A statistical physics framework for optimal learning
This work provides a principled foundation for understanding and designing optimal learning protocols, potentially benefiting researchers in machine learning and AI, though it is incremental in building on existing statistical physics and control theory methods.
The authors tackled the problem of designing optimal learning protocols for neural networks by developing a statistical physics and control theory framework, deriving closed-form equations to track stochastic gradient descent and applying it to cases like optimal curricula and adaptive dropout, achieving interpretable strategies that mediate learning tradeoffs.
Learning is a complex dynamical process shaped by a range of interconnected decisions. Careful design of hyperparameter schedules for artificial neural networks or efficient allocation of cognitive resources by biological learners can dramatically affect performance. Yet, theoretical understanding of optimal learning strategies remains sparse, especially due to the intricate interplay between evolving meta-parameters and nonlinear learning dynamics. The search for optimal protocols is further hindered by the high dimensionality of the learning space, often resulting in predominantly heuristic, difficult to interpret, and computationally demanding solutions. Here, we combine statistical physics with control theory in a unified theoretical framework to identify optimal protocols in prototypical neural network models. In the high-dimensional limit, we derive closed-form ordinary differential equations that track online stochastic gradient descent through low-dimensional order parameters. We formulate the design of learning protocols as an optimal control problem directly on the dynamics of the order parameters with the goal of minimizing the generalization error at the end of training. This framework encompasses a variety of learning scenarios, optimization constraints, and control budgets. We apply it to representative cases, including optimal curricula, adaptive dropout regularization and noise schedules in denoising autoencoders. We find nontrivial yet interpretable strategies highlighting how optimal protocols mediate crucial learning tradeoffs, such as maximizing alignment with informative input directions while minimizing noise fitting. Finally, we show how to apply our framework to real datasets. Our results establish a principled foundation for understanding and designing optimal learning protocols and suggest a path toward a theory of meta-learning grounded in statistical physics.