"Convex Until Proven Guilty": Dimension-Free Acceleration of Gradient Descent on Non-Convex Functions

TL;DR

This paper introduces a dimension-free accelerated gradient descent method for non-convex functions that either converges quickly or provides a certificate of non-convexity, enabling efficient optimization.

Contribution

It presents a novel AGD variant that detects non-convexity and exploits negative curvature for faster convergence on non-convex functions.

Findings

Achieves dimension-free acceleration for non-convex optimization.

Provides convergence guarantees with specific evaluation complexities.

Offers a non-convexity certificate to guide optimization strategies.

Abstract

We develop and analyze a variant of Nesterov's accelerated gradient descent (AGD) for minimization of smooth non-convex functions. We prove that one of two cases occurs: either our AGD variant converges quickly, as if the function was convex, or we produce a certificate that the function is "guilty" of being non-convex. This non-convexity certificate allows us to exploit negative curvature and obtain deterministic, dimension-free acceleration of convergence for non-convex functions. For a function $f$ with Lipschitz continuous gradient and Hessian, we compute a point $x$ with $\|\nabla f(x)\| \le \epsilon$ in $O(\epsilon^{-7/4} \log(1/ \epsilon) )$ gradient and function evaluations. Assuming additionally that the third derivative is Lipschitz, we require only $O(\epsilon^{-5/3} \log(1/ \epsilon) )$ evaluations.