Inexact Derivative-Free Optimization for Bilevel Learning

Variational regularization techniques are dominant in the field of mathematical imaging. A drawback of these techniques is that they are dependent on a number of parameters which have to be set by the user. A by now common strategy to resolve this issue is to learn these parameters from data. While mathematically appealing this strategy leads to a nested optimization problem (known as bilevel optimization) which is computationally very difficult to handle. A key ingredient in solving the upper-level problem is the exact solution of the lower-level problem which is practically infeasible. In this work we propose to solve these problems using inexact derivative-free optimization algorithms which never require to solve the lower-level problem exactly. We provide global convergence and worst-case complexity analysis of our approach, and test our proposed framework on ROF-denoising and learning MRI sampling patterns. Dynamically adjusting the lower-level accuracy yields learned parameters with similar reconstruction quality as high-accuracy evaluations but with dramatic reductions in computational work (up to 100 times faster in some cases).