Abstract

Recently, it has been shown how, on the basis of the usual accelerated gradient method for solving problems of smooth convex optimization, accelerated methods for more complex problems (with a structure) and problems that are solved using various local information about the behavior of a function (stochastic gradient, Hessian, etc.) can be obtained. The term “accelerated methods” here means, on the one hand, the presence of some unified and fairly general way of acceleration. On the other hand, this also means the optimality of the methods, which can often be proved rigorously. In the present work, an attempt is made to construct in the same way a theory of accelerated methods for solving smooth convex-concave saddle-point problems with a structure. The main result of this article is the obtainment of in some sense necessary and sufficient conditions under which the complexity of solving nonlinear convex-concave saddle-point problems with a structure in the number of calculations of the gradients of composites in direct variables is equal in order of magnitude to the complexity of solving bilinear problems with a structure.

中文翻译：

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鞍点问题的加速方法

摘要

最近，已经显示出如何在通常的用于解决平滑凸优化问题的加速梯度方法的基础上，针对更复杂的问题（具有结构）的加速方法，以及如何使用有关行为的各种局部信息解决的问题函数（随机梯度，Hessian等）可以获得。术语“加速方法”在这里一方面意味着存在某种统一且相当普遍的加速方式。另一方面，这也意味着方法的最优性，通常可以严格证明这一点。在本工作中，尝试以相同的方式构造用于解决具有结构的光滑凸凹鞍点问题的加速方法的理论。