Calculating Optimistic Likelihoods Using (Geodesically) Convex Optimization

TL;DR

This paper introduces a method to evaluate optimistic likelihoods within ambiguity sets of distributions using convex optimization, improving classification robustness with synthetic and real data.

Contribution

It proposes a novel approach to compute optimistic likelihoods via geodesic convex optimization, addressing estimation errors in nominal distributions.

Findings

Optimistic likelihoods can be computed efficiently using convex optimization techniques.

The method improves classification performance on synthetic and empirical datasets.

Using ambiguity sets enhances robustness against distribution estimation errors.

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 \emph{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.