Iterated Vector Fields and Conservatism, with Applications to Federated Learning
This work addresses theoretical challenges in federated learning by linking geometric properties of vector fields to optimization equivalence, offering insights for improving convergence in distributed settings.
The paper tackles the problem of whether iterated vector fields are conservative, showing that for certain gradient vector fields in generalized linear models, self-composition preserves conservatism. In federated learning, this leads to novel convergence results when client loss gradients satisfy this condition, while violations cause distinct behavior from centralized optimization.
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.