Gradient Methods with Memory for Minimizing Composite Functions

TL;DR

This paper introduces advanced Gradient Methods with Memory for efficiently solving composite optimization problems, including non-smooth cases, with improved convergence guarantees and practical restart strategies.

Contribution

The work extends Gradient Methods with Memory to composite problems, incorporating inexact auxiliary solutions and adaptive strategies to surpass previous convergence rates.

Findings

Achieves state-of-the-art worst-case rates for composite problems.

Provides a near-optimal restart strategy for sublinear convergence methods.

Supports theoretical results with simulation experiments.

Abstract

The recently introduced Gradient Methods with Memory use a subset of the past oracle information to create an accurate model of the objective function that enables them to surpass the Gradient Method in practical performance. The model introduces an overhead that is substantial on all problems but the smooth unconstrained ones. In this work, we introduce several Gradient Methods with Memory that can solve composite problems efficiently, including unconstrained problems with non-smooth objectives. The auxiliary problem at each iteration still cannot be solved exactly but we show how to alter the model and how to initialize the auxiliary problem solver to ensure that this inexactness does not degrade the convergence guarantees. Moreover, we dynamically increase the convergence guarantees as to provably surpass those of their memory-less counterparts. These properties are preserved when applying acceleration and the containment of inexactness further prevents error accumulation. Our methods are able to estimate key geometry parameters to attain state-of-the-art worst-case rates on many important subclasses of composite problems, where the objective smooth part satisfies a strong convexity condition or a relaxation thereof. In particular, we formulate a near-optimal restart strategy applicable to optimization methods with sublinear convergence guarantees of any order. We support the theoretical results with simulations.