Stochastic three points method for unconstrained smooth minimization

Type
Article

Authors
Bergou, El Houcine
Gorbunov, Eduard
Richtarik, Peter

KAUST Department
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Computer Science Program

Preprint Posting Date
2019-02-10

Online Publication Date
2020-10-01

Print Publication Date
2020-01

Date
2020-10-01

Submitted Date
2019-02-12

Abstract
In this paper we consider the unconstrained minimization problem of a smooth function in Rn in a setting where only function evaluations are possible. We design a novel randomized derivative-free algorithm-the stochastic three points (STP) method-and analyze its iteration complexity. At each iteration, STP generates a random search direction according to a certain fixed probability law. Our assumptions on this law are very mild: Roughly speaking, all laws which do not concentrate all measures on any halfspace passing through the origin will work. For instance, we allow for the uniform distribution on the sphere and also distributions that concentrate all measures on a positive spanning set. Although our approach is designed to not explicitly use derivatives, it covers some first order methods. For instance, if the probability law is chosen to be the Dirac distribution concentrated on the sign of the gradient, then STP recovers the signed gradient descent method. If the probability law is the uniform distribution on the coordinates of the gradient, then STP recovers the randomized coordinate descent method. The complexity of STP depends on the probability law via a simple characteristic closely related to the cosine measure which is used in the analysis of deterministic direct search (DDS) methods. Unlike in DDS, where O(n) (n is the dimension of x) function evaluations must be performed in each iteration in the worst case, our method only requires two new function evaluations per iteration. Consequently, while the complexity of DDS depends quadratically on n, our method depends linearly on n.

Citation
Bergou, E. H., Gorbunov, E., & Richtárik, P. (2020). Stochastic Three Points Method for Unconstrained Smooth Minimization. SIAM Journal on Optimization, 30(4), 2726–2749. doi:10.1137/19m1244378

Acknowledgements
This author received support from the AgreenSkills+ fellowship programme which has received funding from the EU’s Seventh Framework Programme under grant agreement No FP7-609398 (AgreenSkills+ contract)

Publisher
Society for Industrial & Applied Mathematics (SIAM)

Journal
SIAM Journal on Optimization

DOI
10.1137/19M1244378

arXiv
1902.03591

Additional Links
https://epubs.siam.org/doi/10.1137/19M1244378

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