arXiv：cs.LG· Lai Tian, Johannes O. Royset·· 2 天前AI 评分30
随机函数的 Lipschitz 强大数定律
Lipschitzian SLLNs for random functions
AI 导读
研究者证明了随机函数在 Lipschitz 伪度量下的强大数定律，结果在拓扑条件或模型论条件下均成立，后者涵盖 o-minimal 结构中联合可定义的函数并大幅超出该类。应用包括极限次微分与 Clarke 次微分的均匀收敛，以及解的有限样本识别，从而界定出此前负结果所揭示的失效现象不会发生的一大类函数。
正文
Abstract:We prove strong laws of large numbers for random functions in the Lipschitz pseudometric. Our results hold under either a topological or a model-theoretic condition, with the latter encompassing functions jointly definable in o-minimal structures but extending substantially beyond this class. Applications include uniform convergence of limiting and Clarke subdifferentials and finite-sample identification of solutions. Consequently, we identify broad classes of functions for which the failure phenomena revealed by our previous negative results [Tian and Royset, arXiv:2511.16568, 2025] do not occur.
| Comments: | 35 pages |
| Subjects: | Optimization and Control (math.OC); Machine Learning (cs.LG); Statistics Theory (math.ST) |
| Cite as: | arXiv:2607.20411 [math.OC] |
| (or arXiv:2607.20411v2 [math.OC] for this version) | |
| https://doi.org/10.48550/arXiv.2607.20411 arXiv-issued DOI via DataCite |
Submission history
From: Lai Tian [view email]
[v1]
Wed, 22 Jul 2026 17:50:53 UTC (39 KB)
[v2]
Thu, 1 Oct 2026 09:06:48 UTC (45 KB)
来源：arXiv：cs.LG · arxiv.org