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

arXiv: https://arxiv.org/abs/2601.17646 · HTML: https://mlsanity.com/paper/2601.17646/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 17 references that are in the map.
- First posted on arXiv 25 Jan 2026.

## Builds on (most cited first, 5 of 5)

- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Reconciling modern machine learning practice and the bias-variance trade-off](https://mlsanity.com/paper/1812.11118/) (2018)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Improving Reproducibility in Machine Learning Research (A Report from the NeurIPS 2019 Reproducibility Program)](https://mlsanity.com/paper/2003.12206/) (2020)
- [Norm-Based Capacity Control in Neural Networks](https://mlsanity.com/paper/1503.00036/) (2015)

## Abstract (arXiv)

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.