An optimal scheduled learning rate for a randomized Kaczmarz algorithm

We study how the learning rate affects the performance of a relaxed randomized Kaczmarz algorithm for solving A x ≈ b + varepsilon, where A x =b is a consistent linear system and varepsilon has independent mean zero random entries. We derive a learning rate schedule which optimizes a bound on the expected error that is sharp in certain cases; in contrast to the exponential convergence of the standard randomized Kaczmarz algorithm, our optimized bound involves the reciprocal of the Lambert-W function of an exponential.