Beamforming Design Based on Two-Stage Stochastic Optimization for RIS-Assisted Over-the-Air Computation Systems

Over-the-air computation (AirComp) has been recognized as a promising technique of enabling the fusion center (FC) to aggregate the data gleaned from massive distributed wireless devices (WDs). Nevertheless, the computational performance of AirComp is significantly affected by the potentially poor channel conditions between the WDs and FC due to physical obstacles. For mitigating this limitation, we employ reconfigurable intelligent surfaces (RISs) for enhancing the reception quality and, thus, improve the computational performance of AirComp. Moreover, the previous studies of RIS-assisted AirComp tend to rely on the real-time channel state information (CSI), leading to excessive overhead since the number of RIS elements is large. To mitigate the above issue, a mixed-timescale penalty-dual-decomposition (MTPDD) algorithm is proposed, in which the transmit power of each WD, the receive beamforming vector at the FC, and the passive beamforming matrix of the RIS are jointly optimized. We aim to minimize the average computation mean-squared error (MSE) over time with reduced signaling overhead. Specifically, at each time slot, we optimize the short-term transmit power and receive the beamforming vector based on the real-time low-dimensional CSI vectors. In contrast, in each frame, we update the long-term passive RIS beamforming matrix based on the channel statistics. Besides, we analyzed both the convergence and the computational complexity of the proposed algorithms. Simulation results verify the benefits of our proposed MTPDD beamforming algorithm. It is also shown that the performance of the MTPDD algorithm approaches that achieved by the scheme using real-time perfect CSI with reduced signal overhead.