  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 TSAGA algorithm significantly outperforms the state-of-the-art, and is
able to achieve comparable learning performance as the error-free benchmark in
terms of both convergence rate and final test accuracy.
