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On the convergence of stochastic gradient descent for nonlinear inverse problems

Jin, B; Zou, J; Zhou, Z; (2020) On the convergence of stochastic gradient descent for nonlinear inverse problems. SIAM Journal on Optimization , 30 (2) pp. 1421-1450. 10.1137/19M1271798. Green open access

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Abstract

In this work, we analyze the regularizing property of the stochastic gradient descent for the numerical solution of a class of nonlinear ill-posed inverse problems in Hilbert spaces. At each step of the iteration, the method randomly chooses one equation from the nonlinear system to obtain an unbiased stochastic estimate of the gradient and then performs a descent step with the estimated gradient. It is a randomized version of the classical Landweber method for nonlinear inverse problems, and it is highly scalable to the problem size and holds significant potential for solving large-scale inverse problems. Under the canonical tangential cone condition, we prove the regularizing property for a priori stopping rules and then establish the convergence rates under a suitable sourcewise condition and a range invariance condition.

Type: Article
Title: On the convergence of stochastic gradient descent for nonlinear inverse problems
Open access status: An open access version is available from UCL Discovery
DOI: 10.1137/19M1271798
Publisher version: https://doi.org/10.1137/19M1271798
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
UCL classification: UCL
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Computer Science
URI: https://discovery.ucl.ac.uk/id/eprint/10092546
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