The entropic barrier: a simple and optimal universal self-concordant barrier

TL;DR

This paper introduces an explicit universal self-concordant barrier for convex bodies, based on the Cramér transform of the uniform measure, achieving optimal parameters and improving prior results.

Contribution

It provides the first explicit construction of a universal barrier with optimal self-concordance parameter for convex bodies, utilizing geometric and exponential family duality.

Findings

Cramér transform of uniform measure is an (1+o(1)) n-self-concordant barrier

Improves a seminal result of Nesterov and Nemirovski

Provides explicit construction with optimal parameters

Abstract

We prove that the Cram\'er transform of the uniform measure on a convex body in $\mathbb{R}^n$ is a $(1+o(1)) n$-self-concordant barrier, improving a seminal result of Nesterov and Nemirovski. This gives the first explicit construction of a universal barrier for convex bodies with optimal self-concordance parameter. The proof is based on basic geometry of log-concave distributions, and elementary duality in exponential families.