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 \(A x \approx b + \varepsilon\) , 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.Keywordslearning raterandomized Kaczmarzstochastic gradient descentMSC codes65F1060H2565F20