Score-Preserving Continual Lasso under Task Shift

Continual learning must balance the learning of new knowledge with the retention of previously learned knowledge to incrementally learn tasks from a data stream without catastrophic forgetting. While leveraging pretrained models has significantly advanced continual learning, existing methods exhibit a scalability bottleneck when trained sequentially on many tasks, suffering from performance degradation due to inter-task interference and loss of plasticity. Inspired by evidence that sparse fine-tuning achieves performance comparable to full fine-tuning, this paper presents a novel sparsity-driven continual learning framework. Our continual learning method, termed CLARE, operates in two stages: it first identifies a sparse, task-critical parameter mask via a sparsity-inducing objective, then performs mask-constrained fine-tuning by only optimizing parameters selected by the mask. This two-stage sparse adapter mechanism enables all tasks to be accumulated within a shared adapter space while reducing destructive interference across tasks. Extensive experiments demonstrate the scalability of CLARE. On the long task-sequence benchmark Omnibenchmark-1k, CLARE outperforms strong baselines in final accuracy by a large margin, e.g, improving EASE by 4.64% and 13.34% after learning 100 tasks, respectively.

Continual learning (CL) is concerned with learning multiple tasks sequentially without forgetting previously learned tasks. Despite substantial empirical advances over recent years, the theoretical development of CL remains in its infancy. At the heart of developing CL theory lies the challenge that the data distribution varies across tasks, and we argue that properly addressing this challenge requires understanding this variation--dependency among tasks. To explicitly model task dependency, we consider nonlinear regression tasks and propose the assumption that these tasks are dependent in such a way that the data of the current task is a nonlinear transformation of previous data. With this model and under natural assumptions, we prove statistical recovery guarantees (more specifically, bounds on estimation errors) for several CL paradigms in practical use, including experience replay with data-independent regularization and data-independent weights that balance the losses of tasks, replay with data-dependent weights, and continual learning with data-dependent regularization (e.g., knowledge distillation). To the best of our knowledge, our bounds are informative in cases where prior work gives vacuous bounds.

We study continual linear regression in a high-dimensional regime, where a learner observes a sequence of tasks and updates its parameters sequentially. In this setting, we derive an explicit closed-form expression for the generalization loss of a model trained with gradient descent for a fixed number of steps per task, which holds for arbitrary teachers. We prove that the optimal fixed step size decays nearly inversely with the number of tasks, specifically as $\ln T / T$, for a single teacher and, more generally, i.i.d. teachers. Under this scaling, we show that the resulting generalization loss is asymptotically equivalent to that achieved by optimally tuned L2 (isotropic) regularization. Furthermore, we extend our analysis to generalized ridge regression with diagonal regularization matrices, obtaining a closed-form characterization of the generalization loss and identifying necessary and sufficient conditions under which the optimal regularizer is isotropic. Finally, we identify a regime where the optimal step size remains finite even as the number of tasks tends to infinity, thus revealing a non-trivial tradeoff between task variability and implicit regularization, in contrast to the i.i.d. teacher scaling. Our theoretical predictions are corroborated by experiments on both linear models and neural networks, providing practical guidance for the design of continual learning systems.