Quantile-RK and Double Quantile-RK Error Horizon Analysis

Abstract

In solving linear systems of equations of the form $Ax=b$, corruptions present in $b$ affect stochastic iterative algorithms' ability to reach the true solution $x^{\ast }$ to the uncorrupted linear system. The randomized Kaczmarz method converges in expectation to $x^{\ast }$ up to an error horizon dependent on the conditioning of $A$ and the supremum norm of the corruption in $b$. To avoid this error horizon in the sparse corruption setting, previous works have proposed quantile-based adaptations that make iterative methods robust. Our work first establishes a new convergence rate for the quantile-based random Kaczmarz (qRK) and double quantile-based random Kaczmarz (dqRK) methods, which, under certain conditions, improves upon known bounds. We further consider the more practical setting in which the vector $b$ includes both non-sparse ``noise" and sparse ``corruption". Error horizon bounds for qRK and dqRK are derived and shown to produce a smaller error horizon compared to their non-quantile-based counterparts, further demonstrating the advantages of quantile-based methods.