Inexact Derivative-Free Optimization for Bilevel Learning
This addresses the practical infeasibility of exact solutions in bilevel learning for mathematical imaging, offering a computationally efficient method for parameter tuning.
The paper tackles the computational difficulty of bilevel optimization in learning parameters for variational regularization by proposing an inexact derivative-free optimization algorithm that uses approximate lower-level solutions, achieving similar reconstruction quality with up to 100 times faster computation in tests on ROF denoising and MRI sampling patterns.
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. It is common when solving the upper-level problem to assume access to exact solutions 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 exact lower-level problem solutions, but instead assume access to approximate solutions with controllable accuracy, which is achievable in practice. We prove global convergence and a worstcase complexity bound for our approach. We test our proposed framework on ROFdenoising and learning MRI sampling patterns. Dynamically adjusting the lower-level accuracy yields learned parameters with similar reconstruction quality as highaccuracy evaluations but with dramatic reductions in computational work (up to 100 times faster in some cases).