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The impacts of channel estimation errors, inter-cell interference, phaseadjustment cost, and computation cost on an intelligent reflecting surface(IRS)-assisted system are severe in practice but have been ignored forsimplicity in most existing works. In this paper, we investigate amulti-antenna base station (BS) serving a single-antenna user with the help ofa multi-element IRS in the presence of channel estimation errors and inter-cellinterference. We consider imperfect channel state information (CSI) at the BS,i.e., imperfect CSIT, and focus on the robust optimization of the BS'sinstantaneous CSI-adaptive beamforming and the IRS's quasi-static phase shifts.First, we formulate the robust optimization of the BS's instantaneous channelstate information (CSI)-adaptive beamforming and IRS's quasi-static phaseshifts for the ergodic rate maximization as a very challenging two-timescalestochastic non-convex problem. Then, we obtain a closed-form beamformer for anygiven phase shifts and a more tractable single-timescale stochastic non-convexproblem only for phase shifts. Next, we propose a low-complexity stochasticalgorithm to obtain quasi-static phase shifts which correspond to a KKT pointof the single-timescale stochastic problem. It is worth noting that theproposed method offers a closed-form robust instantaneous CSI-adaptivebeamforming design that can promptly adapt to rapid CSI changes over slots anda robust quasi-static phase shift design of low computation and phaseadjustment costs in the presence of channel estimation errors and inter-cellinterference. Finally, numerical results demonstrate the notable gains of theproposed robust joint design over existing ones and reveal the practical valuesof the proposed solutions.

Low-complexity Robust Optimization for an IRS-assisted Multi-Cell Network