On the Convergence of A Data-Driven Regularized Stochastic Gradient Descent for Nonlinear Ill-Posed Problems

2024/03/18 by Zehui Zhou, Zhou, Zehui
Engineering · Mathematics · #47J06 #65J20 #65J22 #Advanced Image Fusion Techniques #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Optimization and Control (math.OC) #Radiative Heat Transfer Studies

paper · pdf · doi:10.48550/arxiv.2403.11787

openalex publication_date 2024/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Stochastic gradient descent (SGD) is a promising method for solving large-scale inverse problems, due to its excellent scalability with respect to data size. In this work, we analyze a new data-driven regularized stochastic gradient descent for the efficient numerical solution of a class of nonlinear ill-posed inverse problems in infinite dimensional Hilbert spaces. At each step of the iteration, the method randomly selects one equation from the nonlinear system combined with a corresponding equation from the learned system based on training data to obtain a stochastic estimate of the gradient and then performs a descent step with the estimated gradient. We prove the regularizing property of this method under the tangential cone condition and a priori parameter choice and then derive the convergence rates under the additional source condition and range invariance conditions. Several numerical experiments are provided to complement the analysis.

Related

• On the Convergence of Stochastic Gradient Descent for Nonlinear Ill-Posed Problems
• Hanke-Raus heuristic rule for iteratively regularized stochastic gradient descent
• Stochastic asymptotical regularization for nonlinear ill-posed problems
• Learning Operators by Regularized Stochastic Gradient Descent with Operator-valued Kernels
• Stochastic gradient descent method with convex penalty for ill-posed problems in Banach spaces