{"title":"The Cost of Nonconvexity in Deterministic Nonsmooth Optimization","authors":"Siyu Kong, A. S. Lewis","doi":"10.1287/moor.2022.0289","DOIUrl":null,"url":null,"abstract":"We study the impact of nonconvexity on the complexity of nonsmooth optimization, emphasizing objectives such as piecewise linear functions, which may not be weakly convex. We focus on a dimension-independent analysis, slightly modifying a 2020 black-box algorithm of Zhang-Lin-Jegelka-Sra-Jadbabaie that approximates an ϵ-stationary point of any directionally differentiable Lipschitz objective using [Formula: see text] calls to a specialized subgradient oracle and a randomized line search. Seeking by contrast a deterministic method, we present a simple black-box version that achieves [Formula: see text] for any difference-of-convex objective and [Formula: see text] for the weakly convex case. Our complexity bound depends on a natural nonconvexity modulus that is related, intriguingly, to the negative part of directional second derivatives of the objective, understood in the distributional sense.Funding: This work was supported by the National Science Foundation [Grant DMS-2006990].","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1287/moor.2022.0289","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 3
Abstract
We study the impact of nonconvexity on the complexity of nonsmooth optimization, emphasizing objectives such as piecewise linear functions, which may not be weakly convex. We focus on a dimension-independent analysis, slightly modifying a 2020 black-box algorithm of Zhang-Lin-Jegelka-Sra-Jadbabaie that approximates an ϵ-stationary point of any directionally differentiable Lipschitz objective using [Formula: see text] calls to a specialized subgradient oracle and a randomized line search. Seeking by contrast a deterministic method, we present a simple black-box version that achieves [Formula: see text] for any difference-of-convex objective and [Formula: see text] for the weakly convex case. Our complexity bound depends on a natural nonconvexity modulus that is related, intriguingly, to the negative part of directional second derivatives of the objective, understood in the distributional sense.Funding: This work was supported by the National Science Foundation [Grant DMS-2006990].
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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