Stochastic Matrix-Free Equilibration

We present a novel method for approximately equilibrating a matrix $A \in {\bf R}^{m \times n}$ using only multiplication by $A$ and $A^T$. Our method is based on convex optimization and projected stochastic gradient descent, using an unbiased estimate of a gradient obtained by a randomized method. Our method provably converges in expectation with an $O(1/t)$ convergence rate and empirically gets good results with a small number of iterations. We show how the method can be applied as a preconditioner for matrix-free iterative algorithms such as LSQR and Chambolle-Cremers-Pock, substantially reducing the iterations required to reach a given level of precision. We also derive a novel connection between equilibration and condition number, showing that equilibration minimizes an upper bound on the condition number over all choices of row and column scalings.