连续随机梯度法:第一部分收敛理论

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY
Max Grieshammer, Lukas Pflug, Michael Stingl, Andrian Uihlein
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引用次数: 1

摘要

在这篇文章中,我们全面概述了连续随机梯度(CSG)方法,包括收敛结果、步长规则和算法见解。我们考虑目标函数需要某种形式的积分的优化问题,例如期望值。由于用固定的积分规则逼近积分可以在增加计算量的同时引入人为的局部解,因此随机优化方案在这种情况下越来越受欢迎。然而,已知的随机梯度型方法通常局限于预期的风险函数,并且固有地需要多次迭代。后者尤其成问题,如果成本函数的评估涉及求解多个状态方程,例如以偏微分方程的形式给出。为了克服这些缺点,最近的一篇文章介绍了CSG方法,该方法通过计算设计相关的积分权重来重用旧的梯度样本信息,以获得更好的近似全梯度。虽然在最初的CSG论文中建立了子序列的收敛性,但在这里,我们提供了恒定步长和armijo型线搜索的CSG的完整收敛分析。此外,还提出了新的积分权值获取方法,将CSG的应用范围扩大到涉及高维积分和分布式数据的问题。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

The continuous stochastic gradient method: part I–convergence theory

The continuous stochastic gradient method: part I–convergence theory

In this contribution, we present a full overview of the continuous stochastic gradient (CSG) method, including convergence results, step size rules and algorithmic insights. We consider optimization problems in which the objective function requires some form of integration, e.g., expected values. Since approximating the integration by a fixed quadrature rule can introduce artificial local solutions into the problem while simultaneously raising the computational effort, stochastic optimization schemes have become increasingly popular in such contexts. However, known stochastic gradient type methods are typically limited to expected risk functions and inherently require many iterations. The latter is particularly problematic, if the evaluation of the cost function involves solving multiple state equations, given, e.g., in form of partial differential equations. To overcome these drawbacks, a recent article introduced the CSG method, which reuses old gradient sample information via the calculation of design dependent integration weights to obtain a better approximation to the full gradient. While in the original CSG paper convergence of a subsequence was established for a diminishing step size, here, we provide a complete convergence analysis of CSG for constant step sizes and an Armijo-type line search. Moreover, new methods to obtain the integration weights are presented, extending the application range of CSG to problems involving higher dimensional integrals and distributed data.

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来源期刊
Accounts of Chemical Research
Accounts of Chemical Research 化学-化学综合
CiteScore
31.40
自引率
1.10%
发文量
312
审稿时长
2 months
期刊介绍: Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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