On the Convergence of A Data-Driven Regularized Stochastic Gradient Descent for Nonlinear Ill-Posed Problems
<|reference_start|>On the Convergence of A Data-Driven Regularized Stochastic Gradient Descent for Nonlinear Ill-Posed Problems: 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.<|reference_end|>
@article{zhou2024on,
    title={On the Convergence of A Data-Driven Regularized Stochastic Gradient
  Descent for Nonlinear Ill-Posed Problems},
    author={Zehui Zhou},
    journal={arXiv preprint arXiv:2403.11787},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.11787},
    primaryClass={math.NA cs.NA math.OC}
}