Comparison-Based Algorithms for One-Dimensional Stochastic Convex Optimization

Xi Chen, Qihang Lin, Zizhuo Wang
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引用次数: 1

Abstract

Stochastic optimization finds a wide range of applications in operations research and management science. However, existing stochastic optimization techniques usually require the information of random samples (e.g., demands in the newsvendor problem) or the objective values at the sampled points (e.g., the lost sales cost), which might not be available in practice. In this paper, we consider a new setup for stochastic optimization, in which the decision maker has access to only comparative information between a random sample and two chosen decision points in each iteration. We propose a comparison-based algorithm (CBA) to solve such problems in one dimension with convex objective functions. Particularly, the CBA properly chooses the two points in each iteration and constructs an unbiased gradient estimate for the original problem. We show that the CBA achieves the same convergence rate as the optimal stochastic gradient methods (with the samples observed). We also consider extensions of our approach to multi-dimensional quadratic problems as well as problems with non-convex objective functions. Numerical experiments show that the CBA performs well in test problems.
基于比较的一维随机凸优化算法
随机优化在运筹学和管理学中有着广泛的应用。然而,现有的随机优化技术通常需要随机样本的信息(例如,报贩问题中的需求)或采样点的客观值(例如,损失的销售成本),这在实践中可能无法获得。在本文中,我们考虑了一种新的随机优化设置,其中决策者在每次迭代中只能访问随机样本和两个选定决策点之间的比较信息。我们提出了一种基于比较的算法(CBA)来解决一维凸目标函数问题。特别是,CBA算法在每次迭代中正确选择两个点,并对原问题构造无偏梯度估计。我们证明了CBA与最优随机梯度方法具有相同的收敛速度(与观察到的样本)。我们也考虑将我们的方法扩展到多维二次问题以及非凸目标函数问题。数值实验表明,CBA在测试问题中表现良好。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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