次梯度方法
数学
有界函数
正多边形
迭代函数
次导数
凸函数
规范(哲学)
歧管(流体力学)
曲率
投影(关系代数)
凸体
应用数学
功能(生物学)
纯数学
数学分析
数学优化
凸壳
凸优化
几何学
算法
工程类
法学
生物
机械工程
进化生物学
政治学
作者
Pascal Bianchi,Walid Hachem,Sholom Schechtman
标识
DOI:10.1287/moor.2021.0194
摘要
In nonsmooth stochastic optimization, we establish the nonconvergence of the stochastic subgradient descent (SGD) to the critical points recently called active strict saddles by Davis and Drusvyatskiy. Such points lie on a manifold M, where the function f has a direction of second-order negative curvature. Off this manifold, the norm of the Clarke subdifferential of f is lower-bounded. We require two conditions on f. The first assumption is a Verdier stratification condition, which is a refinement of the popular Whitney stratification. It allows us to establish a strengthened version of the projection formula of Bolte et al. for Whitney stratifiable functions and which is of independent interest. The second assumption, termed the angle condition, allows us to control the distance of the iterates to M. When f is weakly convex, our assumptions are generic. Consequently, generically, in the class of definable weakly convex functions, SGD converges to a local minimizer. Funding: The work of Sholom Schechtman was supported by “Région Ile-de-France”.
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