In various domains within imaging and data science, particularly when
addressing tasks modeled utilizing the variational regularization approach,
manually configuring regularization parameters presents a formidable challenge.
The difficulty intensifies when employing regularizers involving a large number
of hyperparameters. To overcome this challenge, bilevel learning is employed to
learn suitable hyperparameters. However, due to the use of numerical solvers,
the exact gradient with respect to the hyperparameters is unattainable,
necessitating the use of methods relying on approximate gradients.
State-of-the-art inexact methods a priori select a decreasing summable sequence
of the required accuracy and only assure convergence given a sufficiently small
fixed step size. Despite this, challenges persist in determining the Lipschitz
constant of the hypergradient and identifying an appropriate fixed step size.
Conversely, computing exact function values is not feasible, impeding the use
of line search. In this work, we introduce a provably convergent inexact
backtracking line search involving inexact function evaluations and
hypergradients. We show convergence to a stationary point of the loss with
respect to hyperparameters. Additionally, we propose an algorithm to determine
the required accuracy dynamically. Our numerical experiments demonstrate the
efficiency and feasibility of our approach for hyperparameter estimation in
variational regularization problems, alongside its robustness in terms of the
initial accuracy and step size choices.

使用变分正则化方法，通过双层学习来学习合适的超参数，同时提出了能够收敛到损失函数超参数的稳定点的近似回溯线搜索算法，并在变分正则化问题的超参数估计中展示了其有效性和可行性。