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arXiv · 1611.00756

Accelerated Methods for Non-Convex Optimization

Yair Carmon·John C. Duchi·Oliver Hinder·Aaron Sidford
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Abstract

We present an accelerated gradient method for non-convex optimization problems with Lipschitz continuous first and second derivatives. The method requires time $O(ε^{-7/4} \log(1/ ε) )$ to find an $ε$-stationary point, meaning a point $x$ such that $\|\nabla f(x)\| \le ε$. The method improves upon the $O(ε^{-2} )$ complexity of gradient descent and provides the additional second-order guarantee that $\nabla^2 f(x) \succeq -O(ε^{1/2})I$ for the computed $x$. Furthermore, our method is Hessian free, i.e. it only requires gradient computations, and is therefore suitable for large scale applications.

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Yair Carmon, John C. Duchi, Oliver Hinder, Aaron Sidford. 2017-02-02. Accelerated Methods for Non-Convex Optimization. https://arxiv.org/abs/1611.00756

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