标题： 汉克-劳斯迭代正则化随机梯度下降启发式规则 显示英文标题

标题： Hanke-Raus heuristic rule for iteratively regularized stochastic gradient descent

摘要： 在本工作中，我们提出了一种随机梯度下降方法的新变体，称为迭代正则化随机梯度下降（IRSGD）方法，以解决希尔伯特空间中的非线性不适定问题。 在标准假设下，我们证明了该方法的均方迭代误差对于精确数据收敛到零。 在存在噪声数据的情况下，我们首先提出了一种基于Hanke和Raus建议的方法的启发式参数选择规则（HPCR），然后将IRSGD方法与HPCR结合使用。 具体来说，HPCR在不需要任何噪声水平先验知识的情况下选择正则化参数。 我们证明了在噪声数据情况下，该方法在有限步内终止，并具有正则化特性。 此外，我们在HPCR以及不一致原则下，使用已知的源条件和其他相关条件讨论了该方法的收敛速率。 据我们所知，这是第一篇在无限维希尔伯特空间设置中使用启发式规则建立随机梯度方法的正则化性质和收敛速率的工作。 最后，我们提供了数值实验，以展示所提出方法的实际有效性。 显示英文摘要

摘要： In this work, we present a novel variant of the stochastic gradient descent method termed as iteratively regularized stochastic gradient descent (IRSGD) method to solve nonlinear ill-posed problems in Hilbert spaces. Under standard assumptions, we demonstrate that the mean square iteration error of the method converges to zero for exact data. In the presence of noisy data, we first propose a heuristic parameter choice rule (HPCR) based on the method suggested by Hanke and Raus, and then apply the IRSGD method in combination with HPCR. Precisely, HPCR selects the regularization parameter without requiring any a-priori knowledge of the noise level. We show that the method terminates in finitely many steps in case of noisy data and has regularizing features. Further, we discuss the convergence rates of the method using well-known source and other related conditions under HPCR as well as discrepancy principle. To the best of our knowledge, this is the first work that establishes both the regularization properties and convergence rates of a stochastic gradient method using a heuristic type rule in the setting of infinite-dimensional Hilbert spaces. Finally, we provide the numerical experiments to showcase the practical efficacy of the proposed method.