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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

  • Czech description

Classification

  • Type

    J<sub>SC</sub> - Article in a specialist periodical, which is included in the SCOPUS database

  • CEP classification

  • 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

  • EID of the result in the Scopus database

    2-s2.0-85125866895