We perform a non-asymptotic analysis of the contrastive divergence (CD) algorithm, a training method for unnormalized models. While prior work has established that (for exponential family distributions) the CD iterates asymptotically converge at an $O(n^{-1 / 3})$ rate to the true parameter of the data distribution, we show, under some regularity assumptions, that CD can achieve the parametric rate $O(n^{-1 / 2})$. Our analysis provides results for various data batching schemes, including the fully online and minibatch ones. We additionally show that CD can be near-optimal, in the sense that its asymptotic variance is close to the Cramér-Rao lower bound.

These authors contributed equally to this work.

In this paper, a simple gradient truncation mechanism is proposed to address this issue.

Identifying clients with similar objectives and learning a model-per-cluster is an intuitive and interpretable approach to personalization in federated learning. However, doing so with provable and optimal guarantees has remained an open challenge. We formalize this problem as a stochastic optimization problem, achieving optimal convergence rates for a large class of loss functions. We propose simple iterative algorithms which identify clusters of similar clients and train a personalized model-per-cluster, using local client gradients and flexible constraints on the clusters. The convergence rates of our algorithms asymptotically match those obtained if we knew the true underlying clustering of the clients and are provably robust in the Byzantine setting where some fraction of the clients are malicious.