Iteration Complexity of Fixed-Step Methods by Nesterov and Polyak for Convex Quadratic Functions

TL;DR

This paper analyzes the iteration complexity of Polyak's momentum and Nesterov's accelerated gradient methods on convex quadratic functions, providing explicit bounds and demonstrating their optimality and relationship.

Contribution

It derives explicit iteration bounds for both methods on quadratic functions, showing their optimality and clarifying their connection.

Findings

Bounds are optimal up to a constant factor.

Distance to the solution can be non-monotone.

Establishes a link between the momentum and Nesterov's methods.

Abstract

This note considers the momentum method by Polyak and the accelerated gradient method by Nesterov, both without line search but with fixed step length applied to strictly convex quadratic functions assuming that exact gradients are used and appropriate upper and lower bounds for the extreme eigenvalues of the Hessian matrix are known. Simple 2-d-examples show that the Euclidean distance of the iterates to the optimal solution is non-monotone. In this context an explicit bound is derived on the number of iterations needed to guarantee a reduction of the Euclidean distance to the optimal solution by a factor $\varepsilon$. For both methods the bound is optimal up to a constant factor, it complements earlier asymptotically optimal results for the momentum method, and it establishes another link of the momentum method and Nesterov's accelerated gradient method.