Drift and Dependence: Layer-wise Information-Theoretic Bounds for Replay-Based Continual Learning

2026-08-12

AI Summary

This work addresses the degradation in generalization and catastrophic forgetting in replay-based continual learning, which stem from the coupling between limited memory and optimization trajectories. The authors propose a hierarchical information-theoretic framework that, for the first time, disentangles representation drift from optimization dependency at the layer level. The generalization error is decomposed into a replay-induced distribution shift term and an optimization dependency term, the latter further refined into stability, plasticity, interaction, and residual coupling components. To handle support set mismatch, Wasserstein relaxation is employed to bound distributional drift, while SGLD approximation translates optimization dependency into a trajectory-level log-determinant budget. The framework also introduces a depth-dependent drift-sensitivity trade-off and a curvature-aware gradient alignment metric. Experiments reveal memory scaling laws, a “funnel” phenomenon of internal layer stabilization, and a strong correlation between gradient alignment signals and task forgetting.

Abstract

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.

Problem

• continual learning
• catastrophic forgetting
• replay
• generalization gap
• representation drift

Innovation

• information-theoretic bounds
• representation drift
• optimization dependence
• Wasserstein relaxation
• gradient alignment