Adaptive iterative Hessian sketch via A-optimal subsampling

Abstract

Iterative Hessian sketch (IHS) is an effective sketching method for modeling large-scale data. It was originally proposed by Pilanci and Wainwright (J Mach Learn Res 17(1):1842–1879, 2016) based on randomized sketching matrices. However, it is computationally intensive due to the iterative sketch process. In this paper, we analyze the IHS algorithm under the unconstrained least squares problem setting and then propose a deterministic approach for improving IHS via A-optimal subsampling. Our contributions are threefold: (1) a good initial estimator based on the A-optimal design is suggested; (2) a novel ridged preconditioner is developed for repeated sketching; and (3) an exact line search method is proposed for determining the optimal step length adaptively. Extensive experimental results demonstrate that our proposed A-optimal IHS algorithm outperforms the existing accelerated IHS methods.