Amplify-and-Forward Relaying for Hierarchical Over-the-Air Computation

Over-the-air computation (AirComp) has emerged as a promising technique in future intelligent wireless networks, which enables swift functional computation among distributed wireless devices (WDs) by exploiting the superposition property of wireless channels. This paper studies a new
hierarchical
AirComp network over a large area, in which a set of intermediate relays are exploited to facilitate the massive data aggregation from a large number of WDs. Under this setup, we present a two-phase amplify-and-forward (AF) relaying protocol. In the first phase, the WDs simultaneously send their data to the relays, while in the second phase, the relays amplify the respectively received signals and concurrently forward them to the fusion center (FC) for aggregation. Our objective is to minimize the computational mean squared error (MSE) at the FC, by jointly optimizing the transmit coefficients of the WDs, the AF coefficients of the relays, and the de-noising factor of the FC, subject to their individual transmit power constraints. First, we consider the centralized design with global channel state information (CSI), in which the inter-relay signals can be exploited beneficially for data aggregation. In this case, we develop an alternating-optimization-based algorithm to obtain a high-quality solution to the computational MSE minimization problem. The obtained solution shows that the phase of the transmit coefficient at each WD is opposite to that of the WD-relay-FC channel to ensure the signal phase alignment at the FC, and the transmit power of each WD/relay follows a regularized composite-channel-inversion structure to strike a balance between minimizing the signal-magnitude-misalignment-induced error and the noise-induced error. Next, to reduce the signaling overhead caused by the centralized design, we consider an alternative decentralized design with partial CSI, in which the relays and the FC make their own decisions by only requiring the channel power gain information across different relays. In this case, the relays and FC need to treat the inter-relay signals as harmful interference or noise. Accordingly, we optimize the transmit coefficients of the WDs associated with each relay, and the relay AF coefficients (together with the FC de-noising factor) in an iterative manner, which can be implemented efficiently in a decentralized way. Finally, numerical results show the fast convergence of the proposed centralized and decentralized designs. It is also shown that both designs achieve significant MSE performance gains over benchmark schemes without the joint optimization.