Stochastic optimization using a trust-region method and random models

• 成果类型: Article
• 署名作者: Chen, R.; Menickelly, M.; Scheinberg, K.
• 署名单位: Bosch; Lehigh University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-017-1141-8
• 发表日期: 2018
• 页码: 447-487
• 关键词: derivative-free optimization; approximation; CONVERGENCE

摘要

In this paper, we propose and analyze a trust-region model-based algorithm for solving unconstrained stochastic optimization problems. Our framework utilizes random models of an objective function f(x), obtained from stochastic observations of the function or its gradient. Our method also utilizes estimates of function values to gauge progress that is being made. The convergence analysis relies on requirements that these models and these estimates are sufficiently accurate with high enough, but fixed, probability. Beyond these conditions, no assumptions are made on how these models and estimates are generated. Under these general conditions we show an almost sure global convergence of the method to a first order stationary point. In the second part of the paper, we present examples of generating sufficiently accurate random models under biased or unbiased noise assumptions. Lastly, we present some computational results showing the benefits of the proposed method compared to existing approaches that are based on sample averaging or stochastic gradients.