Stochastic Beamforming for Reconfigurable Intelligent Surface Aided Over-the-Air Computation

Over-the-air computation (AirComp) is a promising technology that is capable of achieving fast data aggregation in Internet of Things (IoT) networks. The mean-squared error (MSE) performance of AirComp is bottlenecked by the unfavorable channel conditions. This limitation can be mitigated by deploying a reconfigurable intelligent surface (RIS), which reconfigures the propagation environment to facilitate the receiving power equalization. The achievable performance of RIS relies on the availability of accurate channel state information (CSI), which however is generally difficult to be obtained. In this paper, we consider an RIS-aided AirComp IoT network, where an access point (AP) aggregates sensing data from distributed devices. Without assuming any prior knowledge on the underlying channel distribution, we formulate a stochastic optimization problem to maximize the probability that the MSE is below a certain threshold. The formulated problem turns out to be non-convex and highly intractable. To this end, we propose a data-driven approach to jointly optimize the receive beamforming vector at the AP and the phase-shift vector at the RIS based on historical channel realizations. After smoothing the objective function by adopting the sigmoid function, we develop an alternating stochastic variance reduced gradient (SVRG) algorithm with a fast convergence rate to solve the problem. Simulation results demonstrate the effectiveness of the proposed algorithm and the importance of deploying an RIS in reducing the MSE outage probability.