Fast MCMC sampling algorithms on polytopes

TL;DR

This paper introduces two novel MCMC sampling algorithms, the Vaidya walk and the John walk, which are based on interior point methods and significantly improve mixing times for sampling from polytopes.

Contribution

The paper presents the Vaidya and John walks, new MCMC algorithms with faster mixing times for sampling from polytopes, based on volumetric barriers and ellipsoids.

Findings

Vaidya walk mixes in (n^{0.5}d^{1.5}) steps, faster than Dikin walk.

Vaidya walk's per-step cost is comparable to Dikin walk.

John walk has a mixing time of (d^{2.5}\,( ext{polylog}(n/d))).

Abstract

We propose and analyze two new MCMC sampling algorithms, the Vaidya walk and the John walk, for generating samples from the uniform distribution over a polytope. Both random walks are sampling algorithms derived from interior point methods. The former is based on volumetric-logarithmic barrier introduced by Vaidya whereas the latter uses John's ellipsoids. We show that the Vaidya walk mixes in significantly fewer steps than the logarithmic-barrier based Dikin walk studied in past work. For a polytope in $\mathbb{R}^d$ defined by $n >d$ linear constraints, we show that the mixing time from a warm start is bounded as $\mathcal{O}(n^{0.5}d^{1.5})$, compared to the $\mathcal{O}(nd)$ mixing time bound for the Dikin walk. The cost of each step of the Vaidya walk is of the same order as the Dikin walk, and at most twice as large in terms of constant pre-factors. For the John walk, we prove an $\mathcal{O}(d^{2.5}\cdot\log^4(n/d))$ bound on its mixing time and conjecture that an improved variant of it could achieve a mixing time of $\mathcal{O}(d^2\cdot\text{polylog}(n/d))$. Additionally, we propose variants of the Vaidya and John walks that mix in polynomial time from a deterministic starting point. The speed-up of the Vaidya walk over the Dikin walk are illustrated in numerical examples.