{"title":"Highly Smooth Zeroth-Order Methods for Solving Optimization Problems under the PL Condition","authors":"A. V. Gasnikov, A. V. Lobanov, F. S. Stonyakin","doi":"10.1134/s0965542524700118","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>In this paper, we study the black box optimization problem under the Polyak–Lojasiewicz (PL) condition, assuming that the objective function is not just smooth, but has higher smoothness. By using “kernel-based” approximations instead of the exact gradient in the Stochastic Gradient Descent method, we improve the best-known results of convergence in the class of gradient-free algorithms solving problems under the PL condition. We generalize our results to the case where a zeroth-order oracle returns a function value at a point with some adversarial noise. We verify our theoretical results on the example of solving a system of nonlinear equations.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Mathematics and Mathematical Physics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0965542524700118","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
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Abstract
In this paper, we study the black box optimization problem under the Polyak–Lojasiewicz (PL) condition, assuming that the objective function is not just smooth, but has higher smoothness. By using “kernel-based” approximations instead of the exact gradient in the Stochastic Gradient Descent method, we improve the best-known results of convergence in the class of gradient-free algorithms solving problems under the PL condition. We generalize our results to the case where a zeroth-order oracle returns a function value at a point with some adversarial noise. We verify our theoretical results on the example of solving a system of nonlinear equations.
期刊介绍：
Computational Mathematics and Mathematical Physics is a monthly journal published in collaboration with the Russian Academy of Sciences. The journal includes reviews and original papers on computational mathematics, computational methods of mathematical physics, informatics, and other mathematical sciences. The journal welcomes reviews and original articles from all countries in the English or Russian language.
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