An optimal scheduled learning rate for a randomized Kaczmarz algorithm

Abstract

We study how the learning rate affects the performance of a relaxed randomized Kaczmarz algorithm for solving $Ax\approx b+\epsilon$, where $Ax=b$ is a consistent linear system and $\epsilon$ 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.

arXiv comment19 pages, 7 figures