Trust-region algorithms: probabilistic complexity and intrinsic noise with applications to subsampling techniques

TL;DR

This paper introduces a trust-region algorithm capable of efficiently finding approximate minimizers of noisy functions, with proven probabilistic complexity bounds for various optimality orders, and discusses its application to subsampling techniques.

Contribution

It provides the first probabilistic complexity analysis for trust-region methods achieving arbitrary optimality orders under noise.

Findings

The algorithm finds an $ ext{ extit{epsilon}}$-approximate minimizer in $ ext{O}( ext{ extit{epsilon}}^{-(q+1)})$ evaluations.

Probabilistic assumptions ensure expected convergence for functions with intrinsic noise.

Degraded guarantees are available when assumptions fail, especially in first-order cases.

Abstract

A trust-region algorithm is presented for finding approximate minimizers of smooth unconstrained functions whose values and derivatives are subject to random noise. It is shown that, under suitable probabilistic assumptions, the new method finds (in expectation) an $\epsilon$-approximate minimizer of arbitrary order $ q \geq 1$ in at most $\mathcal{O}(\epsilon^{-(q+1)})$ inexact evaluations of the function and its derivatives, providing the first such result for general optimality orders. The impact of intrinsic noise limiting the validity of the assumptions is also discussed and it is shown that difficulties are unlikely to occur in the first-order version of the algorithm for sufficiently large gradients. Conversely, should these assumptions fail for specific realizations, then "degraded" optimality guarantees are shown to hold when failure occurs. These conclusions are then discussed and illustrated in the context of subsampling methods for finite-sum optimization.