  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. 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.
