Stochastic three points method for unconstrained smooth minimization
Type
ArticleKAUST Department
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) DivisionComputer Science Program
Date
2020-10-01Preprint Posting Date
2019-02-10Online Publication Date
2020-10-01Print Publication Date
2020-01Submitted Date
2019-02-12Permanent link to this record
http://hdl.handle.net/10754/653110
Metadata
Show full item recordAbstract
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:10.1137/19m1244378Sponsors
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)Journal
SIAM Journal on OptimizationarXiv
1902.03591Additional Links
https://epubs.siam.org/doi/10.1137/19M1244378ae974a485f413a2113503eed53cd6c53
10.1137/19M1244378