Federated Differentially Private Optimization via Low-Rank Preconditioning

We consider differentially private (DP) optimization in the federated setting, where a DP estimate of the feature covariance matrix serves as a Hessian approximation for generalized linear models. Existing central DP methods for covariance estimation have communication cost that grows quadratically in the data dimension, while existing baselines with linear communication cost are only locally private and degrade as the number of clients increases. As a remedy, we propose a federated DP-PCA framework that achieves linear communication cost while providing central DP guarantees under secure aggregation. It consists of a DP rank selection step, a DP power method, and a noisy release of the covariance projected onto the selected low-dimensional subspace. We give error bounds for this covariance estimate and for the resulting low-communication preconditioned DP gradient descent. Empirically, our proposed DP optimization method matches a centralized baseline with quadratic communication cost at a fraction of the communication.