On Adaptive Sketch-and-Project for Solving Linear Systems

We generalize the concept of adaptive sampling rules to the sketch-and-project method for solving linear systems. Analyzing adaptive sampling rules in the sketch-and-project setting yields convergence results that apply to all special cases at once, including the Kaczmarz and coordinate descent. This eliminates the need to separately analyze analogous adaptive sampling rules in each special case. To deduce new sampling rules, we 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 [Z.-Z. Bai and W.-T. Wu, SIAM J. Sci. Comput., 40 (2018), pp. A592--A606]. We provide a global exponential convergence theorem for each sampling rule and show that the max-distance sampling 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 product of the number of sketches with the sketch size is not significantly larger than the number of columns.