Federated Online Prediction from Experts with Differential Privacy: Separations and Regret Speed-ups

We study the problems of differentially private federated online prediction from experts against both *stochastic adversaries* and *oblivious adversaries*. We aim to minimize the average regret on $m$ clients working in parallel over time horizon $T$ with explicit differential privacy (DP) guarantees. With stochastic adversaries, we propose a **Fed-DP-OPE-Stoch** algorithm that achieves $\sqrt{m}$-fold speed-up of the per-client regret compared to the single-player counterparts under both pure DP and approximate DP constraints, while maintaining logarithmic communication costs. With oblivious adversaries, we establish non-trivial lower bounds indicating that *collaboration among clients does not lead to regret speed-up with general oblivious adversaries*. We then consider a special case of the oblivious adversaries setting, where there exists a low-loss expert. We design a new algorithm **Fed-SVT** and show that it achieves an $m$-fold regret speed-up under both pure DP and approximate DP constraints over the single-player counterparts. Our lower bound indicates that Fed-SVT is nearly optimal up to logarithmic factors. Experiments demonstrate the effectiveness of our proposed algorithms. To the best of our knowledge, this is the first work examining the differentially private online prediction from experts in the federated setting.