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.

翻译：联邦学习结合量化与故意添加噪声的无线网络方法，是一种在减少无线资源消耗的同时保护用户差分隐私的有效途径。具体而言，联邦学习过程可融合基于多项分布机制的量化更新，这些更新由多个用户共同贡献。然而，如何优化量化参数、通信资源（如发射功率、带宽和量化比特数）以及添加的噪声，以同时满足差分隐私要求和学习模型性能，仍是一个开放且具有挑战性的问题。本文旨在联合优化量化与二项分布机制参数及通信资源，在无线网络约束和差分隐私要求的条件下最大化收敛速率。为此，我们首先推导出联邦学习在量化/噪声条件下的新型差分隐私预算估计，该估计比现有最优边界更紧凑。随后，我们提供了收敛速率的理论上界。该理论边界被分解为两个部分：全局梯度的方差和二次偏差，前者可通过优化通信资源与量化/噪声参数最小化。上述优化问题最终转化为混合整数非线性规划问题。为求解该问题，我们首先将其转化为一个新问题，并证明该新问题的解即为原问题的最优解。随后，我们提出一种近似算法，可在任意相对误差保证下求解转化后的问题。大量仿真结果表明，在相同的无线资源约束和差分隐私保护要求下，所提出的近似算法能达到与不采用量化/噪声的传统联邦学习相近的精度。该结果能在保护用户隐私的同时实现更高的收敛速率。