Adjusted Scores for Discrete Langevin Algorithms

TL;DR

This paper introduces a theoretical framework for discrete Langevin algorithms, interpreting them as discretizations of continuous-time dynamics on the hypercube, and provides bounds on their convergence properties.

Contribution

It offers a novel interpretation of discrete Langevin algorithms as discretizations of continuous dynamics and derives contraction bounds with and without Metropolis adjustments.

Findings

Discrete Langevin algorithms can be viewed as discretizations of continuous-time dynamics.

Upper bounds for contraction of these algorithms are established.

The approach enhances theoretical understanding of discrete sampling methods.

Abstract

Sampling from discrete distributions is a ubiquitous task in machine learning, recently revisited by the emergence of discrete diffusion models. While Langevin algorithms constitute the state of the art for continuous spaces, discrete versions lack similar theoretical guarantees when the step-size becomes small. In this paper, we address this limitation by interpreting discrete sampling algorithms as discretizations of continuous-time dynamics on the hypercube. In particular, we describe several score functions for discrete algorithms which result in approximations of Glauber dynamics for the correct target distribution. We also compute upper bounds for the contraction of these algorithms, with or without Metropolis adjustment.