Privacy Amplification for Federated Learning via User Sampling and\n Wireless Aggregation

In this paper, we study the problem of federated learning over a wireless\nchannel with user sampling, modeled by a Gaussian multiple access channel,\nsubject to central and local differential privacy (DP/LDP) constraints. It has\nbeen shown that the superposition nature of the wireless channel provides a\ndual benefit of bandwidth efficient gradient aggregation, in conjunction with\nstrong DP guarantees for the users. Specifically, the central DP privacy\nleakage has been shown to scale as $\\mathcal{O}(1/K^{1/2})$, where $K$ is the\nnumber of users. It has also been shown that user sampling coupled with\northogonal transmission can enhance the central DP privacy leakage with the\nsame scaling behavior. In this work, we show that, by join incorporating both\nwireless aggregation and user sampling, one can obtain even stronger privacy\nguarantees. We propose a private wireless gradient aggregation scheme, which\nrelies on independently randomized participation decisions by each user. The\ncentral DP leakage of our proposed scheme scales as $\\mathcal{O}(1/K^{3/4})$.\nIn addition, we show that LDP is also boosted by user sampling. We also present\nanalysis for the convergence rate of the proposed scheme and study the\ntradeoffs between wireless resources, convergence, and privacy theoretically\nand empirically for two scenarios when the number of sampled participants are\n$(a)$ known, or $(b)$ unknown at the parameter server.\n