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Greedy Quasi-Newton Methods with Explicit Superlinear Convergence
In this paper, we study greedy variants of quasi-Newton methods. They are based on the updating formulas from a certain subclass of the Broyden family. In particular, this subclass includes the well-known DFP, BFGS, and SR1 updates. However, in contrast to the classical quasi-Newton methods, which use the difference of successive iterates for updating the Hessian approximations, our methods apply basis vectors, greedily selected so as to maximize a certain measure of progress. For greedy quasi-Newton methods, we establish an explicit nonasymptotic bound on their rate of local superlinear convergence, as applied to minimizing strongly convex and strongly self-concordant functions (and, in particular, to strongly convex functions with Lipschitz continuous Hessian). The established superlinear convergence rate contains a contraction factor, which depends on the square of the iteration counter. We also show that greedy quasi-Newton methods produce Hessian approximations whose deviation from the exact Hessians linearly converges to zero.
Read More: https://epubs.siam.org/doi/abs/10.1137/20M1320651
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