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This paper gives a closed-form theory of generalization loss for continual linear regression trained by gradient descent with a fixed step count per task. Its main result is that, for single or i.i.d. teachers, the optimal step size shrinks roughly as ln T / T, and the resulting loss matches that of optimally tuned isotropic L2 regularization. The authors also show that for some task distributions the optimal step size stays finite as tasks grow, so implicit regularization can be a durable design lever. The experiments on linear models and neural networks are a useful check, but the scope is still high-dimensional linear theory, so readers should treat the guidance as a hypothesis for deep continual learning rather than a settled recipe.
Optimal Explicit and Implicit Regularization Strengths in High-Dimensional Continual Linear Regression
arXiv:2610.20804v1 Announce Type: new Abstract: 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.
Abstract: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.
Machine Learning (cs.LG)
arXiv:2610.20804 [cs.LG]
(or arXiv:2610.20804v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2610.20804 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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