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This paper studies quasi-Newton methods for solvingstrongly-convex-strongly-concave saddle point problems (SPP). We propose greedyand random Broyden family updates for SPP, which have explicit localsuperlinear convergence rate of ${\mathcalO}\big(\big(1-\frac{1}{n\kappa^2}\big)^{k(k-1)/2}\big)$, where $n$ isdimensions of the problem, $\kappa$ is the condition number and $k$ is thenumber of iterations. The design and analysis of proposed algorithm are basedon estimating the square of indefinite Hessian matrix, which is different fromclassical quasi-Newton methods in convex optimization. We also present twospecific Broyden family algorithms with BFGS-type and SR1-type updates, whichenjoy the faster local convergence rate of $\mathcalO\big(\big(1-\frac{1}{n}\big)^{k(k-1)/2}\big)$. Additionally, we extend ouralgorithms to solve general nonlinear equations and prove it enjoys the similarconvergence rate.

Quasi-Newton Methods for Saddle Point Problems and Beyond