具有光滑映射的凸差复合函数的在线优化

中文摘要 AI 辅助

我们研究了一类广泛的结构化非凸非光滑问题的在线优化，其中每个损失是一个凸差函数与一个光滑映射的复合，可行域由同类约束函数定义。我们提出了一种时间平滑近端线性算法和基于近端残差映射的局部遗憾度量。我们证明了该残差是原始问题的一个恰当的平稳性度量：其不动点条件意味着一阶平稳性。我们的分析依赖于由复合凸差约束描述的可行域的切锥特征，这具有独立的研究意义，并且允许通过凸优化预言机计算每次更新，尽管问题是非凸的。我们建立了局部遗憾界和内部凸子问题总数的界。我们还推导了一个误差界，将近端残差与到平稳点的距离联系起来，提供了近似平稳性的定量证明。

英文摘要

We study online optimization for a broad class of structured non-convex non-smooth problems where each loss is a composition of a difference-of-convex function with a smooth mapping, and the feasible region is defined by constraint functions of the same kind. We propose a time-smoothed proximal linear algorithm and a local-regret measure based on a proximal residual mapping. We show that this residual is a proper stationarity measure for the original problem: its fixed-point condition implies first-order stationarity. Our analysis relies on a tangent-cone characterization for a feasible region described by composite difference-of-convex constraints, which is of independent interest and allows each update to be computed via a convex optimization oracle, despite the non-convexity of the problem. We establish a local-regret bound and a bound on the total number of inner convex subproblems. We also derive an error bound connecting the proximal residual to the distance to stationarity, providing a quantitative certificate of approximate stationarity.

发表机构

• Management Science and Engineering, Stanford University（斯坦福大学管理科学与工程系）
• The Daniel J. Epstein Department of Industrial and Systems Engineering, University of Southern California（南加州大学丹尼尔·J·埃普斯坦工业与系统工程系）

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