Randomized Kaczmarz algorithms: Exact MSE analysis and optimal sampling probabilities

Abstract

The Kaczmarz method, or the algebraic reconstruction technique (ART), is a popular method for solving large-scale overdetermined systems of equations. Recently, Strohmer et al. proposed the randomized Kaczmarz algorithm, an improvement that guarantees exponential convergence to the solution. This has spurred much interest in the algorithm and its extensions. We provide in this paper an exact formula for the mean squared error (MSE) in the value reconstructed by the algorithm. We also compute the exponential decay rate of the MSE, which we call the “annealed” error exponent. We show that the typical performance of the algorithm is far better than the average performance. We define the “quenched” error exponent to characterize the typical performance. This is far harder to computethan the annealed error exponent, but we provide an approximation that matches empirical results. We also explore optimizing the algorithm’s row-selection probabilities to speed up the algorithm’s convergence.