Ускоренный спуск по случайному направлению с неевклидовой прокс-структурой

International audienceРассматриваются задачи гладкой выпуклой оптимизации,. для численногорешения которых полный градиент недоступен. В 2011 г. Ю.Е. Нестеровымбыли предложены ускоренные безградиентные методы решения таких задач.Поскольку рассматривались только задачи безусловной оптимизации, то ис-пользовалась евклидова прокс-структура. Однако если заранее знать, например,что решение задачи разреженно, а точнее, что расстояние от точки старта дорешения в 1-норме и в 2-норме близки, то более выгодно выбирать не евклидовупрокс-структуру, связанную с 2-нормой, а прокс-структуру, связанную с 1-нормой. Полное обоснование этого утверждения проводится в статье. Предла-гается ускоренный метод спуска по случайному направлению с неевклидовойпрокс-структурой для решения задачи безусловной оптимизации (в дальней-шем подход предполагается расширить на ускоренный безградиентный метод).Получены оценки скорости сходимости метода. Показаны сложности переносаописанного подхода на задачи условной оптимизации

Gradient-Free Methods for Saddle-Point Problem

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The vast majority of zeroth order optimization methods try to imitate first order methods via some smooth approximation of the gradient. Here, the smaller the smoothing parameter, the smaller the gradient approximation error. We show that for the majority of zeroth order methods this smoothing parameter can however not be chosen arbitrarily small as numerical cancellation errors will dominate. As such, theoretical and numerical performance could differ significantly. Using classical tools from numerical differentiation we will propose a new smoothed approximation of the gradient that can be integrated into general zeroth order algorithmic frameworks. Since the proposed smoothed approximation does not suffer from cancellation errors, the smoothing parameter (and hence the approximation error) can be made arbitrarily small. Sublinear convergence rates for algorithms based on our smoothed approximation are proved. Numerical experiments are also presented to demonstrate the superiority of algorithms based on the proposed approximation.Comment: New: Figure 3.