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.

翻译：经验风险最小化（ERM）的稳定性通常通过单值输出进行研究，而凸非严格损失函数会产生集值最小化器。我们指出Painlevé-Kuratowski上半连续性（PK-u.s.c.）是ERM解对应关系（集合层面Hadamard适定性）的内在稳定性概念，也是解释选择子稳定性的先决条件。随后我们刻画了一个最小非退化定性机制：Mosco一致性扰动与局部有界最小化器可推出PK-u.s.c.、最小值连续性以及消失间隙近最小化器的一致性。二次增长条件可导出显式的定量偏差界。