On the Second-order Convergence Properties of Random Search Methods
Aurelien Lucchi, Antonio Orvieto, Adamos Solomou
Abstract
We study the theoretical convergence properties of random-search methods when optimizing non-convex objective functions without having access to derivatives. We prove that standard random-search methods that do not rely on second-order information converge to a second-order stationary point. However, they suffer from an exponential complexity in terms of the input dimension of the problem. In order to address this issue, we propose a novel variant of random search that exploits negative curvature by only relying on function evaluations. We prove that this approach converges to a second-order stationary point at a much faster rate than vanilla methods: namely, the complexity in terms of the number of function evaluations is only linear in the problem dimension. We test our algorithm empirically and find good agreements with our theoretical results.
BibTeX
@inproceedings{
lucchi2021on,
title={On the Second-order Convergence Properties of Random Search Methods},
author={Aurelien Lucchi and Antonio Orvieto and Adamos Solomou},
booktitle={Advances in Neural Information Processing Systems},
editor={A. Beygelzimer and Y. Dauphin and P. Liang and J. Wortman Vaughan},
year={2021},
url={https://openreview.net/forum?id=1oR_gQGp3Rm}
}