Mirrored Langevin Dynamics

TL;DR

This paper introduces a mirror descent-inspired framework for sampling from constrained distributions, achieving improved convergence rates and practical algorithms, especially for Dirichlet posteriors in LDA, with promising experimental results.

Contribution

It proposes a unified first-order sampling framework that improves convergence rates and extends to stochastic gradients, with specific application to Dirichlet distributions in LDA.

Findings

Achieves $ ilde{O}( ext{epsilon}^{-2}d)$ convergence for strongly convex potentials.

Derives the first non-asymptotic $ ilde{O}( ext{epsilon}^{-2}d^2)$ rate for Dirichlet posterior sampling.

Experimental results demonstrate effectiveness on real LDA datasets.

Abstract

We consider the problem of sampling from constrained distributions, which has posed significant challenges to both non-asymptotic analysis and algorithmic design. We propose a unified framework, which is inspired by the classical mirror descent, to derive novel first-order sampling schemes. We prove that, for a general target distribution with strongly convex potential, our framework implies the existence of a first-order algorithm achieving $\tilde{O}(\epsilon^{-2}d)$ convergence, suggesting that the state-of-the-art $\tilde{O}(\epsilon^{-6}d^5)$ can be vastly improved. With the important Latent Dirichlet Allocation (LDA) application in mind, we specialize our algorithm to sample from Dirichlet posteriors, and derive the first non-asymptotic $\tilde{O}(\epsilon^{-2}d^2)$ rate for first-order sampling. We further extend our framework to the mini-batch setting and prove convergence rates when only stochastic gradients are available. Finally, we report promising experimental results for LDA on real datasets.