A Simple Complexity Lower Bound for Solving $Ax=b$

Abstract

In this note, we provide a short and direct proof that approximately solving $Ax=b$ to relative error $\varepsilon$, where $A$ has condition number $\kappa$ and unrestricted dimension, requires $\Omega(\kappa\log(1/\varepsilon))$ matrix-vector multiplications in the worst case, even for randomized algorithms. This essentially recovers the lower bound of Derezi\'nski, Epperly and Meyer [2026] for this setting, whose elegant and more general approach inspired us to seek a short direct proof. A straightforward reduction implies the classical $\Omega(\sqrt{\kappa}\log(1/\varepsilon))$ lower bound for optimizing strongly convex quadratic functions, applicable to randomized algorithms.