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