On the Outsized Importance of Learning Rates in Local Update Methods

TL;DR

This paper analyzes local update methods in federated and meta-learning, revealing the critical role of learning rates in convergence and proposing a practical automatic decay method to improve performance.

Contribution

It provides a theoretical characterization of local update methods for quadratic objectives and introduces a new automatic learning rate decay technique.

Findings

Proper learning rate tuning can achieve near-optimal results in communication-limited settings.

The choice of client learning rate affects the surrogate loss's condition number and alignment with the true loss.

The proposed automatic learning rate decay improves empirical performance across various tasks.

Abstract

We study a family of algorithms, which we refer to as local update methods, that generalize many federated learning and meta-learning algorithms. We prove that for quadratic objectives, local update methods perform stochastic gradient descent on a surrogate loss function which we exactly characterize. We show that the choice of client learning rate controls the condition number of that surrogate loss, as well as the distance between the minimizers of the surrogate and true loss functions. We use this theory to derive novel convergence rates for federated averaging that showcase this trade-off between the condition number of the surrogate loss and its alignment with the true loss function. We validate our results empirically, showing that in communication-limited settings, proper learning rate tuning is often sufficient to reach near-optimal behavior. We also present a practical method for automatic learning rate decay in local update methods that helps reduce the need for learning rate tuning, and highlight its empirical performance on a variety of tasks and datasets.