Damped stochastic mirror descent for non-smooth ill-posed problems in Banach space
Abstract
We introduce a damped variant of the stochastic mirror descent method for systems of non-smooth nonlinear ill-posed operator equations in Banach spaces. At each iteration, the method samples a random subset of the equations, performs a mirror-descent step in the dual space with an adaptive step size derived from the discrepancy principle, and applies a damping term that prevents the iterates from drifting away from the initial guess. The scheme accommodates non-smooth forward operators through a family of bounded linear surrogates, admits general p-convex penalty functionals which also includes trained convex penalty, allows the noise level to differ from equation to equation, and dispenses with two classical requirements, namely reflexivity of the image space and Gâteaux differentiability of the forward operator. We study its regularization properties under an \textit{a posteriori} stopping rule by proving first that the iteration terminates after finitely many steps almost surely, and then that the mean-square error of the iterates tends to zero as the noise level vanishes. Numerical experiments on a system of non-smooth semilinear inverse source problems and the identification of a diffusion coefficient confirm the theoretical analysis and shows improved convergence results.
Related
- A stochastic mirror descent algorithm for mean-field control problems
- Stochastic Mirror Descent under Dependent Observations
- Adaptive Schauder Stochastic Mirror Descent in Banach Spaces
- Averaged Mirror Descent and Dual Gradient Methods: Convergent Algorithms for Entropic Gromov-Wasserstein Problems
- Exact Theory and Optimal Stepsizes for Stochastic Gradient Descent in Smooth Convex Minimization
BibTeX
@misc{bajpai2026damped,
title = {{Damped stochastic mirror descent for non-smooth ill-posed problems in Banach space}},
author = {Harshit Bajpai and Gaurav Mittal and Ankik Kumar Giri},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38258},
eprint = {2610.38258},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38258},
}
RIS
TY - UNPB
TI - Damped stochastic mirror descent for non-smooth ill-posed problems in Banach space
AU - Harshit Bajpai
AU - Gaurav Mittal
AU - Ankik Kumar Giri
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38258
UR - https://arxiv.org/abs/2610.38258
AB - We introduce a damped variant of the stochastic mirror descent method for systems of non-smooth nonlinear ill-posed operator equations in Banach spaces. At each iteration, the method samples a random subset of the equations, performs a mirror-descent step in the dual space with an adaptive step size derived from the discrepancy principle, and applies a damping term that prevents the iterates from drifting away from the initial guess. The scheme accommodates non-smooth forward operators through a family of bounded linear surrogates, admits general p-convex penalty functionals which also includes trained convex penalty, allows the noise level to differ from equation to equation, and dispenses with two classical requirements, namely reflexivity of the image space and Gâteaux differentiability of the forward operator. We study its regularization properties under an \textit{a posteriori} stopping rule by proving first that the iteration terminates after finitely many steps almost surely, and then that the mean-square error of the iterates tends to zero as the noise level vanishes. Numerical experiments on a system of non-smooth semilinear inverse source problems and the identification of a diffusion coefficient confirm the theoretical analysis and shows improved convergence results.
ER -