Optimal learning protocols via statistical physics and control theory

Learning is a complex dynamical process shaped by many interconnected decisions. Protocols that govern how to tune hyperparameters in artificial networks, or how to allocate cognitive effort in biological learners, can have dramatic effects on performance. Yet our theoretical understanding of optimal learning strategies remains limited, due to the nonlinear nature of learning dynamics and the high dimensionality of the learning space.

In this talk, I will present a framework that combines statistical physics and control theory to identify optimal learning protocols in prototypical neural network models (see Refs. [1,2]). In the high-dimensional limit, we derive closed-form equations for a small set of order parameters that track stochastic gradient descent. This reduction allows to formulate the design of learning protocols—such as curricula, dropout schedules, or noise levels—as an optimal control problem on the dynamics of the order parameters, with the objective of minimizing the final generalization error.

I will discuss applications to both toy models and real datasets, showing how the resulting strategies unveil key learning trade-offs, for example, between exploiting informative directions in the data and limiting noise sensitivity, and how these insights may contribute to a principled theory of meta-learning.

[1] Mignacco, F. and Mori, F., 2025. A statistical physics framework for optimal learning. arXiv preprint arXiv:2507.07907.

[2] Mori, F., Mannelli, S.S. and Mignacco, F., Optimal Protocols for Continual Learning via Statistical Physics and Control Theory. In The Thirteenth International Conference on Learning Representations.

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