In-and-Out: Algorithmic Diffusion for Sampling Convex Bodies

Yunbum Kook, Santosh Vempala, Matthew Shunshi Zhang

Keywords: Sampling, convex bodies, MCMC, uniform sampling

TL;DR: We propose a sampler for the uniform distribution on a convex body, based on forward and backward heat flows.

Abstract: We present a new random walk for uniformly sampling high-dimensional convex bodies. It achieves state-of-the-art runtime complexity with stronger guarantees on the output than previously known, namely in Rényi divergence (which implies TV, $\mathcal{W}_2$, KL, $\chi^2$). The proof departs from known approaches for polytime algorithms for the problem - we utilize a stochastic diffusion perspective to show contraction to the target distribution with the rate of convergence determined by functional isoperimetric constants of the stationary density.

Primary Area: Probabilistic methods (for example: variational inference, Gaussian processes)

Submission Number: 6735