Smoothed Analysis for the Conjugate Gradient Algorithm
This work provides theoretical guarantees for the practical efficiency of the conjugate gradient algorithm under realistic noise conditions, addressing a gap in smoothed analysis for iterative methods.
The paper establishes bounds on the convergence rate of the conjugate gradient algorithm for random perturbations of deterministic positive definite matrices, providing estimates for expected iteration counts in the smoothed analysis framework.
The purpose of this paper is to establish bounds on the rate of convergence of the conjugate gradient algorithm when the underlying matrix is a random positive definite perturbation of a deterministic positive definite matrix. We estimate all finite moments of a natural halting time when the random perturbation is drawn from the Laguerre unitary ensemble in a critical scaling regime explored in Deift et al. (2016). These estimates are used to analyze the expected iteration count in the framework of smoothed analysis, introduced by Spielman and Teng (2001). The rigorous results are compared with numerical calculations in several cases of interest.