On the Convergence of Stochastic Gradient Descent for Linear Inverse Problems in Banach Spaces

TL;DR

This paper extends the theoretical understanding of stochastic gradient descent (SGD) for linear inverse problems from Euclidean spaces to general Banach spaces, proving convergence and regularisation properties.

Contribution

It provides the first convergence analysis of SGD for inverse problems in Banach spaces, demonstrating almost sure convergence and regularisation effects.

Findings

SGD converges almost surely to the minimum norm solution in Banach spaces.

The regularising property of SGD is established with appropriate stopping criteria.

Numerical experiments illustrate the theoretical results.

Abstract

In this work we consider stochastic gradient descent (SGD) for solving linear inverse problems in Banach spaces. SGD and its variants have been established as one of the most successful optimisation methods in machine learning, imaging and signal processing, etc. At each iteration SGD uses a single datum, or a small subset of data, resulting in highly scalable methods that are very attractive for large-scale inverse problems. Nonetheless, the theoretical analysis of SGD-based approaches for inverse problems has thus far been largely limited to Euclidean and Hilbert spaces. In this work we present a novel convergence analysis of SGD for linear inverse problems in general Banach spaces: we show the almost sure convergence of the iterates to the minimum norm solution and establish the regularising property for suitable a priori stopping criteria. Numerical results are also presented to illustrate features of the approach.