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 worst-case complexity bound for our approach. We 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).

变分正则化技术在数学成像领域占主导地位。这些技术的缺点在于它们取决于用户必须设置的许多参数。解决此问题的一种常见策略是从数据中学习这些参数。尽管在数学上很吸引人，但是此策略会导致嵌套的优化问题（称为双级优化），该问题在计算上很难处理。解决上层问题时，通常会假设可以访问下层问题的精确解，这实际上是不可行的。在这项工作中，我们建议使用不精确的无导数优化算法来解决这些问题，这些算法从不需要精确的低层问题解决方案，而是假设可以访问具有可控精度的近似解决方案，在实践中是可以实现的。我们证明了我们的方法具有全局收敛性和最坏情况的复杂性。我们在ROF去噪和学习MRI采样模式上测试了我们提出的框架。动态调整较低级别的精度会产生学习参数，这些参数具有与高精度评估相似的重建质量，但运算量却大大减少（在某些情况下，速度提高了100倍）。