Iterated Vector Fields and Conservatism, with Applications to Federated Learning

TL;DR

This paper investigates the properties of iterated vector fields, especially their conservatism, and applies these insights to analyze and improve federated learning algorithms, providing new convergence results and understanding.

Contribution

It introduces a framework for understanding when federated averaging aligns with gradient descent, based on vector field conservatism, and explores its implications for federated learning.

Findings

Federated averaging equals gradient descent when client gradients are conservative.

Violations of conservatism lead to fundamentally different federated learning behavior.

New convergence results are derived for federated learning under specific vector field conditions.

Abstract

We study whether iterated vector fields (vector fields composed with themselves) are conservative. We give explicit examples of vector fields for which this self-composition preserves conservatism. Notably, this includes gradient vector fields of loss functions associated with some generalized linear models. As we show, characterizing the set of vector fields satisfying this condition leads to non-trivial geometric questions. In the context of federated learning, we show that when clients have loss functions whose gradients satisfy this condition, federated averaging is equivalent to gradient descent on a surrogate loss function. We leverage this to derive novel convergence results for federated learning. By contrast, we demonstrate that when the client losses violate this property, federated averaging can yield behavior which is fundamentally distinct from centralized optimization. Finally, we discuss theoretical and practical questions our analytical framework raises for federated learning.