Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Inexact Restoration via random models for unconstrained noisy optimization

Benedetta Morini, Simone Rebegoldi

TL;DR

This work tackles unconstrained optimization where objective and gradient evaluations are stochastic by recasting the problem into a probabilistic Inexact Restoration framework with random models. The proposed irerm algorithm blends a trust-region method with random first-order models and a probabilistic accuracy mechanism to guarantee meaningful progress toward optimality while controlling infeasibility via a constraint $h(y)=0$. The authors establish that the penalty parameter remains well-behaved, true iterations occur with fixed probability, and the expected iteration count to reach a prescribed gradient accuracy scales as $O(1/\epsilon^2)$. Numerical experiments on noisy nonlinear least-squares problems show irerm achieving competitive results with state-of-the-art methods, sometimes outperforming them on best runs, and illustrating the practical viability of the constrained IR approach for stochastic optimization.

Abstract

We study the Inexact Restoration framework with random models for minimizing functions whose evaluation is subject to errors. We propose a constrained formulation that includes well-known stochastic problems and an algorithm applicable when the evaluation of both the function and its gradient is random and a specified accuracy of such evaluations is guaranteed with sufficiently high probability. The proposed algorithm combines the Inexact Restoration framework with a trust-region methodology based on random first-order models. We analyse the properties of the algorithm and provide the expected number of iterations performed to reach an approximate first-order optimality point. Numerical experiments show that the proposed algorithm compares well with a state-of-the-art competitor.

Inexact Restoration via random models for unconstrained noisy optimization

TL;DR

This work tackles unconstrained optimization where objective and gradient evaluations are stochastic by recasting the problem into a probabilistic Inexact Restoration framework with random models. The proposed irerm algorithm blends a trust-region method with random first-order models and a probabilistic accuracy mechanism to guarantee meaningful progress toward optimality while controlling infeasibility via a constraint . The authors establish that the penalty parameter remains well-behaved, true iterations occur with fixed probability, and the expected iteration count to reach a prescribed gradient accuracy scales as . Numerical experiments on noisy nonlinear least-squares problems show irerm achieving competitive results with state-of-the-art methods, sometimes outperforming them on best runs, and illustrating the practical viability of the constrained IR approach for stochastic optimization.

Abstract

We study the Inexact Restoration framework with random models for minimizing functions whose evaluation is subject to errors. We propose a constrained formulation that includes well-known stochastic problems and an algorithm applicable when the evaluation of both the function and its gradient is random and a specified accuracy of such evaluations is guaranteed with sufficiently high probability. The proposed algorithm combines the Inexact Restoration framework with a trust-region methodology based on random first-order models. We analyse the properties of the algorithm and provide the expected number of iterations performed to reach an approximate first-order optimality point. Numerical experiments show that the proposed algorithm compares well with a state-of-the-art competitor.
Paper Structure (10 sections, 9 theorems, 83 equations, 2 figures, 3 tables, 1 algorithm)

This paper contains 10 sections, 9 theorems, 83 equations, 2 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. An inexact restoration trust region method with random models
  4. Analysis of the proposed algorithm
  5. On the sequence $\{\theta_k\}$
  6. Successful iterations
  7. Iteration complexity
  8. Numerical experiments
  9. Conclusions and future work
  10. Appendix

Key Result

Lemma 4.2

Suppose Assumption ass_fh holds. The following facts hold true.

Figures (2)

  • Figure 1: Best runs of irerm_v1 and storm_v1 in terms of value of $f$ at termination. From left to right: value of true function $f$ vs computational budget and trust-region radius $\delta_k$ vs computational budget. From top to bottom: Problems p5, p10, p11, p12.
  • Figure 2: Best runs of irerm_v2 and storm_v2 in terms of value of $f$ at termination. From left to right: value of true function $f$ vs computational budget and trust-region radius $\delta_k$ vs computational budget. From top to bottom: Problems p5, p10, p11, p12.

Theorems & Definitions (23)

  • Definition 4.1
  • Lemma 4.2
  • proof
  • Definition 4.3
  • Lemma 4.4
  • proof
  • Lemma 4.6
  • proof
  • Lemma 4.7
  • proof
  • ...and 13 more