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

arXiv: https://arxiv.org/abs/1412.1587 · HTML: https://mlsanity.com/paper/1412.1587/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Dec 2014.

## Built on by (most cited first, 15 of 15)

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- [Convergence Rates for the MAP of an Exponential Family and Stochastic Mirror Descent -- an Open Problem](https://mlsanity.com/paper/2111.06826/) (2021)
- [Decomposable Non-Smooth Convex Optimization with Nearly-Linear Gradient Oracle Complexity](https://mlsanity.com/paper/2208.03811/) (2022)
- [A conversion theorem and minimax optimality for continuum contextual bandits](https://mlsanity.com/paper/2406.05714/) (2024)
- [Pliable rejection sampling](https://mlsanity.com/paper/2604.22385/) (2026)
- [A Complete Characterization of Learnability for Adversarial Noisy Bandits](https://mlsanity.com/paper/2605.09200/) (2026)

## Abstract (arXiv)

We prove that the Cram'er transform of the uniform measure on a convex body in ℝn 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.