Stochastic Trust-Region and Direct-Search Methods: A Weak Tail Bound Condition and Reduced Sample Sizing

Abstract

SIAM Journal on Optimization, Volume 34, Issue 2, Page 2067-2092, June 2024.
Abstract. Using tail bounds, we introduce a new probabilistic condition for function estimation in stochastic derivative-free optimization (SDFO) which leads to a reduction in the number of samples and eases algorithmic analyses. Moreover, we develop simple stochastic direct-search and trust-region methods for the optimization of a potentially nonsmooth function whose values can only be estimated via stochastic observations. For trial points to be accepted, these algorithms require the estimated function values to yield a sufficient decrease measured in terms of a power larger than 1 of the algoritmic stepsize. Our new tail bound condition is precisely imposed on the reduction estimate used to achieve such a sufficient decrease. This condition allows us to select the stepsize power used for sufficient decrease in such a way that the number of samples needed per iteration is reduced. In previous works, the number of samples necessary for global convergence at every iteration [math] of this type of algorithm was [math], where [math] is the stepsize or trust-region radius. However, using the new tail bound condition, and under mild assumptions on the noise, one can prove that such a number of samples is only [math], where [math] can be made arbitrarily small by selecting the power of the stepsize in the sufficient decrease test arbitrarily close to 1. In the common random number generator setting, a further improvement by a factor of [math] can be obtained. The global convergence properties of the stochastic direct-search and trust-region algorithms are established under the new tail bound condition.