Mini-batch descent in semiflows

Chair for Dynamics, Control, Machine Learning, and Numerics, Alexander von Humboldt-Professorship, Department of Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany

^* Corresponding Author: martin.hernandez@fau.de

Received: 11 July 2024
Accepted: 6 February 2025

Abstract

This paper investigates the application of mini-batch gradient descent to semiflows (gradient flows). Given a loss function (potential), we introduce a continuous version of mini-batch gradient descent by randomly selecting sub-loss functions over time, defining a piecewise flow. We prove that, under suitable assumptions on the potential generating the semiflow, the mini-batch descent flow trajectory closely approximates the original semiflow trajectory on average. In addition, we study a randomized minimizing movement scheme that also approximates the semiflow of the full loss function. We illustrate the versatility of this approach across various problems, including constrained optimization, sparse inversion, and domain decomposition. Finally, we validate our results with several numerical examples.