Polynomial Preconditioning for Gradient Methods

We study first-order methods with preconditioning for solving structured convex optimization problems. We propose a new family of preconditioners generated by the symmetric polynomials. They provide the first-order optimization methods with a provable improvement of the condition number, cutting the gaps between highest eigenvalues, without explicit knowledge of the actual spectrum. We give a stochastic interpretation of this preconditioning in terms of the coordinate volume sampling and compare it with other classical approaches, including the Chebyshev polynomials. We show how to incorporate a polynomial preconditioning into the Gradient and Fast Gradient Methods and establish their better global complexity bounds. Finally, we propose a simple adaptive search procedure that automatically ensures the best polynomial preconditioning for the Gradient Method, minimizing the objective along a low-dimensional Krylov subspace. Numerical experiments confirm the efficiency of our preconditioning strategies for solving various machine learning problems.