提出非唯一凸损失下经验风险最小化的内在稳定性判据。
A Mosco sufficient condition for intrinsic stability of non-unique convex Empirical Risk Minimization
- 以PK上半连续性定义解集的内在稳定性
- 在Mosco一致扰动下，解集稳定且近似最小值点一致
- 适用于分析非唯一解的优化问题，尤其适合理论研究者
经验风险最小化（ERM）的稳定性通常基于单值输出研究，但凸非严格损失会产生多值最小化解。本文识别出Painlevé-Kuratowski上半连续性（PK-u.s.c.）为ERM解对应关系的内在稳定性概念（集合层面Hadamard适定性），并构成选择稳定性分析的前提。我们刻画了一个最小非退化定性条件：Mosco一致扰动与局部有界最小化解可推出PK-u.s.c.、最小值连续性以及间隙趋零近最小值点的一致性。二次增长条件下可获得显式的定量偏差界。
原文摘要 · Abstract (English)
Empirical risk minimization (ERM) stability is usually studied via single-valued outputs, while convex non-strict losses yield set-valued minimizers. We identify Painlevé-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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