Mirrored Langevin Dynamics

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 O (\epsilon {-2}d) convergence, suggesting that the state-of-the-art 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 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.