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Calculating Optimistic Likelihoods Using (Geodesically) Convex Optimization

Viet Anh Nguyen(viet-anh.nguyen***at***epfl.ch)
Soroosh Shafieezadeh-Abadeh (soroosh.shafiee***at***epfl.ch)
Man-Chung Yue(manchung.yue***at***polyu.edu.hk)
Daniel Kuhn(daniel.kuhn***at***epfl.ch)
Wolfram Wiesemann(ww***at***imperial.ac.uk)

Abstract: A fundamental problem arising in many areas of machine learning is the evaluation of the likelihood of a given observation under different nominal distributions. Frequently, these nominal distributions are themselves estimated from data, which makes them susceptible to estimation errors. We thus propose to replace each nominal distribution with an ambiguity set containing all distributions in its vicinity and to evaluate an optimistic likelihood, that is, the maximum of the likelihood over all distributions in the ambiguity set. When the proximity of distributions is quantified by the Fisher-Rao distance or the Kullback-Leibler divergence, the emerging optimistic likelihoods can be computed efficiently using either geodesic or standard convex optimization techniques. We showcase the advantages of working with optimistic likelihoods on a classification problem using synthetic as well as empirical data.

Keywords: Optimistic Likelihood, Fisher-Rao distance, Kullback-Leibler divergence, Geodesic Convex

Category 1: Stochastic Programming

Category 2: Convex and Nonsmooth Optimization (Generalized Convexity/Monoticity )

Category 3: Linear, Cone and Semidefinite Programming (Semi-definite Programming )


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Entry Submitted: 10/21/2019
Entry Accepted: 10/21/2019
Entry Last Modified: 10/21/2019

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