We consider problems where agents in a network seek a common quantity, measured independently and periodically by each agent through a local time-varying process. Numerous solvers addressing such problems have been developed in the past, featuring various adaptations of the local processing and the consensus step. However, existing solvers still lack support for advanced techniques, such as superiorization and over-the-air function computation (OTA-C). To address this limitation, we introduce a comprehensive framework for the analysis of distributed algorithms by characterizing them using the quasi-Fej\'er type algorithms and an extensive communication model. Under weak assumptions, we prove almost sure convergence of the algorithm to a common estimate for all agents. Moreover, we develop a specific class of algorithms within this framework to tackle distributed optimization problems with time-varying objectives, and, assuming that a time-invariant solution exists, prove its convergence to a solution. We also present a novel OTA-C protocol for consensus step in large decentralized networks, reducing communication overhead and enhancing network autonomy as compared to the existing protocols. The effectiveness of the algorithm, featuring superiorization and OTA-C, is demonstrated in a real-world application of distributed supervised learning over time-varying wireless networks, highlighting its low-latency and energy-efficiency compared to standard approaches.

翻译：考虑网络中智能体通过本地时变过程独立且周期性地测量共同目标量的问题。已有大量求解此类问题的算法，其核心在于本地处理与一致性步骤的不同组合。然而，现有算法仍缺乏对超优化（superiorization）及空中函数计算（OTA-C）等先进技术的支持。为此，我们提出一种面向分布式算法分析的统一框架：通过拟费耶尔型算法（quasi-Fejér type algorithms）与泛化通信模型对算法进行表征。在弱假设条件下，我们证明了算法对所有智能体几乎必然收敛于共同估计值。进一步，在该框架中构造了一类针对时变目标的分布式优化算法，并在假设存在时不变解的条件下，证明了其收敛性。我们同时提出一种适用于大规模去中心化网络的新型OTA-C一致性协议，相较现有协议可显著降低通信开销并提升网络自治能力。最终通过时变无线网络中分布式监督学习的实际应用验证了融合超优化与OTA-C的算法有效性，结果表明该方法相较标准算法具备更低延迟与更高能源效率。