Stability of symmetrically preconditioned conjugate gradient

Prove that when the upper-triangular preconditioner R is obtained by QR factorizing the sketched matrix S A (with S a sketching matrix for A) and the linear system M y = c is formed with M := R^{-T} A^T A R^{-1} and c := R^{-T} A^T b, the conjugate gradient algorithm applied to M y = c in floating-point arithmetic returns a computed vector ŷ whose forward error satisfies a finite-precision stability bound of the form ||ŷ − M^{-1} c|| ≤ C · cond(A) · u · ||c|| for a modest constant C, where u is the unit roundoff. Establishing this bound (the inner-solver guarantee used in the paper) would validate the stability of using conjugate gradient as the inner solver within the proposed randomized least-squares framework.

Background

The paper introduces SPIR (sketch-and-precondition with iterative refinement) as an empirically backward stable variant of sketch-and-precondition. Their meta-stability theorem shows that if the inner solver for the symmetrically preconditioned normal-equation system M y = c achieves a forward error bounded by O(cond(A) * u * ||c||, then two refinement steps yield a backward stable least-squares solution.

To extend SPIR from LSQR to conjugate gradient, the authors formulate a conjecture asserting that conjugate gradient applied to the preconditioned system with R coming from the QR factorization of the sketched matrix S A attains the same inner-solver stability guarantee. They note a substantial gap between empirical behavior and current error bounds for CG/Lanczos in finite precision, and argue that proving this conjecture would yield a fast, parameter-free, backward stable randomized least-squares solver.