Strong Self-Concordance and Sampling

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.