A non-exponential extension of Sanov's theorem via convex duality

TL;DR

This paper extends Sanov's theorem beyond exponential bounds using convex duality, enabling new large deviation results for empirical measures, including heavy-tailed distributions and applications in optimal transport and stochastic optimization.

Contribution

It introduces a non-exponential version of Sanov's theorem based on convex duality, broadening the scope of large deviation principles and applications.

Findings

Non-exponential large deviation bounds for empirical distributions

Convergence rates of empirical measures in Wasserstein distance

Applications to stochastic optimization error estimates

Abstract

This work is devoted to a vast extension of Sanov's theorem, in Laplace principle form, based on alternatives to the classical convex dual pair of relative entropy and cumulant generating functional. The abstract results give rise to a number of probabilistic limit theorems and asymptotics. For instance, widely applicable non-exponential large deviation upper bounds are derived for empirical distributions and averages of i.i.d. samples under minimal integrability assumptions, notably accommodating heavy-tailed distributions. Other interesting manifestations of the abstract results include new results on the rate of convergence of empirical measures in Wasserstein distance, uniform large deviation bounds, and variational problems involving optimal transport costs, as well as an application to error estimates for approximate solutions of stochastic optimization problems. The proofs build on the Dupuis-Ellis weak convergence approach to large deviations as well as the duality theory for convex risk measures.