[论文] Lipschitzian SLLNs for random functions

论文概要

研究领域: ML
作者: Lai Tian, Johannes O. Royset
发布时间: 2026-07-24
arXiv: 2507.18390

中文摘要

我们证明了局部Lipschitz函数在Lipschitz伪度量下的强大数定律。我们的结果在拓扑条件或模型理论条件下成立，后者涵盖了在o-minimal结构中联合可定义的函数，但远远超出了这一类。应用包括极限和Clarke次微分的均匀收敛以及解的有限样本识别。因此，我们确定了广泛的函数类，对于我们之前负面结果中揭示的失败现象[Tian和Royset, arXiv:2511.16568, 2025]不会发生。

原文摘要

We prove strong laws of large numbers for locally Lipschitz 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.

自动采集于 2026-07-24

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