Consistency analysis of bilevel data-driven learning in inverse problems

TL;DR

This paper analyzes the performance of data-driven bilevel optimization for selecting regularization parameters in inverse problems, demonstrating theoretical guarantees and efficient algorithms for both linear and nonlinear cases.

Contribution

It provides a theoretical framework for bilevel data-driven regularization parameter learning, including convergence analysis and practical stochastic gradient algorithms.

Findings

Inverse accuracy is independent of ambient space dimension in linear problems.

Online stochastic gradient schemes converge under certain conditions.

Numerical experiments confirm theoretical results and demonstrate applicability.

Abstract

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 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.