Adaptive Sketch-and-Project Methods for Solving Linear Systems

Abstract

We present new adaptive sampling rules for the sketch-and-project method for solving linear systems. To deduce our new sampling rules, we first show how the progress of one step of the sketch-and-project method depends directly on a sketched residual. Based on this insight, we derive a 1) max-distance sampling rule, by sampling the sketch with the largest sketched residual 2) a proportional sampling rule, by sampling proportional to the sketched residual, and finally 3) a capped sampling rule. The capped sampling rule is a generalization of the recently introduced adaptive sampling rules for the Kaczmarz method [3]. We provide a global linear convergence theorem for each sampling rule and show that the max-distance rule enjoys the fastest convergence. This finding is also verified in extensive numerical experiments that lead us to conclude that the max-distance sampling rule is superior both experimentally and theoretically to the capped sampling rule. We also provide numerical insights into implementing the adaptive strategies so that the per iteration cost is of the same order as using a fixed sampling strategy when the number of sketches times the sketch size is not significantly larger than the number of columns.

A Weighted Randomized Kaczmarz Method for Solving Linear Systems

“…Related Results. This result is related to recent work of Gower, Molitor, Moorman and Needell [11] who discuss non-uniform selection probabilities in the more general framework of Sketch-and-Project methods. In particular, their §7.4.…”

“…discusses the convergence rate of the algorithm considered here for p = 1. The case p = ∞ has been studied by a large number of people; we emphasize the results of Nutini, Sepehey, Laradji, Schmidt, Koepke, Virani [29] and to references in [11].…”

Surrounding the solution of a Linear System of Equations from all sides

“…, the Projection onto Convex Sets Method[2,6,7,11,36] and the Randomized Kaczmarz method[8,9,13,15,14,22,23,24,26,27,29,30,31,32,33,34,35,37,38,39,40,41,42].Strohmer & Vershynin …”

Faster Randomized Block Sparse Kaczmarz by Averaging

“…They showed that as the number of threads increases, the rate of convergence improves and the convergence horizon for inconsistent systems decreases. Another more general class of block methods are sketch-and-project methods [4]. For a linear system Ax = b, sketch-and-project methods iteratively project the current iterate onto the solution space of a sketched subsystem S T Ax = S T b.…”