Heterogeneity Matters even More in Distributed Learning: Study from Generalization Perspective

In this paper, we investigate the effect of data heterogeneity across clients on the performance of distributed learning systems, i.e., one-round Federated Learning, as measured by the associated generalization error. Specifically, $K$ clients have each $n$ training samples generated independently according to a possibly different data distribution, and their individually chosen models are aggregated by a central server. We study the effect of the discrepancy between the clients' data distributions on the generalization error of the aggregated model. First, we establish in-expectation and tail upper bounds on the generalization error in terms of the distributions. In part, the bounds extend the popular Conditional Mutual Information (CMI) bound, which was developed for the centralized learning setting, i.e., $K=1$, to the distributed learning setting with an arbitrary number of clients $K \geq 1$. Then, we connect with information-theoretic rate-distortion theory to derive possibly tighter \textit{lossy} versions of these bounds. Next, we apply our lossy bounds to study the effect of data heterogeneity across clients on the generalization error for the distributed classification problem in which each client uses Support Vector Machines (DSVM). In this case, we establish explicit generalization error bounds that depend explicitly on the data heterogeneity degree. It is shown that the bound gets smaller as the degree of data heterogeneity across clients increases, thereby suggesting that DSVM generalizes better when the dissimilarity between the clients' training samples is bigger. This finding, which goes beyond DSVM, is validated experimentally through several experiments.