Mirrored Langevin Dynamics
This work addresses a fundamental challenge in machine learning for researchers and practitioners dealing with constrained sampling, offering a novel algorithmic approach with substantial theoretical improvements.
The paper tackles the problem of sampling from constrained distributions by proposing a unified framework inspired by mirror descent, achieving a convergence rate of $ ilde{O}(ε^{-2}d)$ for strongly convex potentials, which significantly improves upon the previous state-of-the-art of $ ilde{O}(ε^{-6}d^5)$.
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 $\tilde{O}(ε^{-2}d)$ convergence, suggesting that the state-of-the-art $\tilde{O}(ε^{-6}d^5)$ 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 $\tilde{O}(ε^{-2}d^2)$ 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.