Analysis of Langevin Monte Carlo via convex optimization

Abstract

In this paper, we provide new insights on the Unadjusted Langevin Algorithm. We show that this method can be formulated as a first order optimization algorithm of an objective functional defined on the Wasserstein space of order $2$. Using this interpretation and techniques borrowed from convex optimization, we give a non-asymptotic analysis of this method to sample from logconcave smooth target distribution on ${\mathbb{R}}^d$. Based on this interpretation, we propose two new methods for sampling from a non-smooth target distribution, which we analyze as well. Besides, these new algorithms are natural extensions of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm, which is a popular extension of the Unadjusted Langevin Algorithm. Similar to SGLD, they only rely on approximations of the gradient of the target log density and can be used for large-scale Bayesian inference.

References in corpus (8)

• Rényi Divergence and Kullback-Leibler Divergence
• User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient
• Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis
• Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm
• Convergence of Langevin MCMC in KL-divergence
• The entropy per coordinate of a random vector is highly constrained under convexity conditions
• Optimal Concentration of Information Content For Log-Concave Densities
• Global Convergence of Langevin Dynamics Based Algorithms for Nonconvex Optimization

Cited by in corpus (6)

• Maximum Mean Discrepancy Gradient Flow
• Simulated Tempering Langevin Monte Carlo II: An Improved Proof using Soft Markov Chain Decomposition
• KALE Flow: A Relaxed KL Gradient Flow for Probabilities with Disjoint Support
• Understanding MCMC Dynamics as Flows on the Wasserstein Space
• Fast Convergence of Langevin Dynamics on Manifold: Geodesics meet Log-Sobolev
• Mirrored Langevin Dynamics