Restart of accelerated first order methods with linear convergence under a quadratic functional growth condition

TL;DR

This paper introduces a restart scheme for accelerated first order methods that achieves linear convergence under a quadratic functional growth condition, improving practical performance for a broad class of convex problems.

Contribution

The paper proposes a new restart scheme that guarantees linear convergence for accelerated methods under quadratic functional growth, extending applicability beyond strongly convex functions.

Findings

Achieves linear convergence under quadratic functional growth

Comparable worst-case convergence rate to optimal fixed-rate restart

Demonstrates effectiveness through a model predictive control case study

Abstract

Accelerated first order methods, also called fast gradient methods, are popular optimization methods in the field of convex optimization. However, they are prone to suffer from oscillatory behaviour that slows their convergence when medium to high accuracy is desired. In order to address this, restart schemes have been proposed in the literature, which seek to improve the practical convergence by suppressing the oscillatory behaviour. This paper presents a restart scheme for accelerated first order methods for which we show linear convergence under the satisfaction of a quadratic functional growth condition, thus encompassing a broad class of non-necessarily strongly convex optimization problems. Moreover, the worst-case convergence rate is comparable to the one obtained using a (generally non-implementable) optimal fixed-rate restart strategy. We compare the proposed algorithm with other restart schemes by applying them to a model predictive control case study.