Tikhonov regularization involves minimizing the combination of a data discrepancy term and a regularizing term, and is the standard approach for solving inverse problems. The use of non-convex regularizers, such as those defined by trained neural networks, has been shown to be effective in many cases. However, finding global minimizers in non-convex situations can be challenging, making existing theory inapplicable. A recent development in regularization theory relaxes this requirement by providing convergence based on critical points instead of strict minimizers. This paper investigates convergence rates for the regularization with critical points using Bregman distances. Furthermore, we show that when implementing near-minimization through an iterative algorithm, a finite number of iterations is sufficient without affecting convergence rates.

翻译：Tikhonov正则化通过最小化数据保真项与正则化项的组合来求解反问题，是该领域的标准方法。研究表明，采用非凸正则化器（例如由训练后的神经网络定义的正则化器）在许多情况下具有良好效果。然而，在非凸情形下寻找全局极小值点颇具挑战性，这使得现有理论难以适用。正则化理论的最新进展放宽了该要求，通过基于临界点（而非严格极小值点）的收敛性分析提供了解决方案。本文利用Bregman距离研究了临界点正则化的收敛率。此外，我们证明：当通过迭代算法实现近似最小化时，有限步迭代足以保证收敛率不受影响。