A Mosco sufficient condition for intrinsic stability of non-unique convex Empirical Risk Minimization

TL;DR

This paper introduces a new stability condition for convex ERM solutions with set-valued minimizers, using Painlevé-Kuratowski upper semicontinuity, and provides explicit bounds under quadratic growth.

Contribution

It establishes Mosco consistency as a sufficient condition for intrinsic stability of non-unique convex ERM solutions, extending stability analysis to set-valued cases.

Findings

PK-u.s.c. characterizes intrinsic stability of set-valued ERM solutions.

Mosco-consistent perturbations ensure stability and continuity properties.

Quadratic growth yields explicit deviation bounds for solutions.

Abstract

Empirical risk minimization (ERM) stability is usually studied via single-valued outputs, while convex non-strict losses yield set-valued minimizers. We identify Painlev\'e-Kuratowski upper semicontinuity (PK-u.s.c.) as the intrinsic stability notion for the ERM solution correspondence (set-level Hadamard well-posedness) and a prerequisite to interpret stability of selections. We then characterize a minimal non-degenerate qualitative regime: Mosco-consistent perturbations and locally bounded minimizers imply PK-u.s.c., minimal-value continuity, and consistency of vanishing-gap near-minimizers. Quadratic growth yields explicit quantitative deviation bounds.