Fast parallel sampling under isoperimetry

TL;DR

This paper introduces parallel algorithms for sampling from distributions satisfying isoperimetric inequalities, achieving near-optimal gradient evaluations and providing the first TV distance guarantees in parallel sampling.

Contribution

The paper presents the first parallel sampling algorithms with TV distance guarantees for distributions satisfying a log-Sobolev inequality, enabling efficient RNC reductions for complex combinatorial problems.

Findings

Achieves $ ilde{O}(d)$ gradient evaluations with $ ext{polylog}(d)$ parallel rounds.

Provides the first parallel algorithms with TV distance guarantees.

Enables RNC sampling for Eulerian tours and determinantal point processes.

Abstract

We show how to sample in parallel from a distribution $\pi$ over $\mathbb R^d$ that satisfies a log-Sobolev inequality and has a smooth log-density, by parallelizing the Langevin (resp. underdamped Langevin) algorithms. We show that our algorithm outputs samples from a distribution $\hat\pi$ that is close to $\pi$ in Kullback--Leibler (KL) divergence (resp. total variation (TV) distance), while using only $\log(d)^{O(1)}$ parallel rounds and $\widetilde{O}(d)$ (resp. $\widetilde O(\sqrt d)$) gradient evaluations in total. This constitutes the first parallel sampling algorithms with TV distance guarantees. For our main application, we show how to combine the TV distance guarantees of our algorithms with prior works and obtain RNC sampling-to-counting reductions for families of discrete distribution on the hypercube $\{\pm 1\}^n$ that are closed under exponential tilts and have bounded covariance. Consequently, we obtain an RNC sampler for directed Eulerian tours and asymmetric determinantal point processes, resolving open questions raised in prior works.