Federated averaging (FedAvg) is a popular algorithm for horizontal federated learning (FL), where samples are gathered across different clients and are not shared with each other or a central server. Extensive convergence analysis of FedAvg exists for the discrete iteration setting, guaranteeing convergence for a range of loss functions and varying levels of data heterogeneity. We extend this analysis to the continuous-time setting where the global weights evolve according to a multivariate stochastic differential equation (SDE), which is the first time FedAvg has been studied from the continuous-time perspective. We use techniques from stochastic processes to establish convergence guarantees under different loss functions, some of which are more general than existing work in the discrete setting. We also provide conditions for which FedAvg updates to the server weights can be approximated as normal random variables. Finally, we use the continuous-time formulation to reveal generalization properties of FedAvg.

Automated feature engineering (AutoFE) is used to automatically create new features from original features to improve predictive performance without needing significant human intervention and expertise. Many algorithms exist for AutoFE, but very few approaches exist for the federated learning (FL) setting where data is gathered across many clients and is not shared between clients or a central server. We introduce AutoFE algorithms for the horizontal, vertical, and hybrid FL settings, which differ in how the data is gathered across clients. To the best of our knowledge, we are the first to develop AutoFE algorithms for the horizontal and hybrid FL cases, and we show that the downstream model performance of federated AutoFE is similar to the case where data is held centrally and AutoFE is performed centrally.

Very few methods for hybrid federated learning, where clients only hold subsets of both features and samples, exist. Yet, this scenario is very important in practical settings. We provide a fast, robust algorithm for hybrid federated learning that hinges on Fenchel Duality. We prove the convergence of the algorithm to the same solution as if the model was trained centrally in a variety of practical regimes. Furthermore, we provide experimental results that demonstrate the performance improvements of the algorithm over a commonly used method in federated learning, FedAvg. We also provide privacy considerations and necessary steps to protect client data.