Centroid Approximation for Byzantine-Tolerant Federated Learning

Federated learning allows each client to keep its data locally when training machine learning models in a distributed setting. Significant recent research established the requirements that the input must satisfy in order to guarantee convergence of the training loop. This line of work uses averaging as the aggregation rule for the training models. In particular, we are interested in whether federated learning is robust to Byzantine behavior, and observe and investigate a tradeoff between the average/centroid and the validity conditions from distributed computing. We show that the various validity conditions alone do not guarantee a good approximation of the average. Furthermore, we show that reaching good approximation does not give good results in experimental settings due to possible Byzantine outliers. Our main contribution is the first lower bound of $\min\{\frac{n-t}{t},\sqrt{d}\}$ on the centroid approximation under box validity that is often considered in the literature, where $n$ is the number of clients, $t$ the upper bound on the number of Byzantine faults, and $d$ is the dimension of the machine learning model. We complement this lower bound by an upper bound of $2\min\{n,\sqrt{d}\}$, by providing a new analysis for the case $n<d$. In addition, we present a new algorithm that achieves a $\sqrt{2d}$-approximation under convex validity, which also proves that the existing lower bound in the literature is tight. We show that all presented bounds can also be achieved in the distributed peer-to-peer setting. We complement our analytical results with empirical evaluations in federated stochastic gradient descent and federated averaging settings.