In this paper, we investigate over-the-air model aggregation in a federated edge learning (FEEL) system. We introduce a Markovian probability model to characterize the intrinsic temporal structure of the model aggregation series. With this temporal probability model, we formulate the model aggregation problem as to infer the desired aggregated update given all the past observations from a Bayesian perspective. We develop a message passing based algorithm, termed temporal-structure-assisted gradient aggregation (TSA-GA), to fulfil this estimation task with low complexity and near-optimal performance. We further establish the state evolution (SE) analysis to characterize the behaviour of the proposed TSA-GA algorithm, and derive an explicit bound of the expected loss reduction of the FEEL system under certain standard regularity conditions. In addition, we develop an expectation maximization (EM) strategy to learn the unknown parameters in the Markovian model. We show that the proposed TSA-GA significantly outperforms the state-of-the-art analog compression scheme, and is able to achieve comparable learning performance as the error-free benchmark in terms of final test accuracy.

中文翻译：

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用于空中联合边缘学习的时间结构辅助梯度聚合

在本文中，我们研究了联合边缘学习 (FEEL) 系统中的无线模型聚合。我们引入了马尔可夫概率模型来表征模型聚合系列的内在时间结构。使用这个时间概率模型，我们制定了模型聚合问题，以从贝叶斯的角度给定所有过去的观察结果来推断所需的聚合更新。我们开发了一种基于消息传递的算法，称为时间结构辅助梯度聚合（TSA-GA），以低复杂度和接近最佳的性能完成这一估计任务。我们进一步建立状态演化 (SE) 分析来表征所提出的 TSA-GA 算法的行为，并推导出在某些标准规律条件下 FEEL 系统的预期损失减少的明确界限。此外，我们开发了一种期望最大化 (EM) 策略来学习马尔可夫模型中的未知参数。我们表明，所提出的 TSA-GA 显着优于最先进的模拟压缩方案，并且能够在最终测试准确性方面实现与无错误基准相当的学习性能。