Optimal protocols for continual learning via statistical physics and control theory^*

Abstract Artificial neural networks often struggle with catastrophic forgetting when learning multiple tasks sequentially, as training on new tasks degrades performance on previously learned tasks. Recent theoretical work has addressed this issue by analysing learning curves in synthetic frameworks under predefined training protocols. However, these protocols rely on heuristics and lack a solid theoretical foundation for assessing their optimality. In this paper, we fill this gap by combining exact equations for training dynamics, derived using statistical physics techniques, with optimal control methods. We apply this approach to teacher–student models for continual learning and multi-task problems, obtaining a theory for task-selection protocols that maximises performance while minimising forgetting. Our theoretical analysis offers nontrivial yet interpretable strategies for mitigating catastrophic forgetting, shedding light on how optimal learning protocols modulate established effects, such as the influence of task similarity on forgetting. Finally, we validate our theoretical findings with experiments on real-world data.