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arXiv · 2601.17646

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

Karim Bounja·Lahcen Laayouni·Abdeljalil Sakat
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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.

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Karim Bounja, Lahcen Laayouni, Abdeljalil Sakat. 2026-01-25. A Mosco sufficient condition for intrinsic stability of non-unique convex Empirical Risk Minimization. https://arxiv.org/abs/2601.17646

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