Parameter-free accelerated gradient descent for nonconvex minimization

TL;DR

This paper introduces a parameter-free accelerated gradient descent method for nonconvex functions that achieves optimal complexity bounds without prior knowledge of problem parameters, using novel Hessian estimation techniques.

Contribution

It presents a new parameter-free first-order method with restart mechanisms for nonconvex optimization, avoiding the need for problem-dependent parameter tuning.

Findings

Achieves $O( ext{epsilon}^{-7/4})$ complexity for finding approximate stationary points.

Performs comparably to existing methods in numerical experiments.

Introduces Hessian-free analysis techniques for parameter estimation.

Abstract

We propose a new first-order method for minimizing nonconvex functions with a Lipschitz continuous gradient and Hessian. The proposed method is an accelerated gradient descent with two restart mechanisms and finds a solution where the gradient norm is less than $\epsilon$ in $O(\epsilon^{-7/4})$ function and gradient evaluations. Unlike existing first-order methods with similar complexity bounds, our algorithm is parameter-free because it requires no prior knowledge of problem-dependent parameters, e.g., the Lipschitz constants and the target accuracy $\epsilon$. The main challenge in achieving this advantage is estimating the Lipschitz constant of the Hessian using only first-order information. To this end, we develop a new Hessian-free analysis based on two technical inequalities: a Jensen-type inequality for gradients and an error bound for the trapezoidal rule. Several numerical results illustrate that the proposed method performs comparably to existing algorithms with similar complexity bounds, even without parameter tuning.