Beyond Parameter Protection: Fisher-Shaped EWC for Replay-Free Continual Learning

Weight regularization methods in continual learning (CL) alleviate catastrophic forgetting by assessing and penalizing changes to important model weights. Elastic Weight Consolidation (EWC) is a foundational and widely used approach within this framework that estimates weight importance based on gradients. However, it has consistently shown suboptimal performance. In this paper, we conduct a systematic analysis of importance estimation in EWC from a gradient-based perspective. For the first time, we find that EWC's reliance on the Fisher Information Matrix (FIM) results in gradient vanishing and inaccurate importance estimation in certain scenarios. Our analysis also reveals that Memory Aware Synapses (MAS), a variant of EWC, imposes unnecessary constraints on parameters irrelevant to prior tasks, termed the redundant protection. Consequently, both EWC and its variants exhibit fundamental misalignments in estimating weight importance, leading to inferior performance. To tackle these issues, we propose the Logits Reversal (LR) operation, a simple yet effective modification that rectifies EWC's importance estimation. Specifically, reversing the logit values during the calculation of FIM can effectively prevent both gradient vanishing and redundant protection. Extensive experiments across various CL tasks and datasets show that the proposed method significantly outperforms existing EWC and its variants. Therefore, we refer to it as EWC Done Right (EWC-DR). Code is available at https://github.com/scarlet0703/EWC-DR.

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.

Continual learning regularizers like EWC fight forgetting by penalizing changes from previous-task parameters with per-parameter importance, typically diagonal Fisher values. Per-parameter looks more flexible than per-layer, but each layer's diagonal Fisher is a weak summary of its actual curvature, missing the top-eigenvalue information that controls forgetting. Adversarial bit-flip attacks and Hessian-spectrum studies show that this missing per-layer sensitivity spans orders of magnitude in neural networks. Under a block-diagonal Hessian assumption, the layer-level analogue of EWC's existing diagonal assumption, we prove three things. Forgetting decomposes as a sum of per-layer terms weighted by each layer's top Hessian eigenvalue. Diagonal-Fisher weights cannot recover this eigenvalue. For instance, two layers with identical Fisher averages can have top eigenvalues differing by a factor as large as the layer width. For the same level of forgetting, uniform regularization loses new-task performance by an amount scaling with the layer condition number. Our theoretical analysis leads to a simple recipe: protect early layers strongly, let deeper layers move. We apply this recipe to EWC and SLCA and show clear improvements in average performance and forgetting metrics.