Machine learning in and out of equilibrium

TL;DR

This paper applies a statistical physics framework to analyze the nonequilibrium dynamics of neural network training, revealing universal fluctuation theorems and proposing a new sampling algorithm that outperforms existing methods.

Contribution

It introduces a Fokker-Planck approach to neural network training, demonstrating nonequilibrium fluctuation theorems and developing a novel stochastic gradient Langevin dynamics variant.

Findings

Fluctuation theorems hold in neural network training dynamics.

Stationary state properties depend on training details like minibatch sampling.

The proposed SGWORLD algorithm accelerates Bayesian posterior sampling.

Abstract

The algorithms used to train neural networks, like stochastic gradient descent (SGD), have close parallels to natural processes that navigate a high-dimensional parameter space -- for example protein folding or evolution. Our study uses a Fokker-Planck approach, adapted from statistical physics, to explore these parallels in a single, unified framework. We focus in particular on the stationary state of the system in the long-time limit, which in conventional SGD is out of equilibrium, exhibiting persistent currents in the space of network parameters. As in its physical analogues, the current is associated with an entropy production rate for any given training trajectory. The stationary distribution of these rates obeys the integral and detailed fluctuation theorems -- nonequilibrium generalizations of the second law of thermodynamics. We validate these relations in two numerical examples, a nonlinear regression network and MNIST digit classification. While the fluctuation theorems are universal, there are other aspects of the stationary state that are highly sensitive to the training details. Surprisingly, the effective loss landscape and diffusion matrix that determine the shape of the stationary distribution vary depending on the simple choice of minibatching done with or without replacement. We can take advantage of this nonequilibrium sensitivity to engineer an equilibrium stationary state for a particular application: sampling from a posterior distribution of network weights in Bayesian machine learning. We propose a new variation of stochastic gradient Langevin dynamics (SGLD) that harnesses without replacement minibatching. In an example system where the posterior is exactly known, this SGWORLD algorithm outperforms SGLD, converging to the posterior orders of magnitude faster as a function of the learning rate.