Restarts subject to approximate sharpness: A parameter-free and optimal scheme for first-order methods

TL;DR

This paper introduces a parameter-free restart scheme for first-order optimization methods that adapts to approximate sharpness, improving robustness and maintaining optimal convergence rates even with unknown problem-specific constants.

Contribution

The authors develop a novel restart scheme that works with approximate sharpness, does not require known constants, and applies broadly to various first-order methods.

Findings

Achieves optimal convergence rates for different first-order methods.

Works without knowledge of problem-specific sharpness constants.

Demonstrates robustness in noisy or relaxed model scenarios.

Abstract

Sharpness is an almost generic assumption in continuous optimization that bounds the distance from minima by objective function suboptimality. It facilitates the acceleration of first-order methods through restarts. However, sharpness involves problem-specific constants that are typically unknown, and restart schemes typically reduce convergence rates. Moreover, these schemes are challenging to apply in the presence of noise or with approximate model classes (e.g., in compressive imaging or learning problems), and they generally assume that the first-order method used produces feasible iterates. We consider the assumption of approximate sharpness, a generalization of sharpness that incorporates an unknown constant perturbation to the objective function error. This constant offers greater robustness (e.g., with respect to noise or relaxation of model classes) for finding approximate minimizers. By employing a new type of search over the unknown constants, we design a restart scheme that applies to general first-order methods and does not require the first-order method to produce feasible iterates. Our scheme maintains the same convergence rate as when the constants are known. The convergence rates we achieve for various first-order methods match the optimal rates or improve on previously established rates for a wide range of problems. We showcase our restart scheme in several examples and highlight potential future applications and developments of our framework and theory.