M-IHS: An accelerated randomized preconditioning method avoiding costly matrix decompositions

Abstract

Momentum Iterative Hessian Sketch (▪) techniques, a group of solvers for large scale regularized linear Least Squares (LS) problems, are proposed and analyzed in detail. The proposed ▪ techniques are obtained by incorporating Polyak's heavy ball acceleration into the Iterative Hessian Sketch algorithm and they provide significant improvements over the randomized preconditioning techniques. By solving the linear systems arising in the sub-problems during the iterations approximately, the proposed techniques are capable of avoiding all matrix decompositions and inversions, which is one of the main advantages over the alternative solvers such as the Blendenpik and the LSRN. Similar to the Chebyshev semi-iterations, the ▪ variants do not use any inner products and eliminate the corresponding synchronization steps in hierarchical or distributed memory systems, yet the ▪ converges faster than the Chebyshev semi-iteration based solvers. Lower bounds on the required sketch size for various randomized distributions are established through the error analyses. Unlike the previously proposed approaches to produce a solution approximation, the proposed ▪ techniques can use sketch sizes that are proportional to the statistical dimension which is always smaller than the rank of the coefficient matrix.