Geodesic Walks in Polytopes

TL;DR

This paper introduces the geodesic walk, a novel sampling algorithm for polytopes that significantly improves mixing time bounds by simulating a stochastic differential equation on a Riemannian manifold.

Contribution

It presents the first walk that surpasses the quadratic mixing barrier for high-dimensional polytopes, with efficient implementation and analysis on general Hessian manifolds.

Findings

Mixing time of O*(mn^{3/4}) steps for polytopes

First walk to break quadratic barrier in high dimension

Efficient implementation of each step via solving an ODE

Abstract

We introduce the geodesic walk for sampling Riemannian manifolds and apply it to the problem of generating uniform random points from polytopes in R^n specified by m inequalities. The walk is a discrete-time simulation of a stochastic differential equation (SDE) on the Riemannian manifold equipped with the metric induced by the Hessian of a convex function; each step is the solution of an ordinary differential equation (ODE). The resulting sampling algorithm for polytopes mixes in O*(mn^{3/4}) steps. This is the first walk that breaks the quadratic barrier for mixing in high dimension, improving on the previous best bound of O*(mn) by Kannan and Narayanan for the Dikin walk. We also show that each step of the geodesic walk (solving an ODE) can be implemented efficiently, thus improving the time complexity for sampling polytopes. Our analysis of the geodesic walk for general Hessian manifolds does not assume positive curvature and might be of independent interest.