Iteration Complexity and Finite-Time Efficiency of Adaptive Sampling Trust-Region Methods for Stochastic Derivative-Free Optimization

TL;DR

This paper introduces an improved adaptive sampling trust-region method for stochastic derivative-free optimization, achieving optimal iteration complexity without strong assumptions and demonstrating practical computational benefits.

Contribution

The paper develops local diagonal Hessian models and integrates direct search to enhance ASTRO-DF's finite-time performance and remove dependence on problem dimension.

Findings

Proves $ ext{O}(rac{1}{ ext{epsilon}^2})$ iteration complexity almost surely.

Shows computational advantages of coordinate-based direct search in early iterations.

Demonstrates improved practical performance over existing methods.

Abstract

Adaptive sampling with interpolation-based trust regions or ASTRO-DF is a successful algorithm for stochastic derivative-free optimization with an easy-to-understand-and-implement concept that guarantees almost sure convergence to a first-order critical point. To reduce its dependence on the problem dimension, we present local models with diagonal Hessians constructed on interpolation points based on a coordinate basis. We also leverage the interpolation points in a direct search manner whenever possible to boost ASTRO-DF's performance in a finite time. We prove that the algorithm has a canonical iteration complexity of $\mathcal{O}(\epsilon^{-2})$ almost surely, which is the first guarantee of its kind without placing assumptions on the quality of function estimates or model quality or independence between them. Numerical experimentation reveals the computational advantage of ASTRO-DF with coordinate direct search due to saving and better steps in the early iterations of the search.