Double Momentum and Error Feedback for Clipping with Fast Rates and Differential Privacy

Strong Differential Privacy (DP) and Optimization guarantees are two desirable properties for a method in Federated Learning (FL). However, existing algorithms do not achieve both properties at once: they either have optimal DP guarantees but rely on restrictive assumptions such as bounded gradients/bounded data heterogeneity, or they ensure strong optimization performance but lack DP guarantees. To address this gap in the literature, we propose and analyze a new method called Clip21-SGD2M based on a novel combination of clipping, heavy-ball momentum, and Error Feedback. In particular, for non-convex smooth distributed problems with clients having arbitrarily heterogeneous data, we prove that Clip21-SGD2M has optimal convergence rate and also near optimal (local-)DP neighborhood. Our numerical experiments on non-convex logistic regression and training of neural networks highlight the superiority of Clip21-SGD2M over baselines in terms of the optimization performance for a given DP-budget.