Analytic theory of dropout regularization

Dropout is a regularization technique widely used in training artificial neural networks to mitigate overfitting. It consists of dynamically deactivating subsets of the network during training to promote more robust representations. Despite its widespread adoption, dropout probabilities are often selected heuristically, and theoretical explanations of its success remain sparse. Here, we analytically study dropout in two-layer neural networks trained with online stochastic gradient descent. In the high-dimensional limit, we derive a set of ordinary differential equations that fully characterize the evolution of the network during training and capture the effects of dropout. We obtain a number of exact results describing the generalization error and the optimal dropout probability at short, intermediate, and long training times. Our analysis shows that dropout reduces detrimental correlations between hidden nodes, mitigates the impact of label noise, and that the optimal dropout probability increases with the level of noise in the data. Our results are validated by extensive numerical simulations.

Optimal Protocols for Continual Learning via Statistical Physics and Control Theory

Artificial neural networks often struggle with catastrophic forgetting when learning multiple tasks sequentially, as training on new tasks degrades the performance on previously learned ones. Recent theoretical work has addressed this issue by analysing learning curves in synthetic frameworks under predefined training protocols. However, these protocols relied on heuristics and lacked a solid theoretical foundation assessing their optimality. In this paper, we fill this gap 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 maximising performance while minimising forgetting. Our theoretical analysis offers non-trivial yet interpretable strategies for mitigating catastrophic forgetting, shedding light on how optimal learning protocols can modulate established effects, such as the influence of task similarity on forgetting. Finally, we validate our theoretical findings on real-world data.