Communications in Mathematical Sciences

Volume 20 (2022)

Number 1

Consistency analysis of bilevel data-driven learning in inverse problems

Pages: 123 – 164



Neil K. Chada (Applied Mathematics and Computational Science Program, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia)

Claudia Schillings (Institute of Mathematics, University of Mannheim, Germany)

Xin T. Tong (Department of Mathematics, National University of Singapore)

Simon Weissmann (Interdisciplinary Center for Scientific Computing, University of Heidelberg, Germany)


One fundamental problem when solving inverse problems is how to find regularization parameters. This article considers solving this problem using data-driven bilevel optimization, i.e. we consider the adaptive learning of the regularization parameter from data by means of optimization. This approach can be interpreted as solving an empirical risk minimization problem, and we analyze its performance in the large data sample size limit for general nonlinear problems. We demonstrate how to implement our framework on linear inverse problems, where we can further show that the inverse accuracy does not depend on the ambient space dimension. To reduce the associated computational cost, online numerical schemes are derived using the stochastic gradient descent method. We prove convergence of these numerical schemes under suitable assumptions on the forward problem. Numerical experiments are presented illustrating the theoretical results and demonstrating the applicability and efficiency of the proposed approaches for various linear and nonlinear inverse problems, including Darcy flow, the eikonal equation, and an image denoising example.


bilevel optimization, statistical consistency, inverse problems, stochastic gradient descent, Tikhonov regularization

2010 Mathematics Subject Classification

35R30, 62F12, 65K10, 90C15

Received 7 January 2021

Received revised 5 June 2021

Accepted 5 June 2021

Published 10 December 2021