Robust Design for Reconfigurable Intelligent Surface Assisted Over-the-Air Computation

Distributed data aggregation is a critical design aspect in future Internet-of-Things (IoT) networks. Over-the-air computation (AirComp) is capable of achieving ultra-fast data aggregation by exploiting the superposition property of wireless channel. However, the performance of AirComp, measured by the mean-squared-error (MSE), is generally restricted by the unfavorable channel conditions and relies on the availability of perfect channel state information (CSI). In this paper, we propose to use reconfigurable intelligent surface (RIS) to assist the wireless data aggregation in IoT networks via AirComp in the presence of imperfect CSI. By taking into account the constraints of the transmit power at the devices and the unit modulus of the RIS, we formulate an optimization problem to jointly optimize the transmit power of IoT devices, the beamforming vector at the access point, and the phase-shift matrix at the RIS under the expectation-based channel uncertainty model. We present an alternating optimization method to solve this nonconvex problem. In each iteration, the transmit power and the receive beamformer are updated according to Karush-Kuhn-Tucker conditions and a closed-form solution, respectively. Moreover, we also develop a difference-of-convex algorithm to tackle the nonconvex rank-one constraint in the problem of optimizing the phase-shift matrix. Simulation results illustrate the robustness of the proposed algorithm in terms of minimizing the AirComp distortion.