One fundamental problem when solving inverse problems is how to find regularization parameters. This article considers solving this problem using data-driven bilevel optimization, i.e. we consider the adaptive learning of the regularization parameter from data by means of optimization. This approach can be interpreted as solving an empirical risk minimization problem, and we analyze its performance in the large data sample size limit for general nonlinear problems. To reduce the associated computational cost, online numerical schemes are derived using the stochastic gradient method. We prove convergence of these numerical schemes under suitable assumptions on the forward problem. Numerical experiments are presented illustrating the theoretical results and demonstrating the applicability and efficiency of the proposed approaches for various linear and nonlinear inverse problems, including Darcy flow, the eikonal equation, and an image denoising example.

中文翻译：

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逆问题中双层数据驱动学习的一致性分析

求解逆问题时的一个基本问题是如何找到正则化参数。本文考虑使用数据驱动的双层优化来解决这个问题，即我们考虑通过优化的方式从数据中自适应学习正则化参数。这种方法可以解释为解决经验风险最小化问题，我们分析了它在一般非线性问题的大数据样本大小限制下的性能。为了减少相关的计算成本，使用随机梯度方法导出在线数值方案。我们在前向问题的适当假设下证明了这些数值方案的收敛性。