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arXiv · 2103.00434

Solving smooth min-min and min-max problems by mixed oracle algorithms

Egor Gladin·Abdurakhmon Sadiev·Alexander Gasnikov·Pavel Dvurechensky·Aleksandr Beznosikov·Mohammad Alkousa
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Abstract

In this paper, we consider two types of problems that have some similarity in their structure, namely, min-min problems and min-max saddle-point problems. Our approach is based on considering the outer minimization problem as a minimization problem with inexact oracle. This inexact oracle is calculated via inexact solution of the inner problem, which is either minimization or a maximization problem. Our main assumptions are that the problem is smooth and the available oracle is mixed: it is only possible to evaluate the gradient w.r.t. the outer block of variables which corresponds to the outer minimization problem, whereas for the inner problem only zeroth-order oracle is available. To solve the inner problem we use accelerated gradient-free method with zeroth-order oracle. To solve the outer problem we use either inexact variant of Vaydya's cutting-plane method or a variant of accelerated gradient method. As a result, we propose a framework that leads to non-asymptotic complexity bounds for both min-min and min-max problems. Moreover, we estimate separately the number of first- and zeroth-order oracle calls which are sufficient to reach any desired accuracy.

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Egor Gladin, Abdurakhmon Sadiev, Alexander Gasnikov, Pavel Dvurechensky, Aleksandr Beznosikov, Mohammad Alkousa. 2021-02-28. Solving smooth min-min and min-max problems by mixed oracle algorithms. https://doi.org/10.1007/978-3-030-86433-0_2

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