This paper presents NCOTA-DGD, a Decentralized Gradient Descent (DGD) algorithm that combines local gradient descent with a novel Non-Coherent Over-The-Air (NCOTA) consensus scheme to solve distributed machine-learning problems over wirelessly-connected systems. NCOTA-DGD leverages the waveform superposition properties of the wireless channels: it enables simultaneous transmissions under half-duplex constraints, by mapping local optimization signals to a mixture of preamble sequences, and consensus via non-coherent combining at the receivers. NCOTA-DGD operates without channel state information at transmitters and receivers, and leverages the average channel pathloss to mix signals, without explicit knowledge of the mixing weights (typically known in consensus-based optimization algorithms). It is shown both theoretically and numerically that, for smooth and strongly-convex problems with fixed consensus and learning stepsizes, the updates of NCOTA-DGD converge in Euclidean distance to the global optimum with rate $\mathcal O(K^{-1/4})$ for a target of $K$ iterations. NCOTA-DGD is evaluated numerically over a logistic regression problem, showing faster convergence vis-\`a-vis running time than implementations of the classical DGD algorithm over digital and analog orthogonal channels.

翻译：本文提出NCOTA-DGD算法，一种结合局部梯度下降与新型非相干空中共识（NCOTA）的去中心化梯度下降（DGD）算法，用于解决无线连接系统上的分布式机器学习问题。NCOTA-DGD利用无线信道的波形叠加特性：通过将局部优化信号映射为前导序列的混合，在半双工约束下实现同时传输，并在接收端通过非相干合并达成共识。该算法无需在发射端和接收端获取信道状态信息，且利用平均信道路径损耗实现信号混合，无需显式知晓混合权重（这在基于共识的优化算法中通常是已知的）。理论分析与数值实验表明，对于固定共识和固定学习步长的光滑强凸问题，NCOTA-DGD的更新在欧氏距离上以$\mathcal O(K^{-1/4})$的速率收敛到全局最优解（$K$为目标迭代次数）。在逻辑回归问题上的数值评估显示，与通过数字和模拟正交信道实现的经典DGD算法相比，NCOTA-DGD在收敛速度与运行时间性能比上具有明显优势。