  Reconfigurable intelligent surface (RIS) as an emerging cost-effective
technology can enhance the spectrum- and energy-efficiency of wireless
networks. In this paper, we consider an RIS-aided green edge inference system,
where the inference tasks generated from resource-limited mobile devices (MDs)
are uploaded to and cooperatively performed at multiple resource-enhanced base
stations (BSs). Taking into account both the computation and uplink/downlink
transmit power consumption, we formulate an overall network power consumption
minimization problem, which calls for the joint design of the set of tasks
performed by each BS, transmit and receive beamforming vectors of the BSs,
transmit power of the MDs, and uplink/downlink phase-shift matrices at the RIS.
Such a problem is a mixed combinatorial optimization problem with nonconvex
constraints and is highly intractable. To address the challenge of the
combinatorial objective, a group sparse reformulation is proposed by exploiting
the group sparsity structure of the beamforming vectors, while a
block-structured optimization (BSO) approach is proposed to decouple the
optimization variables. Finally, we propose a BSO with mixed $\ell_{1,2}$-norm
and difference-of-convex-functions (DC) based three-stage framework to solve
the problem, where the mixed $\ell_{1,2}$-norm is adopted to induce the group
sparsity of beamforming vectors and DC is adopted to effectively handle the
nonconvex rank-one constraint after matrix lifting. Numerical results
demonstrate the supreme gain of deploying an RIS and confirm the effectiveness
of the proposed algorithm over the baseline algorithms.
