Strong Self-Concordance and Sampling

TL;DR

This paper introduces strong self-concordance and a symmetric parameter to improve sampling algorithms in high-dimensional convex bodies, leading to faster mixing times and new theoretical insights.

Contribution

It develops a new interior-point theory with strong self-concordance, enabling improved mixing time bounds for Dikin walk-based sampling methods.

Findings

Dikin walk mixes in O(nar{ u}) steps with strong self-concordance

First walk to mix in O(n^2) steps for arbitrary polytopes

Strong self-concordance holds for Lee-Sidford barrier and relates to the KLS conjecture

Abstract

Motivated by the Dikin walk, we develop aspects of an interior-point theory for sampling in high dimension. Specifically, we introduce a symmetric parameter and the notion of strong self-concordance. These properties imply that the corresponding Dikin walk mixes in $\tilde{O}(n\bar{\nu})$ steps from a warm start in a convex body in $\mathbb{R}^{n}$ using a strongly self-concordant barrier with symmetric self-concordance parameter $\bar{\nu}$. For many natural barriers, $\bar{\nu}$ is roughly bounded by $\nu$, the standard self-concordance parameter. We show that this property and strong self-concordance hold for the Lee-Sidford barrier. As a consequence, we obtain the first walk to mix in $\tilde{O}(n^{2})$ steps for an arbitrary polytope in $\mathbb{R}^{n}$. Strong self-concordance for other barriers leads to an interesting (and unexpected) connection -- for the universal and entropic barriers, it is implied by the KLS conjecture.