On the Convergence Rate of Quasi-Newton Methods onStrongly Convex Functions with Lipschitz Gradient : научное издание

Krutikov Vladimir; Tovbis Elena; Stanimirović Predrag; Kazakovtsev Lev

Дополнительные ссылки: eLibrary

База данных: Каталог библиотеки СФУ

Библиографическое описание: On the Convergence Rate of Quasi-Newton Methods onStrongly Convex Functions with Lipschitz Gradient : научное издание / V. V. Krutikov [et al.]. - Текст : непосредственный // Mathematics. - 2023. - Т. 11, № 23. - P. 4715. - This work was supported by the Ministry of Science and Higher Education of the Russian Federation (Grant No. 075-15-2022-1121). - ISSN 2227-7390, DOI 10.3390/math11234715.

Аннотация: The main results of the study of the convergence rate of quasi-Newton minimization methods were obtained under the assumption that the method operates in the region of the extremum of the function, where there is a stable quadratic representation of the function. Methods based on the quadratic model of the function in the extremum area show significant advantages over classical gradient methods. When solving a specific problem using the quasi-Newton method, a huge number of iterations occur outside the extremum area, unless there is a stable quadratic approximation of the function. In this paper, we study the convergence rate of quasi-Newton-type methods on strongly convex functions with a Lipschitz gradient, without using local quadratic approximations of a function based on the properties of its Hessian. We proved that quasi-Newton methods converge on strongly convex functions with a Lipschitz gradient with the rate of a geometric progression, while the estimate of the convergence rate improves with the increasing number of iterations, which reflects the fact that the learning (adaptation) effect accumulates as the method operates. Another important fact discovered during the theoretical study is the ability of quasi-Newton methods to eliminate the background that slows down the convergence rate. This elimination is achieved through a certain linear transformation that normalizes the elongation of function level surfaces in different directions. All studies were carried out without any assumptions regarding the matrix of second derivatives of the function being minimized.

Год издания: 2023

Авторы: Krutikov Vladimir , Tovbis Elena , Stanimirović Predrag , Kazakovtsev Lev

Источник: Mathematics

Выпуск: Т. 11, № 23

Номера страниц: 4715

Держатель оригинала документа: Kemerovo State University, Reshetnev Siberian State University of Science and Technology, Siberian Federal University, University of Niš

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On the Convergence Rate of Quasi-Newton Methods onStrongly Convex Functions with Lipschitz Gradient : научное издание
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