Quasi-Newton Methods for Saddle Point Problems

• URL: http://arxiv.org/abs/2111.02708v1
• Date: Thu, 4 Nov 2021 09:34:00 GMT
• Title: Quasi-Newton Methods for Saddle Point Problems
• Abstract summary: We propose a variant general greedy Broyden family update for indefinite superlinear Obig-frac1nkappa2big. We also present two specific Broyden family algorithms with BFGtype SR1-type, which enjoy the faster local convergence rate.
• Score: 15.11757597288305
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: This paper studies quasi-Newton methods for solving strongly-convex-strongly-concave saddle point problems (SPP). We propose a variant of general greedy Broyden family update for SPP, which has explicit local superlinear convergence rate of ${\mathcal O}\big(\big(1-\frac{1}{n\kappa^2}\big)^{k(k-1)/2}\big)$, where $n$ is dimensions of the problem, $\kappa$ is the condition number and $k$ is the number of iterations. The design and analysis of proposed algorithm are based on estimating the square of indefinite Hessian matrix, which is different from classical quasi-Newton methods in convex optimization. We also present two specific Broyden family algorithms with BFGS-type and SR1-type updates, which enjoy the faster local convergence rate of $\mathcal O\big(\big(1-\frac{1}{n}\big)^{k(k-1)/2}\big)$.

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