Random tilts to find stationary points in stochastic convex optimization

2026-09-15

AI Summary

本文研究了随机凸函数的驻点问题，通过正则化经验风险最小化结合随机倾斜扰动的方法，获得了d维问题给定n个观测值下的驻点残差阶为sqrt(d/n)。

Abstract

We consider the problem of finding stationary points of stochastic convex functions and related variational inequalities. For each, we show that regularized empirical risk minimization, coupled with a random tilting perturbation, obtains stationarity residual order $\sqrt{d/n}$ for $d$-dimensional problems given $n$ observations. We present a few complementary results that show that some dimension dependence is necessary, in distinction from standard stochastic optimization and empirical risk minimization, by providing minimax lower bounds scaling as $\sqrt{\log d / n}$.

Problem

• stochastic convex optimization
• stationary points
• empirical risk minimization

Innovation

• random tilting perturbation
• stationary points
• stochastic convex optimization
• regularized empirical risk minimization