Exploiting problem structure in optimization under uncertainty via online convex optimization
Exploiting problem structure in optimization under uncertainty via online convex optimization
复制标题
通过在线凸优化在不确定性下优化问题结构
作者：
Nam Ho;F. Kılınç
在本文中，我们考虑了两种考虑优化模型中不确定性的范式：稳健优化(RO)和联合估计-优化(JEO)。我们考察了这些问题的高效和可伸缩迭代一阶方法的最新进展，并表明这些迭代方法可以通过在线凸优化(OCO)的透镜来看待。标准的OCO框架因其在动态、不确定、甚至是敌对环境中处理决策的能力而获得了巨大的成功。然而，我们感兴趣的应用程序通过对标准OCO假设的三个简单修改，为OCO提供了进一步的灵活性：我们引入了加权后悔和在线鞍点问题两个新概念，并研究了做出前瞻性(预测性)决策的可能性。我们的分析表明，在OCO框架中引入的这些灵活性在适用时都会产生重大影响。例如，在强凸的情况下，最小化未加权后悔的最优界是O(log(T)/T)Documentclass[12pt]{Minimal}usepackage{amsath}usepackage{wa ysym}usepackage{amsfonts}usepackage{amsbsy}usepackage{mathsfs}usepackage{mathrsfs}usepackage{upgreek}setlong{oddsidemarin}{-69pt}例如在{Document}$$O(mathop{mathm{log}}(T)/T)$$end{Document}中，我们证明了当我们考虑加权后悔时，O(1/T)的界是可能的。类似地，对于平滑的情况，考虑1-lookhead决策将导致O(1/T)界限，而在标准OCO设置中，O(1/T)Documentclass[12pt]{Minimum}usepackage{amsath}usepackage{wa ysym}usepackage{amsfonts}usepackage{amsbsy}usepackage{mathsfs}usepackage{upgreek}setlong{oddsidemargin}{-69pt}例如{Document}$O(1/sqrt{T})$end{Document}。因此，这些OCO工具有助于利用函数的结构属性，并提高RO和JEO的收敛速度。在某些情况下，我们对RO和JEO的结果与相应问题类中最已知或最优的比率匹配，而不存在数据不确定性。
In this paper, we consider two paradigms that are developed to account for uncertainty in optimization models: robust optimization (RO) and joint estimation-optimization (JEO). We examine recent developments on efficient and scalable iterative first-order methods for these problems, and show that these iterative methods can be viewed through the lens of online convex optimization (OCO). The standard OCO framework has seen much success for its ability to handle decision-making in dynamic, uncertain, and even adversarial environments. Nevertheless, our applications of interest present further flexibility in OCO via three simple modifications to standard OCO assumptions: we introduce two new concepts of weighted regret and online saddle point problems and study the possibility of making lookahead (anticipatory) decisions. Our analyses demonstrate that these flexibilities introduced into the OCO framework have significant consequences whenever they are applicable. For example, in the strongly convex case, minimizing unweighted regret has a proven optimal bound of O(log(T)/T)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$O(mathop {mathrm{log}}(T)/T)$$end{document}, whereas we show that a bound of O(1 / T) is possible when we consider weighted regret. Similarly, for the smooth case, considering 1-lookahead decisions results in a O(1 / T) bound, compared to O(1/T)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$O(1/sqrt{T})$$end{document} in the standard OCO setting. Consequently, these OCO tools are instrumental in exploiting structural properties of functions and results in improved convergence rates for RO and JEO. In certain cases, our results for RO and JEO match the best known or optimal rates in the corresponding problem classes without data uncertainty.