Decentralized Gradient Descent (DGD) is a popular algorithm used to solve decentralized optimization problems in diverse domains such as remote sensing, distributed inference, multi-agent coordination, and federated learning. Yet, executing DGD over wireless systems affected by noise, fading and limited bandwidth presents challenges, requiring scheduling of transmissions to mitigate interference and the acquisition of topology and channel state information -- complex tasks in wireless decentralized systems. This paper proposes a DGD algorithm tailored to wireless systems. Unlike existing approaches, it operates without inter-agent coordination, topology information, or channel state information. Its core is a Non-Coherent Over-The-Air (NCOTA) consensus scheme, exploiting a noisy energy superposition property of wireless channels. With a randomized transmission strategy to accommodate half-duplex operation, transmitters map local optimization signals to energy levels across subcarriers in an OFDM frame, and transmit concurrently without coordination. It is shown that received energies form a noisy consensus signal, whose fluctuations are mitigated via a consensus stepsize. NCOTA-DGD leverages the channel pathloss for consensus formation, without explicit knowledge of the mixing weights. It is shown that, for the class of strongly-convex problems, the expected squared distance between the local and globally optimum models vanishes with rate $\mathcal O(1/\sqrt{k})$ after $k$ iterations, with a proper design of decreasing stepsizes. Extensions address a broad class of fading models and frequency-selective channels. Numerical results on an image classification task depict faster convergence vis-\`a-vis running time than state-of-the-art schemes, especially in densely deployed networks.

翻译：去中心化梯度下降（DGD）是一种广泛应用于遥感、分布式推理、多智能体协调和联邦学习等领域的求解去中心化优化问题的流行算法。然而，在受噪声、衰落和有限带宽影响的无线系统上执行DGD面临挑战，需要调度传输以减轻干扰并获取拓扑和信道状态信息——这些在无线去中心化系统中是复杂任务。本文提出了一种专为无线系统定制的DGD算法。与现有方法不同，该算法无需智能体间协调、拓扑信息或信道状态信息即可运行。其核心是一种非相干空中（NCOTA）共识方案，利用了无线信道的噪声能量叠加特性。通过采用随机化传输策略以适应半双工操作，发射器将局部优化信号映射到OFDM帧中多个子载波上的能量电平，并无需协调地并发传输。研究表明，接收能量形成带噪声的共识信号，其波动通过共识步长得以缓解。NCOTA-DGD利用信道路径损耗进行共识形成，无需显式知道混合权重。研究表明，对于强凸问题类别，在适当设计递减步长的情况下，经过k次迭代后，局部与全局最优模型之间的期望平方距离以𝒪(1/√k)的速率消失。扩展方法适用于广泛的衰落模型和频率选择性信道。在图像分类任务上的数值结果表明，与现有最先进方案相比，特别是在密集部署网络中，该方法在运行时间下实现了更快的收敛速度。