Federated Learning You May Communicate Less Often!

We investigate the generalization error of statistical learning models in a Federated Learning (FL) setting. Specifically, we study the evolution of the generalization error with the number of communication rounds between the clients and the parameter server, i.e., the effect on the generalization error of how often the local models as computed by the clients are aggregated at the parameter server. We establish PAC-Bayes and rate-distortion theoretic bounds on the generalization error that account explicitly for the effect of the number of rounds, say $ R \in \mathbb{N}$, in addition to the number of participating devices $K$ and individual datasets size $n$. The bounds, which apply in their generality for a large class of loss functions and learning algorithms, appear to be the first of their kind for the FL setting. Furthermore, we apply our bounds to FL-type Support Vector Machines (FSVM); and we derive (more) explicit bounds on the generalization error in this case. In particular, we show that the generalization error of FSVM increases with $R$, suggesting that more frequent communication with the parameter server diminishes the generalization power of such learning algorithms. Combined with that the empirical risk generally decreases for larger values of $R$, this indicates that $R$ might be a parameter to optimize in order to minimize the population risk of FL algorithms. Moreover, specialized to the case $R=1$ (sometimes referred to as "one-shot" FL or distributed learning) our bounds suggest that the generalization error of the FL setting decreases faster than that of centralized learning by a factor of $\mathcal{O}(\sqrt{\log(K)/K})$, thereby generalizing recent findings in this direction to arbitrary loss functions and algorithms. The results of this paper are also validated on some experiments.