Gradient-Free Methods for Saddle-Point Problem

• URL: http://arxiv.org/abs/2005.05913v4
• Date: Sun, 11 Sep 2022 14:38:00 GMT
• Title: Gradient-Free Methods for Saddle-Point Problem
• Authors: Aleksandr Beznosikov, Abdurakhmon Sadiev, Alexander Gasnikov
• Abstract summary: We generalize the approach Gasnikov et. al., 2017, which allows to solve (stochastic) convex optimization problems with an inexact gradient-free oracle. Our approach reduces $fracnlog n$ times the required number of oracle calls. In the second part of the paper, we analyze the case when such an assumption cannot be made, we propose a general approach on how to modernize the method to solve this problem.
• Score: 125.99533416395765
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In the paper, we generalize the approach Gasnikov et. al, 2017, which allows to solve (stochastic) convex optimization problems with an inexact gradient-free oracle, to the convex-concave saddle-point problem. The proposed approach works, at least, like the best existing approaches. But for a special set-up (simplex type constraints and closeness of Lipschitz constants in 1 and 2 norms) our approach reduces $\frac{n}{\log n}$ times the required number of oracle calls (function calculations). Our method uses a stochastic approximation of the gradient via finite differences. In this case, the function must be specified not only on the optimization set itself, but in a certain neighbourhood of it. In the second part of the paper, we analyze the case when such an assumption cannot be made, we propose a general approach on how to modernize the method to solve this problem, and also we apply this approach to particular cases of some classical sets.

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