First- and Second-Order High Probability Complexity Bounds for Trust-Region Methods with Noisy Oracles

TL;DR

This paper develops convergence guarantees with high probability bounds for a modified trust-region method that handles noisy, biased, and inconsistent stochastic oracles, extending the theoretical understanding of stochastic optimization.

Contribution

It introduces a new trust-region algorithm with relaxed acceptance criteria and analyzes its convergence under more general noisy oracle conditions, providing high probability complexity bounds.

Findings

Exponential tail bounds on iteration complexity for convergence.

Algorithm performs well on noisy derivative-free optimization problems.

Theoretical results are validated through numerical experiments.

Abstract

In this paper, we present convergence guarantees for a modified trust-region method designed for minimizing objective functions whose value and gradient and Hessian estimates are computed with noise. These estimates are produced by generic stochastic oracles, which are not assumed to be unbiased or consistent. We introduce these oracles and show that they are more general and have more relaxed assumptions than the stochastic oracles used in prior literature on stochastic trust-region methods. Our method utilizes a relaxed step acceptance criterion and a cautious trust-region radius updating strategy which allows us to derive exponentially decaying tail bounds on the iteration complexity for convergence to points that satisfy approximate first- and second-order optimality conditions. Finally, we present two sets of numerical results. We first explore the tightness of our theoretical results on an example with adversarial zeroth- and first-order oracles. We then investigate the performance of the modified trust-region algorithm on standard noisy derivative-free optimization problems.