Improved Bounds for Discretization of Langevin Diffusions: Near-Optimal Rates without Convexity

TL;DR

This paper improves the theoretical understanding of Langevin diffusion discretization, achieving near-optimal convergence rates without requiring convexity, and enhances sampling algorithms' efficiency.

Contribution

It provides a new analysis that removes the need for global contractivity and improves convergence rates from linear to quadratic in step size.

Findings

KL divergence rate improved from O(η) to O(η^2)

Polynomial dependence on time horizon achieved

Applicable to various sampling and learning algorithms

Abstract

We present an improved analysis of the Euler-Maruyama discretization of the Langevin diffusion. Our analysis does not require global contractivity, and yields polynomial dependence on the time horizon. Compared to existing approaches, we make an additional smoothness assumption, and improve the existing rate from $O(\eta)$ to $O(\eta^2)$ in terms of the KL divergence. This result matches the correct order for numerical SDEs, without suffering from exponential time dependence. When applied to algorithms for sampling and learning, this result simultaneously improves all those methods based on Dalayan's approach.