On the Concentration of the Minimizers of Empirical Risks

TL;DR

This paper develops concentration inequalities for the set of empirical risk minimizers in broad metric space estimation problems, including unbounded and non-convex cases, highlighting stability conditions that lead to parametric convergence rates.

Contribution

It introduces a general framework for concentration of empirical risk minimizers applicable to complex metric space problems, extending beyond traditional bounded or convex settings.

Findings

Established concentration bounds for barycenters on curved metric spaces

Demonstrated stability leading to parametric convergence rates

Validated assumptions on diverse estimation problems

Abstract

Obtaining guarantees on the convergence of the minimizers of empirical risks to the ones of the true risk is a fundamental matter in statistical learning. Instead of deriving guarantees on the usual estimation error, the goal of this paper is to provide concentration inequalities on the distance between the sets of minimizers of the risks for a broad spectrum of estimation problems. In particular, the risks are defined on metric spaces through probability measures that are also supported on metric spaces. A particular attention will therefore be given to include unbounded spaces and non-convex cost functions that might also be unbounded. This work identifies a set of assumptions allowing to describe a regime that seem to govern the concentration in many estimation problems, where the empirical minimizers are stable. This stability can then be leveraged to prove parametric concentration rates in probability and in expectation. The assumptions are verified, and the bounds showcased, on a selection of estimation problems such as barycenters on metric space with positive or negative curvature, subspaces of covariance matrices, regression problems and entropic-Wasserstein barycenters.