# On the Convergence of Stochastic Gradient Descent for Nonlinear Ill-Posed Problems

arXiv: https://arxiv.org/abs/1907.03132 · HTML: https://mlsanity.com/paper/1907.03132/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Jul 2019.

## Built on by (most cited first, 4 of 4)

- [Statistical Learning and Inverse Problems: A Stochastic Gradient Approach](https://mlsanity.com/paper/2209.14967/) (2022)
- [Controlling the Flow: Stability and Convergence for Stochastic Gradient Descent with Decaying Regularization](https://mlsanity.com/paper/2505.11434/) (2025)
- [On the Convergence of Stochastic Gradient Descent for Linear Inverse Problems in Banach Spaces](https://mlsanity.com/paper/2302.05197/) (2023)
- [Generative Prior-Guided Neural Interface Reconstruction for 3D Electrical Impedance Tomography](https://mlsanity.com/paper/2505.16487/) (2025)

## Abstract (arXiv)

In this work, we analyze the regularizing property of the stochastic gradient descent for the efficient numerical solution of a class of nonlinear ill-posed inverse problems in Hilbert spaces. At each step of the iteration, the method randomly chooses one equation from the nonlinear system to obtain an unbiased stochastic estimate of the gradient, and then performs a descent step with the estimated gradient. It is a randomized version of the classical Landweber method for nonlinear inverse problems, and it is highly scalable to the problem size and holds significant potentials for solving large-scale inverse problems. Under the canonical tangential cone condition, we prove the regularizing property for a priori stopping rules, and then establish the convergence rates under suitable sourcewise condition and range invariance condition.