Federated Learning (FL) is a distributed learning paradigm that preserves privacy by eliminating the need to exchange raw data during training. In its prototypical edge instantiation with underlying wireless transmissions enabled by analog over-the-air computing (AirComp), referred to as \emph{over-the-air FL (AirFL)}, the inherent channel noise plays a unique role of \emph{frenemy} in the sense that it degrades training due to noisy global aggregation while providing a natural source of randomness for privacy-preserving mechanisms, formally quantified by \emph{differential privacy (DP)}. It remains, nevertheless, challenging to effectively harness such channel impairments, as prior arts, under assumptions of either simple channel models or restricted types of loss functions, mostly considering (local) DP enhancement with a single-round or non-convergent bound on privacy loss. In this paper, we study AirFL over multiple-access fading channels with a multi-antenna base station (BS) subject to user-level DP requirements. Despite a recent study, which claimed in similar settings that artificial noise (AN) must be injected to ensure DP in general, we demonstrate, on the contrary, that DP can be gained as a \emph{perk} even \emph{without} employing any AN. Specifically, we derive a novel bound on DP that converges under general bounded-domain assumptions on model parameters, along with a convergence bound with general smooth and non-convex loss functions. Next, we optimize over receive beamforming and power allocations to characterize the optimal convergence-privacy trade-offs, which also reveal explicit conditions in which DP is achievable without compromising training. Finally, our theoretical findings are validated by extensive numerical results.

翻译：联邦学习（FL）是一种分布式学习范式，通过在训练过程中无需交换原始数据来保护隐私。在其典型的边缘实现中，通过模拟空中计算（AirComp）实现底层无线传输，称为空中联邦学习（AirFL），固有信道噪声扮演着独特的朋友兼敌人角色：一方面因噪声全局聚合而降低训练质量，另一方面为隐私保护机制提供了天然的随机性来源，这可由差分隐私（DP）进行形式化量化。然而，有效利用此类信道损伤仍具挑战性，因为现有技术在简单信道模型或受限损失函数类型的假设下，大多仅考虑（局部）DP增强，且仅提供单轮或非收敛的隐私损失界限。本文研究具有用户级DP要求的多天线基站（BS）下多址衰落信道上的AirFL。尽管近期研究在类似设置中声称通常必须注入人工噪声（AN）以确保DP，但我们证明，即使不采用任何AN，DP仍可作为额外收益获得。具体而言，我们推导出在模型参数一般有界域假设下收敛的新型DP界限，以及具有一般光滑非凸损失函数的收敛界限。接着，我们通过优化接收波束成形和功率分配来刻画最优的收敛-隐私权衡关系，并揭示在不影响训练前提下实现DP的显式条件。最后，我们通过大量数值结果验证了理论发现。