Stochastic Gradient Descent Based Variational Inference for Infinite-Dimensional Inverse Problems

Abstract

This paper introduces two variational inference (VI) approaches for infinite-dimensional inverse problems, developed through gradient descent with a constant learning rate. The proposed methods enable efficient approximate sampling from the target posterior distribution using a constant-rate stochastic gradient descent (cSGD) iteration. Specifically, we introduce a randomization strategy that incorporates stochastic gradient noise, allowing the cSGD iteration to be viewed as a discrete-time process. This transformation establishes key relationships between the covariance operators of the approximate and true posterior distributions, thereby validating cSGD as a VI method. We also investigate the regularization properties of the cSGD iteration and provide a theoretical analysis of the discretization error between the approximated posterior mean and the true background function. Building on this framework, we develop a preconditioned version of cSGD to further improve sampling efficiency. Finally, we apply the proposed methods to two practical inverse problems: one governed by a simple smooth equation and the other by the steady-state Darcy flow equation. Numerical results confirm our theoretical findings and compare the sampling performance of the two approaches for solving linear and non-linear inverse problems.

Keywords

Inverse problems / Infinite-dimensional variational inference (iVI) / Bayesian analysis for functions / Partial differential equations (PDEs) / Stochastic gradient descent (SGD) / 65L09 / 35R30 / 49N45 / 62F15

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Funding

National Natural Science Foundation of China(12322116)

UK Research and Innovation

China Scholarship Council

RIGHTS & PERMISSIONS

Shanghai University