Expected complexity analysis of stochastic direct-search

This work presents the convergence rate analysis of stochastic variants of the broad class of direct-search methods of directional type. It introduces an algorithm designed to optimize differentiable objective functions f whose values can only be computed through a stochastically noisy blackbox. The proposed stochastic directional direct-search (SDDS) algorithm accepts new iterates by imposing a sufficient decrease condition on so called probabilistic estimates of the corresponding unavailable objective function values. The accuracy of such estimates is required to hold with a sufficiently large but fixed probability beta. The analysis of this method utilizes an existing supermartingale-based framework proposed for the convergence rates analysis of stochastic optimization methods that use adaptive step sizes. It aims to show that the expected number of iterations required to drive the norm of the gradient of f below a given threshold epsilon is bounded in O((epsilon) -p/min(p-1,1) /(2 beta - 1)) with p > 1. Unlike prior analysis using the same aforementioned framework such as those of stochastic trust-region methods and stochastic line search methods, SDDS does not use any gradient information to find descent directions. However, its convergence rate is similar to those of both latter methods with a dependence on epsilon that also matches that of the broad class of deterministic directional direct-search methods which accept new iterates by imposing a sufficient decrease condition.

Automated summary

This work introduces an algorithm designed to optimize differentiable objective functions computed through a stochastically noisy blackbox. The analysis aims to show the expected number of iterations required to drive the norm of the gradient below a given threshold. The proposed method's convergence rate is similar to other stochastic optimization methods and deterministic direct-search methods with a dependence on epsilon.