An Accelerated Randomized Kaczmarz Algorithm

Abstract

The randomized Kaczmarz ($\RK$) algorithm is a simple but powerful approach for solving consistent linear systems $Ax=b$. This paper proposes an accelerated randomized Kaczmarz ($\ARK$) algorithm with better convergence than the standard $\RK$ algorithm on ill conditioned problems. The per-iteration cost of $\RK$ and $\ARK$ are similar if $A$ is dense, but $\RK$ is much more able to exploit sparsity in $A$ than is $\ARK$. To deal with the sparse case, an efficient implementation for $\ARK$, called $\SARK$, is proposed. A comparison of convergence rates and average per-iteration complexities among $\RK$, $\ARK$, and $\SARK$ is given, taking into account different levels of sparseness and conditioning. Comparisons with the leading deterministic algorithm --- conjugate gradient applied to the normal equations --- are also given. Finally, the analysis is validated via computational testing.