A Stochastic Levenberg--Marquardt Method Using Random Models with Complexity Results
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F22%3A00357703" target="_blank" >RIV/68407700:21230/22:00357703 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.1137/20M1366253" target="_blank" >https://doi.org/10.1137/20M1366253</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1137/20M1366253" target="_blank" >10.1137/20M1366253</a>
Alternative languages
Result language
angličtina
Original language name
A Stochastic Levenberg--Marquardt Method Using Random Models with Complexity Results
Original language description
Globally convergent variants of the Gauss--Newton algorithm are often the methods of choice to tackle nonlinear least-squares problems. Among such frameworks, Levenberg--Marquardt and trust-region methods are two well-established, similar paradigms. Both schemes have been studied when the Gauss--Newton model is replaced by a random model that is only accurate with a given probability. Trust-region schemes have also been applied to problems where the objective value is subject to noise: this setting is of particular interest in fields such as data assimilation, where efficient methods that can adapt to noise are needed to account for the intrinsic uncertainty in the input data. In this paper, we describe a stochastic Levenberg--Marquardt algorithm that handles noisy objective function values and random models, provided sufficient accuracy is achieved in probability. Our method relies on a specific scaling of the regularization parameter that allows us to leverage existing results for trust-region algorithms. Moreover, we exploit the structure of our objective through the use of a family of stationarity criteria tailored to least-squares problems. Provided the probability of accurate function estimates and models is sufficiently large, we bound the expected number of iterations needed to reach an approximate stationary point, which generalizes results based on using deterministic models or noiseless function values. We illustrate the link between our approach and several applications related to inverse problems and machine learning.
Czech name
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Czech description
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Classification
Type
J<sub>SC</sub> - Article in a specialist periodical, which is included in the SCOPUS database
CEP classification
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OECD FORD branch
10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
Result continuities
Project
<a href="/en/project/EF16_019%2F0000765" target="_blank" >EF16_019/0000765: Research Center for Informatics</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2022
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
SIAM/ASA Journal on Uncertainty Quantification
ISSN
2166-2525
e-ISSN
2166-2525
Volume of the periodical
10
Issue of the periodical within the volume
1
Country of publishing house
US - UNITED STATES
Number of pages
30
Pages from-to
507-536
UT code for WoS article
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EID of the result in the Scopus database
2-s2.0-85125866895