Rigorous dynamical mean field theory for stochastic gradient descent methods

TL;DR

This paper derives exact high-dimensional asymptotic equations for first-order gradient methods like SGD, connecting them to dynamical mean-field theory from physics, and providing tools for analyzing their behavior on Gaussian data.

Contribution

It introduces a rigorous framework linking gradient descent algorithms to dynamical mean-field theory, including non-separable updates and datasets with non-identity covariance.

Findings

Derived closed-form equations for high-dimensional asymptotics of gradient methods

Connected these equations to dynamical mean-field theory from physics

Provided numerical implementations for SGD with various batch sizes and learning rates

Abstract

We prove closed-form equations for the exact high-dimensional asymptotics of a family of first order gradient-based methods, learning an estimator (e.g. M-estimator, shallow neural network, ...) from observations on Gaussian data with empirical risk minimization. This includes widely used algorithms such as stochastic gradient descent (SGD) or Nesterov acceleration. The obtained equations match those resulting from the discretization of dynamical mean-field theory (DMFT) equations from statistical physics when applied to gradient flow. Our proof method allows us to give an explicit description of how memory kernels build up in the effective dynamics, and to include non-separable update functions, allowing datasets with non-identity covariance matrices. Finally, we provide numerical implementations of the equations for SGD with generic extensive batch-size and with constant learning rates.