The Entropic Barrier: Exponential Families, Log-Concave Geometry, and Self-Concordance
成果类型：
Article
署名作者：
Bubeck, Sebastien; Eldan, Ronen
署名单位：
Microsoft; Weizmann Institute of Science
刊物名称：
MATHEMATICS OF OPERATIONS RESEARCH
ISSN/ISSBN：
0364-765X
DOI：
10.1287/moor.2017.0923
发表日期：
2019
页码：
264-276
关键词：
摘要：
We prove that the Cramer transform of the uniform measure on a convex body in 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. As a side result, our calculations also show that the universal barrier of Nesterov and Nemirovski is exactly n-self-concordant on convex cones.