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arXiv · 1802.10174

Mirrored Langevin Dynamics

Ya-Ping Hsieh·Ali Kavis·Paul Rolland·Volkan Cevher
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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.

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Ya-Ping Hsieh, Ali Kavis, Paul Rolland, Volkan Cevher. 2018-02-27. Mirrored Langevin Dynamics. https://arxiv.org/abs/1802.10174

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