发表机构
Stanford University（斯坦福大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究随机凸优化问题中寻找驻点，提出以目标函数次微分含小元素为更强概念，借助维度理论分解次微分图像，让随机采样保留其“片段”，实现类似近端点方法的有效应用。
AI 中文摘要
我们考虑寻找随机凸优化问题驻点的问题。我们所要求的不是诸如驻点接近度保证或莫罗包络的小梯度等驻点替代物，而是一个更强的概念：目标函数的次微分实际包含一个小元素。该准则并不平凡，因为凸函数的次微分即使在最优值的任意小邻域内也不能一致收敛。我们的收敛保证依赖维度理论来分解凸函数次微分的图像，展示随机采样如何保留这些图像的“片段”，并允许有效应用类似近端点的方法。
英文摘要
We consider the problem of finding stationary points for stochastic convex optimization problems. Rather than surrogates to stationarity, such as a proximity-to-stationarity guarantee or small gradient of the Moreau envelope, we ask for a stronger notion: that the subdifferential of the objective actually contains a small element. This criterion is non-trivial, because subdifferentials of convex functions fail to converge uniformly, even in arbitrarily small neighborhoods of the optimum. Our convergence guarantees rely on dimension theory to decompose the graph of the subdifferential of a convex function, showing how stochastic sampling preserves "pieces" of these graphs, and allowing effective application of proximal-point-like methods.
