Universal Barrier is $n$-Self-Concordant

TL;DR

This paper proves that the universal barrier's self-concordance parameter on any n-dimensional convex domain is at most n, providing a tight bound that improves previous results and introduces new moment inequalities for s-concave distributions.

Contribution

It establishes a tight upper bound of n for the self-concordance parameter of the universal barrier on convex domains, improving prior bounds and introducing novel moment inequalities.

Findings

The self-concordance parameter is bounded by n for any n-dimensional convex domain.

The bound is tight, matching the lower limit.

New moment inequalities for s-concave distributions are developed.

Abstract

This paper shows that the self-concordance parameter of the universal barrier on any $n$-dimensional proper convex domain is upper bounded by $n$. This bound is tight and improves the previous $O(n)$ bound by Nesterov and Nemirovski. The key to our main result is a pair of new, sharp moment inequalities for $s$-concave distributions, which could be of independent interest.