figure 2 shows that SAM can still obtain a low loss solution on task 2, but the merged solution falls into a less favorable region for old tasks 0 and 1. This indicates that optimizing current task sharpness cannot well capture the cross-task interference caused by sequential LoRA updates. In contrast, TSP constructs the perturbation from one label-disjoint subset and evaluates it on the other subset. As a

Strategy	10S-ImageNet-R $A_{Last} \uparrow$	10S-ImageNet-R $A_{Avg} \uparrow$	10S-ImageNet-A $A_{Last} \uparrow$	10S-ImageNet-A $A_{Avg} \uparrow$
w/o TSP	$80.36 \pm0.40$	$85.78 \pm0.44$	$61.35 \pm1.06$	$70.23 \pm2.04$
SAM [5]	$79.55 \pm0.17$	$84.92 \pm0.44$	$60.46 \pm1.51$	$69.00 \pm1.98$
TSP (Ours)	$81.34 \pm0.16$	$86.53 \pm0.51$	$62.45 \pm1.34$	$71.02 \pm1.82$

result, our method achieves the best performance, improving $\mathcal{A}_{Last}$ over SAM by 1.79%/1.99% on ImageNet-R/ImageNet-A.

Effect of TSP on forgetting. TSP aims to reduce the sensitivity of the current task update to the parameter change introduced by the subsequent merge. As shown in Table 6, adding TSP consistently improves total accuracy and reduces forgetting across incremental stages on both ImageNet-R and ImageNet-A. The final forgetting decreases from 7.76% to 5.77% on ImageNet-R and from 14.01% to 10.04% on ImageNet-A. This supports our motivation that stabilizing the current task update can alleviate forgetting during continual merging. More hyper-parameter analyses and results with different backbones are provided in Appendix § 10.5 and Appendix § 10.6.

Dataset

Metrics

Methods

Task 0

Task 1

Task 2

Task 3

Task 4

Task 5

Task 6

Task 7

Task 8

Task 9

ImageNet-R

Accuracy $\uparrow$

w/o TSP

$94.96 \pm1.23$

$91.31 \pm1.22$

$89.48 \pm0.20$

$86.91 \pm0.67$

$84.73 \pm0.64$

$83.68 \pm0.20$

$83.00 \pm0.61$

$82.13 \pm0.53$

$81.24 \pm0.56$

$80.36 \pm0.40$

w/ TSP

$95.14 \pm1.07$

$92.14 \pm0.86$

$89.93 \pm0.85$

$87.58 \pm0.56$

$85.86 \pm0.91$

$84.49 \pm0.61$

$83.67 \pm0.42$

$82.98 \pm0.40$

$82.14 \pm0.54$

$81.34 \pm0.16$

$\Delta$

$+0.18$

$+0.83$

$+0.45$

$+0.67$

$+1.13$

$+0.81$

$+0.67$

$+0.85$

$+0.90$

$+0.98$

Forgetting $\downarrow$

w/o TSP

–

$4.98 \pm0.90$

$4.87 \pm0.60$

$6.16 \pm0.58$

$7.00 \pm0.59$

$7.34 \pm0.20$

$7.26 \pm0.67$

$7.34 \pm0.49$

$7.43 \pm0.44$

$7.76 \pm0.38$

w/ TSP

–

$3.17 \pm0.60$

$3.79 \pm0.86$

$4.40 \pm0.58$

$4.89 \pm0.78$

$5.41 \pm0.24$

$5.46 \pm0.62$

$5.49 \pm0.88$

$5.52 \pm0.82$

$5.77 \pm0.64$

$\Delta$

–

$-1.81$

$-1.08$

$-1.76$

$-2.11$

$-1.93$

$-1.80$

$-1.85$

$-1.91$

$-1.99$

ImageNet-A

Accuracy $\uparrow$

w/o TSP

$86.56 \pm1.34$

$80.06 \pm1.70$

$74.96 \pm1.79$

$71.95 \pm3.44$

$69.65 \pm3.18$

$67.65 \pm2.97$

$64.79 \pm2.13$

$63.60 \pm1.98$

$61.75 \pm1.88$

$61.35 \pm1.06$

w/ TSP

$86.70 \pm0.30$

$81.60 \pm0.54$

$75.84 \pm2.67$

$72.28 \pm2.89$

$70.06 \pm2.83$

$68.60 \pm2.70$

$65.83 \pm2.63$

$64.07 \pm1.58$

$62.79 \pm1.64$

$62.45 \pm1.34$

$\Delta$

$+0.14$

$+1.54$

$+0.88$

$+0.33$

$+0.41$

$+0.95$

$+1.04$

$+0.47$

$+1.04$

$+1.10$

Forgetting $\downarrow$

w/o TSP

–

$13.54 \pm6.63$

$11.43 \pm1.91$

$12.49 \pm1.06$

$12.71 \pm0.85$

$12.27 \pm0.15$

$13.88 \pm0.72$

$13.47 \pm0.44$

$14.19 \pm1.52$

$14.01 \pm0.90$

w/ TSP

–

$10.56 \pm0.60$

$9.19 \pm2.12$

$9.98 \pm1.49$

$9.95 \pm1.04$

$9.24 \pm0.91$

$10.52 \pm1.26$

$10.77 \pm0.69$

$10.49 \pm0.86$

$10.04 \pm1.29$

$\Delta$

–

$-2.98$

$-2.24$

$-2.51$

$-2.76$

$-3.03$

$-3.36$

$-2.70$

$-3.70$

$-3.97$

Computational and Memory Overhead. Table 7 compares the overhead of HIM and TSP on 10S-ImageNet-A. HIM requires 54.0 MB of additional Fisher storage, independent of the number of tasks, and 3.8 s per task for Fisher estimation. TSP increases the training time per batch from 0.12 s to 0.34 s, but reduces GPU memory usage from 4560 MiB to 2881 MiB. These costs are limited to training, with no additional inference overhead. Further results on the effect of LoRA on overhead and accuracy are provided in Appendix § 10.§ 2.

Methods	GPU Memory (MiB)	Time/Batch (s)	Extra Memory (MB)	$\mathcal{A}_{Last} \uparrow$
Seq-FT	4560	0.12	0.0	$52.56_{\pm2.08}$
TSP	2881	0.34	0.0	$56.57_{\pm2.31}$
HIM	4560	0.12	54.0	$61.35_{\pm1.06}$
HIM w/ SAM	4568	0.26	54.0	$60.46_{\pm1.51}$
HIM w/ TSP	2881	0.34	54.0	$62.45_{\pm1.34}$

Further Analysis

Further analysis of HIM and TSP. We further analyze the behaviors of HIM and TSP using the task 1 accuracy against task 0 accuracy and the corresponding loss landscape visualizations. For the loss landscapes, we construct a 2D plane to compare the scalar interpolation path with our merged solution. Let $W_1$ denote the model obtained by directly adding the task 1 update to $W_0$, and define the scalar interpolation direction as $\mathbf{u}=W_1-W_0$. Let $\mathbf{d}=W_{\mathrm{ours}}-W_0$ denote the displacement from $W_0$ to our selected merged solution. We then remove from $\mathbf{d}$ its component along $\mathbf{u}$ and obtain the residual direction $\mathbf{r}=\mathbf{d}-\frac{\langle\mathbf{d},\mathbf{u}\rangle}{\langle\mathbf{u},\mathbf{u}\rangle}\mathbf{u}$ and $\mathbf{v}=\mathbf{r}/\lVert\mathbf{r}\rVert_2$, where $\mathbf{v}$ is orthogonal to the scalar interpolation direction. The loss is visualized on the plane $W(x,y)=W_0+x\mathbf{u}+y\mathbf{v}$.

Conclusion

In this work, we show that different parameters in LoRA update may prefer different merge strengths. Motivated by this, we propose $M^{2}$LE, which directly solves for the target merged update, yielding an analytic element-wise modulation of the current LoRA update. $M^{2}$LE first derives Hessian Information-guided Adaptive Merging (HIM), an analytic element-wise modulation rule under a joint loss objective over old and new tasks. It further introduces Task Simulation Perturbation (TSP) to simulate cross-task interference during training and improve the stability during subsequent merging. Overall, $M^{2}$LE offers a new view for continual learning with LoRA, showing that adapting merge strengths for different LoRA elements provides an effective way to balance stability and plasticity.

Acknowledgments and Disclosure of Funding

This work was supported in part by the National Key R&D Program of China under Grant No.2023YFA1008600, in part by the National Natural Science Foundation of China under Grants 62576262, in part by the Natural Science Foundation Basic Research Program of Shaanxi Province under grant 2026JC-YXQN-202, in part by Scientific and Technological Innovation Teams in Shaanxi Province under Grant 2025RS-CXTD-011, in part by the Shaanxi Province Core Technology Research and Development Project under Grant 2024QY2G-JGHX-11, in part by the Natural Science Basic Research Program of Shaanxi under Grant 2026SYS-SYSZD-105, in part by the Key Research and Development Program of Shaanxi under Grant 2026JC-ZDGG-42, in part by the Establishment Fund of the State Key Laboratory of Wakefulness/Sleep and Cognition (Chongqing Institute for Brain and Intelligence), in part by the Fundamental Research Funds for the Central Universities under Grant QTZX25083 and QTZX26138.

Training Pipeline

Figure 5 illustrates the overall framework of $M^2$LE, and the training pipeline of $M^2$LE is outlined in Algorithm 1. At task $t$, we start from the inference parameter $\mathbf{W}_{t-1}$ and adapt the frozen backbone on the current data $\mathcal{D}_t$ with LoRA, where the current update is $\Delta\mathbf{W}_t = \mathbf{B}_t\mathbf{A}_t$. During current task training, TSP simulates cross-task interference with label-disjoint subsets, encouraging the learned update to remain stable for subsequent merging. After convergence, HIM estimates the current task Fisher $\mathbf{F}_t$ and obtains the target merged update $\Delta\mathbf{W}_t^*$ by Eq. (9). The merged update is then integrated into the inference model, and the Fisher statistics are accumulated for later tasks.

Algorithm 1 Training pipeline of the proposed Merging Strength Modulation via Joint Loss Estimation ($M^2$LE).

Input: Task sequence $\{\mathcal{D}_1,\ldots,\mathcal{D}_T\}$, where $\mathcal{D}_t = \{(x_i^t,y_i^t)\}_{i=1}^{n_t}$; ViT model $f(\cdot)$ with weight $\mathbf{W}_0$; LoRA $\Delta\mathbf{W}_t = \mathbf{B}_t\mathbf{A}_t$ in each layer; accumulated Fisher statistics $\mathbf{F}_{1:t-1}$ in each layer.

Output: The adapted inference parameters $\mathbf{W}_t$ for each layer in the ViT model.

1. Initialization: $\mathbf{F}_{1:t-1} \leftarrow\mathbf{0}_{d_o \times d_i}$.

2. for task $t = 1$ to $T$ do

3. Freeze the inference weight $\mathbf{W}_{t-1}$ and attach current-task LoRA modules.

4. for each LoRA layer do

5. Initialize current-task LoRA parameters $\mathbf{A}_t$ and $\mathbf{B}_t$.

6. // During Training: Task Simulation Perturbation for Merging

7. repeat

8. Sample a mini-batch $S$ from $\mathcal{D}_t$ and split it into two label-disjoint subsets $S_1$ and $S_2$;

9. Update trainable parameters with the dual TSP objective in Eq. (14) and Eq. (15).

10. until Convergence on $\mathcal{D}_t$

11. // After Training: Hessian Information-guided Adaptive Merging

12. for each LoRA layer do

13. Given the trained update $\Delta\mathbf{W}_t$, estimate $\mathbf{F}_t$ on $\mathcal{D}_t$ and compute the HIM merged update $\Delta\mathbf{W}_t^*$ by Eq. (9).

14. Update the inference model weight $\mathbf{W}_t \leftarrow\mathbf{W}_{t-1} + \Delta\mathbf{W}_t^*$;

15. Update accumulated Fisher $\mathbf{F}_{1:t} \leftarrow\mathbf{F}_{1:t-1} + \mathbf{F}_t$;

Theoretical Analysis

Starting from Eq. (7), we solve the following quadratic problem over the candidate merged update $\Delta\widehat{w}_t$:

We first expand the current task term and separate the parts that depend on $\Delta\widehat{w}_t$:

The last term in Eq. (B.0.2) is constant with respect to $\Delta\widehat{w}_t$ and can be ignored when solving the minimization problem. Thus, the derivative of Eq. (B.0.1) with respect to $\Delta\widehat{w}_t$ is:

By setting Eq. (B.0.3) to zero and denoting the solution by $\Delta w_t^*$, we obtain:

Rearranging the terms gives the linear system, which corresponds to Eq. (8):

More Detailed Experimental Results

Forgetting metric

We measure forgetting after task $t \ge1$ as $\operatorname{Forgetting}_t=\frac{1}{t}\sum_{i=0}^{t-1}\left(\max_{j\in\{i,\ldots,t-1\}}a_{j,i}-a_{t,i}\right)$, where $a_{j,i}$ denotes the accuracy on task $i$ after learning task $j$.

Effect of Different LoRA Ranks on M²LE Performance

We study the effect of rank in M²LE on 20S-ImageNet-R in Table 8. We vary the LoRA rank $r \in\{4, 8, 16, 32, 64\}$ and denote each variant as M²LE-$r=r$. Across all ranks, M²LE achieves competitive performance with only a small number of trainable parameters, ranging from 0.15M to 2.36M. It also keeps the inference overhead unchanged, with 0.00M extra parameters and 17.60 GFLOPs, since the learned components are merged into the backbone after training. For each LoRA layer, the maintained Fisher statistic $F_{1:t}$ has size $d_o \times d_i$, so the extra storage remains 54.0MB for all ranks once the layer dimensions are fixed.

Methods	Venue	Extra Memory	Trainable Params	Training GFLOPs	Extra Params	Inference GFLOPs	$A_{\mathrm{Last}} \uparrow$	$A_{\mathrm{Avg}} \uparrow$
Methods based on the ViT-21K model (86M).
CODA [31]	CVPR-23	15.4MB	3.84M (4.46%)	35.17	3.84M (4.46%)	35.17	$69.96 \pm0.50$	$75.34 \pm0.85$
SLCA [51]	ICCV-23	450.6MB	86M (100%)	17.60	0.00M (0.00%)	17.60	$74.63 \pm1.55$	$79.92 \pm1.29$
SSIAT [35]	CVPR-24	450.6MB	1.19M (1.38%)	17.81	1.19M (1.38%)	17.81	$75.67 \pm0.24$	$82.30 \pm0.36$
InfLoRA [18]	CVPR-24	11.3MB	0.37M (0.43%)	17.67	0.00M (0.00%)	17.67	$71.01 \pm0.45$	$77.28 \pm0.45$
VPT-NSP [22]	NIPS-24	28.4MB	0.09M (0.10%)	17.72	0.09M (0.10%)	17.72	$75.69 \pm0.61$	$81.87 \pm0.59$
SD-LoRA [45]	ICLR-25	14.1MB	0.37M (0.43%)	20.51	0.00M (0.00%)	17.72	$75.26 \pm0.37$	$80.22 \pm0.72$
VPT-CPG [23]	AAAI-25	28.4MB	1.66M (1.93%)	17.72	1.66M (1.93%)	17.72	$75.33 \pm0.37$	$80.22 \pm0.72$
M2LE–$r=4$ (Ours)	–	54.0MB	0.15M (0.17%)	17.59	0.00M (0.00%)	17.60	$76.92 \pm0.20$	$83.28 \pm0.25$
M2LE–$r=8$ (Ours)	–	54.0MB	0.29M (0.34%)	17.62	0.00M (0.00%)	17.60	$77.32 \pm0.55$	$83.68 \pm0.51$
M2LE–$r=16$ (Ours)	–	54.0MB	0.59M (0.69%)	17.68	0.00M (0.00%)	17.60	$78.01 \pm0.13$	$84.07 \pm0.43$
M2LE–$r=32$ (Ours)	–	54.0MB	1.18M (1.37%)	17.80	0.00M (0.00%)	17.60	$78.62 \pm0.43$	$84.62 \pm0.26$
M2LE–$r=64$ (Ours)	–	54.0MB	2.36M (2.74%)	18.03	0.00M (0.00%)	17.60	$78.20 \pm0.30$	$84.66 \pm0.70$

As $r$ increases from 4 to 32, $A_{Last}$ improves from 76.92 to 78.62, while $A_{Avg}$ increases from 83.28 to 84.62. Further increasing the rank to 64 brings only marginal improvement in $A_{Avg}$ (84.66) but slightly decreases $A_{Last}$ to 78.20, together with higher training cost. Thus, we adopt $r = 32$ as a practical trade-off between accuracy and efficiency in the remaining experiments.

Analysis of HIM Variants

We further analyze several HIM variants that retain different numbers of past tasks in the old task term. For a more explicit comparison, we define a truncated HIM variant by retaining only the most recent $k$ past tasks in the old task loss impact:

Here, $k \in\{1,3,5,7\}$ controls the number of past tasks used in the old task term. Correspondingly, this variant replaces the accumulated Fisher statistic in Eq. (9) with $\mathbf{F}_{t-k:t-1} = \sum_{i=\max(1,t-k)}^{t-1}\mathbf{F}_i$, while full HIM uses $\mathbf{F}_{1:t-1} = \sum_{i=1}^{t-1}\mathbf{F}_i$. When $k = 1$, the variant uses only task $t-1$, corresponding to the HIM-last setting discussed in the main text.

As shown in Table 9, HIM-last gives much lower performance, i.e., 76.23/83.81 on 10S-ImageNet-R and 58.17/68.84 on 10S-ImageNet-A. When more past tasks are retained, the performance consistently improves, e.g., HIM-$k = 5$ reaches 80.04/85.73 and 60.92/70.14 on the two datasets, respectively. Full HIM achieves the best results of 80.36/85.78 and 61.35/70.23, indicating that using a broader set of past tasks provides more complete constraints for preserving previously learned knowledge during merging. These results also validate the use of $\mathbf{F}_{1:t-1}$ in HIM, which aggregates information from all past tasks rather than relying only on recent ones.

Variant	10S-ImageNet-R $A_{\mathrm{Last}} \uparrow$	10S-ImageNet-R $A_{\mathrm{Avg}} \uparrow$	10S-ImageNet-A $A_{\mathrm{Last}} \uparrow$	10S-ImageNet-A $A_{\mathrm{Avg}} \uparrow$
HIM-last ($k = 1$)	$76.23 \pm0.76$	$83.81 \pm0.61$	$58.17 \pm2.38$	$68.84 \pm1.93$
HIM-$k = 3$	$78.88 \pm0.62$	$85.37 \pm0.50$	$60.81 \pm1.10$	$69.97 \pm2.28$
HIM-$k = 5$	$80.04 \pm0.35$	$85.73 \pm0.47$	$60.92 \pm0.97$	$70.14 \pm2.48$
HIM-$k = 7$	$80.21 \pm0.45$	$85.77 \pm0.45$	$61.16 \pm1.15$	$70.17 \pm2.42$
HIM-full (Ours)	$80.36 \pm0.40$	$85.78 \pm0.44$	$61.35 \pm1.06$	$70.23 \pm2.04$

Analysis of Historical Fisher Reuse

We further analyze the accumulated Fisher statistics in HIM on 10S-ImageNet-R. HIM estimates each task’s Fisher at its pre-merge solution while the data are available, and retains only the accumulated statistic $\mathbf{F}_{1:t-1}$. For comparison, Recomputed Fisher revisits previous task data before each merge and re-estimates their accumulated Fisher at its current merged model. The recomputed statistic is used for merging and subsequent learning, so the two methods follow their respective model trajectories. We measure their Fisher similarity by $S_t = \frac{1}{|\mathcal{L}|}\sum_{\ell\in\mathcal{L}}\cos(\mathbf{F}_{1:t-1}^{\ell},\widetilde{\mathbf{F}}_{1:t-1}^{\ell})$, where $\mathcal{L}$ denotes the set of layers with Fisher statistics, and $F^\ell_{1:t-1}$ and $\widetilde{F}^\ell_{1:t-1}$ are the accumulated statistics from HIM and Recomputed Fisher, respectively.

As shown in Table 10, $S_t$ remains at least 0.986 across the evaluated stages, indicating similar directions of the accumulated Fisher statistics. Recomputed Fisher achieves similar accuracy to HIM, with a maximum difference of 0.28 performance and the same final $\mathcal{A}_{Last}$ of 80.67%. These results support using the accumulated historical Fisher in HIM, avoiding repeated access to previous task data with little impact on performance over the evaluated task sequence.

Metric	Task 0	Task 1	Task 2	Task 3	Task 4	Task 5	Task 6	Task 7	Task 8	Task 9
$S_t \uparrow$	–	1.000	0.988	0.986	0.986	0.987	0.988	0.987	0.987	0.986
HIM $\mathcal{A}_{Last} \uparrow$	96.66	92.81	89.41	87.39	84.69	83.84	83.18	82.62	81.68	80.67
Recomputed Fisher $\mathcal{A}_{Last} \uparrow$	96.66	92.81	89.36	87.11	84.76	83.70	83.15	82.70	81.68	80.67

Hyper-Parameter Analysis

We conduct experiments to investigate how the key hyper-parameters $\lambda$, $\rho$, and $\beta$ affect performance on 10S-ImageNet-R and 10S-ImageNet-A. Here, $\lambda$ is the trade-off coefficient in the merge objective in Eq. (9), $\rho$ controls the perturbation magnitude in Eq. (11), and $\beta$ balances the perturbed and clean losses in Eq. (13). We use the default settings $\lambda= 5$, $\rho= 0.5$, and $\beta= 0.8$ on ImageNet-R, and $\lambda= 5$, $\rho= 0.3$, and $\beta= 0.6$ on ImageNet-A. For each analysis, we vary only one hyper-parameter while fixing the other two at their default values. The results are shown in Figure 6 and Figure 7. All results are averaged over three random seeds, and we report the mean with error bars.

Impact of $\lambda$. The hyper-parameter $\lambda$ controls the relative weight of the current task term in the merge objective, and thus adjusts the balance between old task retention and current task adaptation. A too-small $\lambda$ (e.g., $\lambda= 1$) underweights the current task term and limits new task adaptation, whereas an overly large $\lambda$ (e.g., $\lambda\ge10$) makes the merged update overly biased toward the current task and weakens old task retention. Overall, the results remain stable for $1 \le\lambda\le10$, with the best trade-off achieved around $\lambda= 3$–$5$. Based on this observation, we set $\lambda= 5$.

Impact of $\rho$. The hyper-parameter $\rho$ controls the perturbation magnitude in the TSP. A too-small $\rho$ (e.g., $\rho= 0.1$) leads to a weak cross-subset probing effect, whereas a too-large $\rho$ (e.g., $\rho\ge0.8$) over-amplifies the perturbation. Overall, the results remain stable for $0.1 \le\rho\le1.0$, with the best performance achieved at moderate values. Accordingly, we set $\rho= 0.5$ on ImageNet-R and $\rho= 0.3$ on ImageNet-A.

Impact of $\beta$. The hyper-parameter $\beta$ controls the relative weight of the perturbed loss and the clean loss in the TSP. A too-small $\beta$ (e.g., $\beta= 0.2$) underweights the perturbation signal, whereas a too-large $\beta$ (e.g., $\beta= 1.0$) makes the perturbation effect overly strong and is less favorable overall. Overall, the results remain stable for $0.2 \le\beta\le1.0$, and the best results are obtained at moderate values. Based on the results, we set $\beta= 0.8$ on ImageNet-R and $\beta= 0.6$ on ImageNet-A.

Experimental Results Using the CLIP Model as the Backbone

To examine whether M²LE can be transferred beyond a single visual backbone, we further conduct experiments with CLIP as the pre-trained model. The results on six benchmarks are summarized in Table 11 and Table 12.

Method	ImageNet-R	ImageNet-R	ImageNet-R	ImageNet-R	CIFAR100	CIFAR100	CIFAR100	CIFAR100	CUB	CUB	CUB	CUB
	B0 Inc10	B0 Inc10	B50 Inc10	B50 Inc10	B0 Inc10	B0 Inc10	B50 Inc10	B50 Inc10	B0 Inc20	B0 Inc20	B100 Inc20	B100 Inc20
	$A_{Last}$	$A_{Avg}$	$A_{Last}$	$A_{Avg}$	$A_{Last}$	$A_{Avg}$	$A_{Last}$	$A_{Avg}$	$A_{Last}$	$A_{Avg}$	$A_{Last}$	$A_{Avg}$
CoOp [56]	37.52	60.73	39.77	54.20	24.24	47.00	24.12	41.23	8.57	27.61	10.14	24.03
ZS-CLIP [29]	77.17	83.37	77.17	79.57	71.38	81.81	71.38	76.49	63.06	74.38	63.06	67.96
L2P [43]	66.52	75.97	66.77	72.82	73.03	82.74	73.61	81.14	57.93	70.87	66.12	75.64
DualPrompt [42]	66.65	76.21	67.58	73.22	72.44	81.63	72.57	80.12	57.46	69.89	64.84	71.40
CODA-Prompt [31]	68.95	77.69	68.05	73.71	73.43	82.43	71.58	78.69	62.98	73.12	62.21	73.95
SimpleCIL [54]	74.48	81.06	74.48	76.84	76.63	84.15	76.63	80.20	77.52	83.81	77.52	79.75
RAPF [12]	79.62	86.28	79.21	84.03	79.04	86.19	78.25	82.35	62.77	79.09	62.93	72.82
PROOF† [55]	77.25	82.69	77.03	80.56	76.29	84.88	75.93	81.85	72.60	79.94	71.80	76.43
M²LE (Ours)	82.05	87.16	82.65	85.24	80.52	87.88	83.12	87.80	80.03	86.92	80.41	84.25

Method	Aircraft	Aircraft	Aircraft	Aircraft	Cars	Cars	Cars	Cars	UCF	UCF	UCF	UCF
	B0 Inc10	B0 Inc10	B50 Inc10	B50 Inc10	B0 Inc10	B0 Inc10	B50 Inc10	B50 Inc10	B0 Inc20	B0 Inc20	B100 Inc20	B100 Inc20
	$A_{Last}$	$A_{Avg}$	$A_{Last}$	$A_{Avg}$	$A_{Last}$	$A_{Avg}$	$A_{Last}$	$A_{Avg}$	$A_{Last}$	$A_{Avg}$	$A_{Last}$	$A_{Avg}$
CoOp [56]	7.14	14.54	7.77	13.05	21.65	36.46	20.87	37.40	33.46	47.85	24.74	42.02
ZS-CLIP [29]	17.22	26.66	17.22	21.70	76.37	82.60	76.37	78.32	67.64	75.50	67.64	71.44
L2P [43]	28.29	47.19	32.13	44.07	61.82	76.63	65.64	76.37	76.43	86.34	76.62	83.95
DualPrompt [42]	25.83	44.30	33.57	46.07	62.94	76.26	67.55	76.88	75.82	85.21	76.35	84.31
CODA-Prompt [31]	27.69	45.98	32.28	45.14	66.47	80.21	64.19	75.06	80.14	87.76	75.03	83.04
SimpleCIL [54]	48.09	59.24	48.09	53.05	86.85	92.04	86.85	88.96	85.68	90.44	85.68	88.12
RAPF [12]	23.61	50.38	25.44	40.47	62.85	82.89	63.19	75.87	86.33	92.28	81.55	90.31
PROOF† [55]	39.03	47.33	45.36	54.35	84.06	86.87	82.66	88.68	81.47	87.04	81.85	86.15
M²LE (Ours)	60.43	70.63	63.85	69.50	90.57	94.39	90.91	92.40	90.53	94.38	91.28	94.81

Datasets and Evaluation Metrics. Following existing CLIP-based continual learning studies [12, 55], we evaluate M²LE on ImageNet-R [9], CIFAR100 [15], CUB [37], Aircraft [24], Cars [14], and UCF [32]. ImageNet-R contains 200 ImageNet categories with different artistic renditions, such as sketches, cartoons, and paintings. CIFAR100 includes 100 object categories with 60,000 natural images. CUB, Aircraft, and Cars are fine-grained recognition datasets, covering bird species, aircraft model variants, and car categories, respectively. UCF101 is an action recognition benchmark with 101 human action classes. For Aircraft, Cars, and UCF, we follow prior protocols and sample 100 classes for continual learning evaluation. The split notation “B-x Inc-y” denotes that the first stage contains $x$ classes and each later stage introduces $y$ new classes. When $x = 0$, all tasks contain the same number of classes.

Implementation Details of $M^{2}$LE based on CLIP. We build $M^{2}$LE upon SimpleCIL [54], a strong training-free baseline that freezes the pre-trained model and constructs a prototypical classifier from downstream features. Specifically, we keep one frozen CLIP branch $f_{\mathrm{pt}}(\cdot)$ together with its prototype classifier $\mathbf{W}_{\mathrm{pt}}$, and sequentially adapt another CLIP branch $f_{\mathrm{ft}}(\cdot)$ using $M^{2}$LE. The adapted branch is trained with the CLIP-style image–text matching objective [55], where the text classifier is denoted as $\mathbf{W}_{\mathrm{text}}$. At inference time, we combine the predictions from the frozen branch and the adapted branch:

Comparison Methods. We compare $M^{2}$LE with a wide range of CLIP-based continual learning baselines, including prompt-based methods CoOp [56], L2P [43], DualPrompt [42], and CODAPrompt [31]; CLIP-oriented methods RAPF [12] and PROOF [55]; the training-free SimpleCIL [54]; and the zero-shot CLIP baseline [29]. For fair comparison, all methods use the CLIP backbone pre-trained on LAION-400M, and no experience replay is used.

Results and Analysis. As shown in Table 11 and Table 12, $M^{2}$LE achieves the best performance across all CLIP-based settings, showing that it can generalize well to vision-language pre-trained models. Some adaptation baselines even perform worse than ZS-CLIP or SimpleCIL, indicating that directly updating CLIP may disturb its pre-trained knowledge. In contrast, $M^{2}$LE enables effective layer-wise continual adaptation through adaptive LoRA merging: HIM assigns element-wise merge strengths to better balance old and new knowledge, while TSP improves the stability of the learned update for later merging. Moreover, $M^{2}$LE is replay-free and does not require class-wise feature statistics, making it efficient and scalable as the label space grows.

Limitations

Although $M^{2}$LE shows strong performance across diverse continual learning benchmarks, its effectiveness still depends on the quality and adaptability of the underlying pre-trained backbone. When the backbone provides less transferable representations or is not well suited to low-rank updates, the benefit of LoRA-based continual adaptation may be reduced.

Potential Impact

$M^{2}$LE achieves efficient continual adaptation by updating only low-rank trainable parameters, which makes it suitable for long task sequences. However, this efficiency-oriented design may not always be optimal for accuracy-critical applications, such as medical imaging or other safety-sensitive scenarios. In such cases, practitioners should carefully evaluate whether the resulting performance meets the precision requirements of the target domain.

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