Quasi-Newton Methods for Saddle Point Problems
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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 𝒪((1-1/nκ^2)^k(k-1)/2), where n is dimensions of the problem, κ 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 𝒪((1-1/n)^k(k-1)/2).
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