Complexity-optimal and parameter-free first-order methods for finding stationary points of composite optimization problems

TL;DR

This paper introduces a parameter-free accelerated proximal descent method for nonconvex composite optimization that achieves optimal iteration complexity without requiring knowledge of Lipschitz constants, supported by theoretical analysis and numerical experiments.

Contribution

It presents a novel parameter-free method with optimal complexity bounds for finding stationary points in nonconvex composite optimization problems.

Findings

Achieves optimal iteration complexity bounds up to logarithmic factors.

Does not require prior knowledge of Lipschitz constants.

Demonstrates practical viability through numerical experiments.

Abstract

This paper develops and analyzes an accelerated proximal descent method for finding stationary points of nonconvex composite optimization problems. The objective function is of the form $f+h$ where $h$ is a proper closed convex function, $f$ is a differentiable function on the domain of $h$, and $\nabla f$ is Lipschitz continuous on the domain of $h$. The main advantage of this method is that it is "parameter-free" in the sense that it does not require knowledge of the Lipschitz constant of $\nabla f$ or of any global topological properties of $f$. It is shown that the proposed method can obtain an $\varepsilon$-approximate stationary point with iteration complexity bounds that are optimal, up to logarithmic terms over $\varepsilon$, in both the convex and nonconvex settings. Some discussion is also given about how the proposed method can be leveraged in other existing optimization frameworks, such as min-max smoothing and penalty frameworks for constrained programming, to create more specialized parameter-free methods. Finally, numerical experiments are presented to support the practical viability of the method.