Complexity of Minimizing Regularized Convex Quadratic Functions

TL;DR

This paper analyzes the iteration complexity of gradient methods for convex quadratic functions with regularization, establishing optimal convergence rates and tight bounds for various classes of methods, especially in the context of cubic regularization.

Contribution

It provides new convergence rate bounds for gradient and multi-step methods on regularized convex quadratics, including tight lower bounds and optimality results.

Findings

Gradient descent achieves $O(N^{-p/(p-2)})$ convergence rate.

Fast gradient method attains the optimal $O(N^{-2p/(p-2)})$ rate.

For cubic regularization ($p=3$), the optimal rate is $O(N^{-6})$.

Abstract

In this work, we study the iteration complexity of gradient methods for minimizing convex quadratic functions regularized by powers of Euclidean norms. We show that, due to the uniform convexity of the objective, gradient methods have improved convergence rates. Thus, for the basic gradient descent with a novel step size, we prove a convergence rate of $O(N^{-p/(p - 2)})$ for the functional residual, where $N$ is the iteration number and $p > 2$ is the power of the regularization term. We also show that this rate is tight by establishing a corresponding lower bound for one-step first-order methods. Then, for the general class of all multi-step methods, we establish that the rate of $O(N^{-2p/(p-2)})$ is optimal, providing a sharp analysis of the minimization of uniformly convex regularized quadratic functions. This rate is achieved by the fast gradient method. A special case of our problem class is $p=3$, which is the minimization of cubically regularized convex quadratic functions. It naturally appears as a subproblem at each iteration of the cubic Newton method. Therefore, our theory shows that the rate of $O(N^{-6})$ is optimal in this case. We also establish new lower bounds on minimizing the gradient norm within our framework.