Improved bounds for discretization of Langevin diffusions: Near-optimal rates without convexity

Abstract

Discretizations of the Langevin diffusion have been proven very useful for developing and analyzing algorithms for sampling and stochastic optimization. We present an improved non-asymptotic 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 in discretization step size from $O(\mathrm{\eta })$ to $O({\mathrm{\eta }}^2)$ in terms of the KL divergence. This result matches the correct order for numerical SDEs, without suffering from exponential time dependence. When applied to MCMC, this result simultaneously improves on the analyses of a range of sampling algorithms that are based on Dalalyan’s approach.