Explaining the effects of non-convergent sampling in the training of Energy-Based Models

TL;DR

This paper analyzes how non-convergent Markov chain sampling affects the training of Energy-Based Models, revealing that short-run sampling can reproduce data statistics without reaching equilibrium.

Contribution

It provides a first-principles explanation for using short, non-convergent runs in EBM training and explores their implications for model behavior and sampling efficiency.

Findings

Short runs can reproduce empirical data statistics without equilibrium.

Theoretical analysis of two solvable models clarifies non-convergent sampling effects.

Numerical tests on ConvNet EBM and Boltzmann machine validate predictions.

Abstract

In this paper, we quantify the impact of using non-convergent Markov chains to train Energy-Based models (EBMs). In particular, we show analytically that EBMs trained with non-persistent short runs to estimate the gradient can perfectly reproduce a set of empirical statistics of the data, not at the level of the equilibrium measure, but through a precise dynamical process. Our results provide a first-principles explanation for the observations of recent works proposing the strategy of using short runs starting from random initial conditions as an efficient way to generate high-quality samples in EBMs, and lay the groundwork for using EBMs as diffusion models. After explaining this effect in generic EBMs, we analyze two solvable models in which the effect of the non-convergent sampling in the trained parameters can be described in detail. Finally, we test these predictions numerically on a ConvNet EBM and a Boltzmann machine.