Regularized Momentum Iterative Hessian Sketch for Large Scale Linear System of Equations

In this article, Momentum Iterative Hessian Sketch (M-IHS) techniques, a group of solvers for large scale linear Least Squares (LS) problems, are proposed and analyzed in detail. The proposed techniques are obtained by incorporating the Heavy Ball Acceleration into the Iterative Hessian Sketch algorithm and they provide significant improvements over the randomized preconditioning techniques. Through the error analyses of the M-IHS variants, lower bounds on the sketch size for various randomized distributions to converge at a pre-determined rate with a constant probability are established. The bounds present the best results in the current literature for obtaining a solution approximation and they suggest that the sketch size can be chosen proportional to the statistical dimension of the regularized problem regardless of the size of the coefficient matrix. The statistical dimension is always smaller than the rank and it gets smaller as the regularization parameter increases. By using approximate solvers along with the iterations, the M-IHS variants 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 M-IHS variants do not use any inner products and eliminate the corresponding synchronizations steps in hierarchical or distributed memory systems, yet the M-IHS converges faster than the Chebyshev Semi-iteration based solvers.