“Convex Until Proven Guilty”: Dimension-Free Acceleration of Gradient Descent on Non-Convex Functions
Yair Carmon, John C. Duchi, Oliver Hinder, Aaron Sidford
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.
BibTeX
@InProceedings{pmlr-v70-carmon17a,
title = {``{C}onvex Until Proven Guilty'': Dimension-Free Acceleration of Gradient Descent on Non-Convex Functions},
author = {Yair Carmon and John C. Duchi and Oliver Hinder and Aaron Sidford},
booktitle = {Proceedings of the 34th International Conference on Machine Learning},
pages = {654--663},
year = {2017},
editor = {Precup, Doina and Teh, Yee Whye},
volume = {70},
series = {Proceedings of Machine Learning Research},
month = {06--11 Aug},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v70/carmon17a/carmon17a.pdf},
url = {https://proceedings.mlr.press/v70/carmon17a.html},
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.}
}