In nonsmooth, nonconvex stochastic optimization, understanding the uniform convergence of subdifferential mappings is crucial for analyzing stationary points of sample average approximations of risk as they approach the population risk. Yet, characterizing this convergence remains a fundamental challenge. This work introduces a novel perspective by connecting the uniform convergence of subdifferential mappings to that of subgradient mappings as empirical risk converges to the population risk. We prove that, for stochastic weakly-convex objectives, and within any open set, a uniform bound on the convergence of subgradients -- chosen arbitrarily from the corresponding subdifferential sets -- translates to a uniform bound on the convergence of the subdifferential sets itself, measured by the Hausdorff metric. Using this technique, we derive uniform convergence rates for subdifferential sets of stochastic convex-composite objectives. Our results do not rely on key distributional assumptions in the literature, which require the population and finite sample subdifferentials to be continuous in the Hausdorff metric, yet still provide tight convergence rates. These guarantees lead to new insights into the nonsmooth landscapes of such objectives within finite samples.

翻译：在非光滑、非凸随机优化中，理解次微分映射的一致收敛性对于分析风险样本平均逼近的平稳点趋近总体风险时的行为至关重要。然而，刻画这种收敛性仍是一个基础性难题。本文通过建立次微分映射的一致收敛性与次梯度映射的一致收敛性之间的关联，提出了一种新颖的研究视角，其中经验风险收敛于总体风险。我们证明，对于随机弱凸目标函数，在任意开集内，从相应次微分集合中任意选取的次梯度收敛的一致界，可转化为次微分集合本身在豪斯多夫度量下收敛的一致界。利用这一技术，我们推导了随机凸复合目标函数次微分集合的一致收敛速率。我们的结果不依赖于文献中要求总体与有限样本次微分在豪斯多夫度量下连续的关键分布假设，却仍能给出紧致的收敛速率。这些保证为理解此类目标函数在有限样本内的非光滑景观提供了新的见解。