On the Second-order Convergence Properties of Random Search Methods

Aurelien Lucchi, ANTONIO ORVIETO, Adamos Solomou

Keywords: random search, derivative-free, optimization, non-convex, saddle

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.

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Code: https://github.com/adamsolomou/second-order-random-search

Community Implementations: [ 1 code implementation](https://www.catalyzex.com/paper/arxiv:2110.13265/code)