Formalizing Forgetting: Theory of Forgetting for Online Continual Learning

Understanding catastrophic forgetting, in which learning new tasks degrades performance on previously learned tasks, remains a fundamental challenge in continual learning (CL). Existing theoretical analyses often rely on specific model or task structures and offer limited insight into how iterative optimization affects forgetting. To address this gap, we develop an algorithmic stability framework for expected population forgetting under SGD without prescribing a particular model architecture. Specifically, we decompose forgetting into intra-task and cross-task generalization gaps and empirical forgetting. To bound the generalization gaps, we track how a sample perturbation affects updates within a task and propagates through subsequent tasks. We consider constant, intra-task decaying, and continuously decaying learning-rate schedules. Comparing the two decaying schedules allows us to characterize how restarting learning-rate decay at task boundaries affects the propagation of historical sample perturbations. For empirical forgetting, we derive bounds for all three schedules by retaining cumulative gradient alignment along the optimization trajectory, without requiring convexity. Combining these results yields upper bounds on expected forgetting that capture the effects of training set size, the number of updates per task, the total number of tasks, learning-rate schedules, and cumulative gradient alignment. Finally, we conduct experiments on Split CIFAR-100 and Split MiniImageNet to examine the empirical implications of our theoretical findings.

A central challenge in continual learning is forgetting: the loss of performance on previously learned tasks after learning new ones. Prior theory has analyzed forgetting under random orderings of fixed task collections in overparameterized linear regression. We shift the focus from task order to task distribution, asking how its structure determines forgetting. In the linear setting with a shared solution, i.i.d. task sampling, and sequential exact fitting, we derive an exact operator identity expressing historical forgetting directly in terms of the task distribution. Building on this identity, we establish an exponential decay guarantee for expected historical forgetting under every fixed task distribution in finite dimensions, characterize its asymptotic behavior, and relate decay to the distribution's coverage of observable directions. For an individual learned task, we show that subsequent tasks can collectively support recovery without exact revisits. We derive a lower bound on recovery time and construct a task distribution attaining its inverse-coverage scaling.

Continual learning must absorb new tasks without erasing old ones, and replay---mixing a small buffer of past examples into current training---is among the most effective remedies for catastrophic forgetting. Yet its generalization behavior is shaped by two coupled effects that existing analyses fold into a single hypothesis-level quantity: finite memory replaces each past distribution with an empirical proxy, and repeated reuse couples the buffer, the current data, and the final hypothesis through a shared optimization trajectory. We develop a layer-wise information-theoretic framework that separates these effects at every depth. Our main result decomposes the expected generalization gap into a replay-induced representation drift and an optimization-dependence term, the latter further resolved into stability, plasticity, interaction, and residual-coupling components. Two refinements make the framework operational. A Wasserstein relaxation of the drift term, valid under support mismatch, yields a depth-dependent drift--sensitivity trade-off whose minimizer identifies which interior layer to stabilize. An SGLD instantiation of the optimization term reduces it to a trajectory-level log-determinant budget, exposing a curvature-aware gradient-alignment statistic that serves as an online diagnostic of task-wise forgetting. Controlled and benchmark experiments confirm the predicted memory scaling, the interior funnel, and the alignment signal's link to forgetting.