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

TL;DR

This paper introduces a novel stochastic diffusion-based random walk for efficiently sampling high-dimensional convex bodies, providing improved theoretical guarantees and convergence rates in various divergence metrics.

Contribution

It proposes a new diffusion-based sampling algorithm with superior runtime and convergence guarantees, departing from traditional polytime methods.

Findings

Achieves state-of-the-art runtime complexity

Guarantees convergence in Rényi divergence and other metrics

Utilizes a diffusion perspective linked to isoperimetric constants

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 R\'enyi 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 target distribution.