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This paper develops and analyzes an accelerated proximal descent method forfinding stationary points of nonconvex composite optimization problems. Theobjective function is of the form $f+h$ where $h$ is a proper closed convexfunction, $f$ 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 methodis that it is "parameter-free" in the sense that it does not require knowledgeof the Lipschitz constant of $\nabla f$ or of any global topological propertiesof $f$. It is shown that the proposed method can obtain an$\varepsilon$-approximate stationary point with iteration complexity boundsthat are optimal, up to logarithmic terms over $\varepsilon$, in both theconvex and nonconvex settings. Some discussion is also given about how theproposed 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, numericalexperiments are presented to support the practical viability of the method.

Complexity-optimal and Parameter-free First-order Methods for Finding Stationary Points of Composite Optimization Problems