The Mixing Time of the Dikin Walk in a Polytope - A Simple Proof

TL;DR

This paper provides a simple proof that the Dikin walk, a random walk used for sampling and optimization in polytopes, mixes in time proportional to the product of the number of inequalities and the dimension.

Contribution

The paper offers a simplified proof of the mixing time bound for the Dikin walk in polytopes, improving understanding and potential application efficiency.

Findings

Mixing time of Dikin walk is O(mn) for an n-dimensional polytope with m inequalities.

Simplified proof enhances theoretical understanding of the Dikin walk's efficiency.

Supports algorithms for sampling and optimization in high-dimensional polytopes.

Abstract

We study the mixing time of the Dikin walk in a polytope - a random walk based on the log-barrier from the interior point method literature. This walk, and a close variant, were studied by Narayanan (2016) and Kannan-Narayanan (2012). Bounds on its mixing time are important for algorithms for sampling and optimization over polytopes. Here, we provide a simple proof of their result that this random walk mixes in time O(mn) for an n-dimensional polytope described using m inequalities.