非强凸时变优化的加速近端梯度方法

中文摘要 AI 辅助

本文研究紧域上、具有有界时变特性且目标函数为凸但不一定强凸的时变复合优化问题。作为基准，我们首先分析时变近端梯度（TV-PG）方法，证明当δ→0时，其渐近函数值间隙以O(δ^(1/2))为界，其中δ衡量目标函数的时变程度，该对δ的依赖关系在TV-PG的最坏情况下是紧的。我们的主要贡献是一种时变加速近端梯度方法，它将函数值间隙的渐近上界改进为O(δ^(2/3))；该方法基于一种新颖的加速方案，其动量参数由当前迭代确定，不显式依赖迭代索引。我们通过基于势函数的统一分析，为基准方法和加速方法建立了界。在合成问题及使用真实世界数据集的自回归问题上的数值实验，验证了所提算法的跟踪性能。

英文摘要

This paper studies time-varying composite optimization problems with convex but not necessarily strongly convex objective functions, on compact domains and with bounded temporal variability. As a baseline, we first analyze the time-varying proximal gradient (TV-PG) method. We show that its asymptotic function value gap is bounded by $O(δ^{1/2})$ as $δ\to0$, where $δ$ measures the temporal variability of the objective function. This dependence on $δ$ is tight in the worst case for TV-PG. Our main contribution is a time-varying accelerated proximal gradient method that improves the asymptotic upper bound on the function value gap to $O(δ^{2/3})$. The method builds on a novel acceleration scheme whose momentum parameter is determined by the current iterates, with no explicit dependence on the iteration index. We establish the bounds for both the baseline and accelerated methods through a unified analysis based on potential functions. Numerical experiments on a synthetic problem and autoregression problems with real-world datasets demonstrate the tracking performance of the proposed algorithm.

发表机构

• The University of Tokyo（东京大学）
• RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）

机构由 AI 辅助整理，请以论文原文为准。