Mini-batch descent in semiflows

TL;DR

This paper introduces a continuous mini-batch gradient descent method for semiflows, proving its approximation accuracy and demonstrating its effectiveness across various optimization problems with numerical validation.

Contribution

It develops a novel continuous mini-batch descent framework for semiflows and proves its approximation properties, extending the applicability of gradient methods to complex flows.

Findings

Mini-batch descent flow closely approximates original semiflow trajectories.

The approach applies to constrained optimization, sparse inversion, and domain decomposition.

Numerical examples validate the theoretical results.

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 \textit{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.