  Federated Learning (FL) with quantization and deliberately added noise over
wireless networks is a promising approach to preserve user differential privacy
(DP) while reducing wireless resources. Specifically, an FL process can be
fused with quantized Binomial mechanism-based updates contributed by multiple
users. However, optimizing quantization parameters, communication resources
(e.g., transmit power, bandwidth, and quantization bits), and the added noise
to guarantee the DP requirement and performance of the learned FL model remains
an open and challenging problem. This article aims to jointly optimize the
quantization and Binomial mechanism parameters and communication resources to
maximize the convergence rate under the constraints of the wireless network and
DP requirement. To that end, we first derive a novel DP budget estimation of
the FL with quantization/noise that is tighter than the state-of-the-art bound.
We then provide a theoretical bound on the convergence rate. This theoretical
bound is decomposed into two components, including the variance of the global
gradient and the quadratic bias that can be minimized by optimizing the
communication resources, and quantization/noise parameters. The resulting
optimization turns out to be a Mixed-Integer Non-linear Programming (MINLP)
problem. To tackle it, we first transform this MINLP problem into a new problem
whose solutions are proved to be the optimal solutions of the original one. We
then propose an approximate algorithm to solve the transformed problem with an
arbitrary relative error guarantee. Extensive simulations show that under the
same wireless resource constraints and DP protection requirements, the proposed
approximate algorithm achieves an accuracy close to the accuracy of the
conventional FL without quantization/noise. The results can achieve a higher
convergence rate while preserving users' privacy.
