CG vs. Randomized Coordinate Descent for Polynomially Decaying Eigenvalues

Determine the asymptotic behavior, as n → ∞, of the stopping times T_CG(A_n,b_n,ε) and T_RCD(A_n,b_n,ε) for unpreconditioned conjugate gradient and randomized coordinate descent applied to the strictly positive definite linear system with A_n = U Λ_n U^*, where U is Haar unitary and Λ_n = diag(1^{-p},2^{-p},…,n^{-p}), and with right-hand side b_n = A_n z_n for z_n uniform on the unit sphere; quantify which method requires fewer epochs to reach ε-accuracy.

Background

The authors contrast classical Krylov methods with sketch-and-project methods, noting that their relative performance can depend delicately on the spectral distribution. They formulate a stylized random matrix model with polynomially decaying eigenvalues to probe this difference.

Understanding the comparative epoch complexity of CG and RCD on this model would clarify when modern randomized methods can outperform traditional Krylov solvers.