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

Yair Carmon, John C. Duchi, Oliver Hinder, Aaron Sidford
Proceedings of the 34th International Conference on Machine Learning, PMLR 70:654-663, 2017.

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.

Cite this Paper

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.} }
Endnote
%0 Conference Paper %T “Convex Until Proven Guilty”: Dimension-Free Acceleration of Gradient Descent on Non-Convex Functions %A Yair Carmon %A John C. Duchi %A Oliver Hinder %A Aaron Sidford %B Proceedings of the 34th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2017 %E Doina Precup %E Yee Whye Teh %F pmlr-v70-carmon17a %I PMLR %P 654--663 %U https://proceedings.mlr.press/v70/carmon17a.html %V 70 %X 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.
APA
Carmon, Y., Duchi, J.C., Hinder, O. & Sidford, A.. (2017). “Convex Until Proven Guilty”: Dimension-Free Acceleration of Gradient Descent on Non-Convex Functions. Proceedings of the 34th International Conference on Machine Learning, in Proceedings of Machine Learning Research 70:654-663 Available from https://proceedings.mlr.press/v70/carmon17a.html.

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