StoMADS: Stochastic blackbox optimization using probabilistic estimates

TL;DR

StoMADS is a novel stochastic optimization algorithm that extends the deterministic MADS method to handle noisy blackbox functions, ensuring convergence to stationary points with probabilistic guarantees.

Contribution

It introduces a stochastic variant of MADS that uses probabilistic function estimates and proves convergence to stationary points under noise.

Findings

Converges to Clarke stationary points with probability one.

Uses probabilistic estimates with fixed accuracy and variance conditions.

Employs martingale theory for convergence proof.

Abstract

This work introduces StoMADS, a stochastic variant of the mesh adaptive direct-search (MADS) algorithm originally developed for deterministic blackbox optimization. StoMADS considers the unconstrained optimization of an objective function f whose values can be computed only through a blackbox corrupted by some random noise following an unknown distribution. The proposed method is based on an algorithmic framework similar to that of MADS and uses random estimates of function values obtained from stochastic observations since the exact deterministic computable version of f is not available. Such estimates are required to be accurate with a sufficiently large but fixed probability and satisfy a variance condition. The ability of the proposed algorithm to generate an asymptotically dense set of search directions is then exploited to show convergence to a Clarke stationary point of f with probability one, using martingale theory.