Consistency analysis of bilevel data-driven learning in inverse problems

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