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The Entropic Barrier: Exponential Families, Log-Concave Geometry, and Self-Concordance

Author

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  • Sébastien Bubeck

    (Microsoft Research Redmond, Redmond, Washington 98052)

  • Ronen Eldan

    (Weizmann Institute of Science, Rehovot 76100, Israel)

Abstract

We prove that the Cramér 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.

Suggested Citation

  • Sébastien Bubeck & Ronen Eldan, 2019. "The Entropic Barrier: Exponential Families, Log-Concave Geometry, and Self-Concordance," Mathematics of Operations Research, INFORMS, vol. 44(1), pages 264-276, February.
    • Handle: RePEc:inm:ormoor:v:44:y:2019:i:1:p:264-276
    DOI: 10.1287/moor.2017.0923
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    References listed on IDEAS

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    1. Ravindran Kannan & Hariharan Narayanan, 2012. "Random Walks on Polytopes and an Affine Interior Point Method for Linear Programming," Mathematics of Operations Research, INFORMS, vol. 37(1), pages 1-20, February.
    2. Roland Hildebrand, 2014. "Canonical Barriers on Convex Cones," Mathematics of Operations Research, INFORMS, vol. 39(3), pages 841-850, August.
    3. Thomas M. Cover, 1991. "Universal Portfolios," Mathematical Finance, Wiley Blackwell, vol. 1(1), pages 1-29, January.
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