Inexact Derivative-Free Optimization for Bilevel Learning

TL;DR

This paper introduces an inexact derivative-free optimization method for bilevel learning problems in imaging, enabling efficient parameter learning without requiring exact lower-level solutions, thus significantly reducing computational costs.

Contribution

It proposes a novel inexact derivative-free approach for bilevel optimization that guarantees convergence and reduces computational effort in imaging applications.

Findings

Achieves similar reconstruction quality with much less computation.

Reduces computational time by up to 100 times in experiments.

Provides theoretical convergence guarantees for the proposed method.

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).