Inexact Derivative-Free Optimization for Bilevel Learning

• URL: http://arxiv.org/abs/2006.12674v2
• Date: Tue, 8 Dec 2020 23:10:15 GMT
• Title: Inexact Derivative-Free Optimization for Bilevel Learning
• Authors: Matthias J. Ehrhardt, Lindon Roberts
• Abstract summary: Variational regularization techniques are dominant in the field of mathematical imaging. A by now common strategy to resolve this issue is to learn these parameters from data. It is common when solving the upper-level problem to assume access to exact solutions of the lower-level problem, which is practically infeasible. We propose to solve these problems using inexact derivative-free optimization algorithms which never require exact lower-level problem solutions.
• Score: 0.27074235008521236
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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).

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