Convergence of trust-region methods based on probabilistic models

Afonso Bandeira, K. Scheinberg, L. N. Vicente

Research output: Contribution to journalArticle

Abstract

In this paper we consider the use of probabilistic or random models within a classical trust-region framework for optimization of deterministic smooth general nonlinear functions. Our method and setting differs from many stochastic optimization approaches in two principal ways. Firstly, we assume that the value of the function itself can be computed without noise, in other words, that the function is deterministic. Second, we use random models of higher quality than those produced by the usual stochastic gradient methods. In particular, a first order model based on random approximation of the gradient is required to provide sufficient quality of approximation with probability ≥ 1/2. This is in contrast with stochastic gradient approaches, where the model is assumed to be "correct" only in expectation. As a result of this particular setting, we are able to prove convergence, with probability one, of a trust-region method which is almost identical to the classical method. Moreover, the new method is simpler than its deterministic counterpart as it does not require a criticality step. Hence we show that a standard optimization framework can be used in cases when models are random and may or may not provide good approximations, as long as "good" models are more likely than "bad" models. Our results are based on the use of properties of martingales. Our motivation comes from using random sample sets and interpolation models in derivative-free optimization. However, our framework is general and can be applied with any source of uncertainty in the model. We discuss various applications for our methods in the paper.

Original languageEnglish (US)
Pages (from-to)1238-1264
Number of pages27
JournalSIAM Journal on Optimization
Volume24
Issue number3
DOIs
StatePublished - 2014

Fingerprint

Trust Region Method
Probabilistic Model
Stochastic Gradient
Model
Approximation
Derivative-free Optimization
Trust Region
Statistical Models
Optimization
Stochastic Optimization
Stochastic Methods
Gradient Method
Gradient methods
Criticality
Nonlinear Function
Martingale
Interpolate
Likely
Interpolation
Model-based

Keywords

  • Derivativefree optimization
  • Global convergence
  • Probabilistic models
  • Trust-region methods
  • Unconstrained optimizat ion

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science

Cite this

Convergence of trust-region methods based on probabilistic models. / Bandeira, Afonso; Scheinberg, K.; Vicente, L. N.

In: SIAM Journal on Optimization, Vol. 24, No. 3, 2014, p. 1238-1264.

Research output: Contribution to journalArticle

Bandeira, A, Scheinberg, K & Vicente, LN 2014, 'Convergence of trust-region methods based on probabilistic models', SIAM Journal on Optimization, vol. 24, no. 3, pp. 1238-1264. https://doi.org/10.1137/130915984
Bandeira, Afonso ; Scheinberg, K. ; Vicente, L. N. / Convergence of trust-region methods based on probabilistic models. In: SIAM Journal on Optimization. 2014 ; Vol. 24, No. 3. pp. 1238-1264.
@article{d387afc87bef4bf380c320c252c6d96c,
title = "Convergence of trust-region methods based on probabilistic models",
abstract = "In this paper we consider the use of probabilistic or random models within a classical trust-region framework for optimization of deterministic smooth general nonlinear functions. Our method and setting differs from many stochastic optimization approaches in two principal ways. Firstly, we assume that the value of the function itself can be computed without noise, in other words, that the function is deterministic. Second, we use random models of higher quality than those produced by the usual stochastic gradient methods. In particular, a first order model based on random approximation of the gradient is required to provide sufficient quality of approximation with probability ≥ 1/2. This is in contrast with stochastic gradient approaches, where the model is assumed to be {"}correct{"} only in expectation. As a result of this particular setting, we are able to prove convergence, with probability one, of a trust-region method which is almost identical to the classical method. Moreover, the new method is simpler than its deterministic counterpart as it does not require a criticality step. Hence we show that a standard optimization framework can be used in cases when models are random and may or may not provide good approximations, as long as {"}good{"} models are more likely than {"}bad{"} models. Our results are based on the use of properties of martingales. Our motivation comes from using random sample sets and interpolation models in derivative-free optimization. However, our framework is general and can be applied with any source of uncertainty in the model. We discuss various applications for our methods in the paper.",
keywords = "Derivativefree optimization, Global convergence, Probabilistic models, Trust-region methods, Unconstrained optimizat ion",
author = "Afonso Bandeira and K. Scheinberg and Vicente, {L. N.}",
year = "2014",
doi = "10.1137/130915984",
language = "English (US)",
volume = "24",
pages = "1238--1264",
journal = "SIAM Journal on Optimization",
issn = "1052-6234",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "3",

}

TY - JOUR

T1 - Convergence of trust-region methods based on probabilistic models

AU - Bandeira, Afonso

AU - Scheinberg, K.

AU - Vicente, L. N.

PY - 2014

Y1 - 2014

N2 - In this paper we consider the use of probabilistic or random models within a classical trust-region framework for optimization of deterministic smooth general nonlinear functions. Our method and setting differs from many stochastic optimization approaches in two principal ways. Firstly, we assume that the value of the function itself can be computed without noise, in other words, that the function is deterministic. Second, we use random models of higher quality than those produced by the usual stochastic gradient methods. In particular, a first order model based on random approximation of the gradient is required to provide sufficient quality of approximation with probability ≥ 1/2. This is in contrast with stochastic gradient approaches, where the model is assumed to be "correct" only in expectation. As a result of this particular setting, we are able to prove convergence, with probability one, of a trust-region method which is almost identical to the classical method. Moreover, the new method is simpler than its deterministic counterpart as it does not require a criticality step. Hence we show that a standard optimization framework can be used in cases when models are random and may or may not provide good approximations, as long as "good" models are more likely than "bad" models. Our results are based on the use of properties of martingales. Our motivation comes from using random sample sets and interpolation models in derivative-free optimization. However, our framework is general and can be applied with any source of uncertainty in the model. We discuss various applications for our methods in the paper.

AB - In this paper we consider the use of probabilistic or random models within a classical trust-region framework for optimization of deterministic smooth general nonlinear functions. Our method and setting differs from many stochastic optimization approaches in two principal ways. Firstly, we assume that the value of the function itself can be computed without noise, in other words, that the function is deterministic. Second, we use random models of higher quality than those produced by the usual stochastic gradient methods. In particular, a first order model based on random approximation of the gradient is required to provide sufficient quality of approximation with probability ≥ 1/2. This is in contrast with stochastic gradient approaches, where the model is assumed to be "correct" only in expectation. As a result of this particular setting, we are able to prove convergence, with probability one, of a trust-region method which is almost identical to the classical method. Moreover, the new method is simpler than its deterministic counterpart as it does not require a criticality step. Hence we show that a standard optimization framework can be used in cases when models are random and may or may not provide good approximations, as long as "good" models are more likely than "bad" models. Our results are based on the use of properties of martingales. Our motivation comes from using random sample sets and interpolation models in derivative-free optimization. However, our framework is general and can be applied with any source of uncertainty in the model. We discuss various applications for our methods in the paper.

KW - Derivativefree optimization

KW - Global convergence

KW - Probabilistic models

KW - Trust-region methods

KW - Unconstrained optimizat ion

UR - http://www.scopus.com/inward/record.url?scp=84910601452&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84910601452&partnerID=8YFLogxK

U2 - 10.1137/130915984

DO - 10.1137/130915984

M3 - Article

AN - SCOPUS:84910601452

VL - 24

SP - 1238

EP - 1264

JO - SIAM Journal on Optimization

JF - SIAM Journal on Optimization

SN - 1052-6234

IS - 3

ER -