Elastic Weight Consolidation Done Right for Continual Learning

Continual Learning (CL) with large-scale pre-trained models (PTMs) has recently gained wide attention, shifting the focus from training from scratch to continually adapting PTMs. This has given rise to a promising paradigm: parameter-efficient continual learning (PECL), where task interference is typically mitigated by assigning a task-specific module during training, such as low-rank adapters. However, weight regularization techniques, such as Elastic Weight Consolidation (EWC)-a key strategy in CL-remain underexplored in this new paradigm. In this paper, we revisit weight regularization in low-rank CL as a new perspective for mitigating task interference in PECL. Unlike existing low-rank CL methods, we mitigate task interference by regularizing a shared low-rank update through EWC, thereby keeping the storage requirement and inference costs constant regardless of the number of tasks. Our proposed method EWC-LoRA leverages a low-rank representation to estimate parameter importance over the full-dimensional space. This design offers a practical, computational- and memory-efficient solution for CL with PTMs, and provides insights that may inform the broader application of regularization techniques within PECL. Extensive experiments on various benchmarks demonstrate the effectiveness of EWC-LoRA, achieving a stability-plasticity trade-off superior to existing low-rank CL approaches. These results indicate that, even under low-rank parameterizations, weight regularization remains an effective mechanism for mitigating task interference. Code is available at: https://github.com/yaoyz96/low-rank-cl.

Sequential task updates are fundamental to continual learning, but their recency bias can impose a lasting performance cost. We study this cost in an overparameterized linear-regression model with i.i.d. task sampling. We prove that distribution-level forgetting and population loss converge to the same stationary limit. We quantify the additional loss incurred by sequential exact fitting, or the sequential price. In more homogeneous task geometries, it equals the intrinsic loss asymptotically attained by joint training, making the total loss twice as large. We further analyze fixed-strength elastic weight consolidation (EWC) under general task curvatures and characterize its stationary sequential price at every regularization strength. Under strong regularization, the price decays inversely with EWC strength while the mean-square coupling horizon grows proportionally. Experiments on Jester and Rotated MNIST support the predicted sequential price and its reduction by EWC, with quantitative agreement on real-world tasks satisfying the theory's assumptions and qualitative agreement under nonlinear finite-step training.

Weight-space regularization methods such as Elastic Weight Consolidation (EWC) are the standard approach to catastrophic forgetting in continual learning. However, those methods tend to underperform when applied to large language models. We argue that such underperformance can be partly explained by the ``polysemantic'' nature of large language models: per-weight importance estimates utilized by EWC-style regularization are too coarse and cannot isolate the knowledge that needs protection. In this paper, we propose regularizing instead in the model's activation space, using pretrained Sparse Autoencoders (SAEs) as a monosemantic feature dictionary. From the perspective of constrained optimization, we derive a new loss function that uses the SAE feature dictionary to explicitly balance stability and plasticity, and show that EWC is a special case in the one-sided weight-space penalty setting. Unlike replay-based methods that store or revisit examples from earlier tasks, our method requires no previous-task data after mask construction: current-task data is used to compute a compact SAE feature mask, and only this mask is retained for later training. Further, since the feature space has significantly lower dimensionality than the parameter space, the proposed method is more memory efficient. On the TRACE and MedCL continual learning benchmarks, the method achieves the strongest result among approaches without introducing task-specific architectural components, also surpassing traditional weight-space regularization methods like EWC. Beyond performance comparisons, we provide empirical evidence for the polysemanticity thesis: task-relevant representations are linearly separable in the SAE feature basis but indistinguishable from chance in the weight basis, and weight-space protection is nearly non-selective at the concept level.