In this paper, we propose a feedback reduction scheme for full-duplex relay-aided multiuser networks. The proposed scheme permits the base station (BS) to obtain channel state information (CSI) from a subset of strong users under substantially reduced feedback overhead. More specifically, we cast the problem of user identification and CSI estimation as a block sparse signal recovery problem in compressive sensing (CS). Using existing CS block recovery algorithms, we first obtain the identity of the strong users and then estimate their CSI using the best linear unbiased estimator (BLUE). To minimize the effect of noise on the estimated CSI, we introduce a back-off strategy that optimally backs-off on the noisy estimated CSI and derive the error covariance matrix of the post-detection noise. In addition to this, we provide exact closed form expressions for the average maximum equivalent SNR at the destination user. Numerical results show that the proposed algorithm drastically reduces the feedback air-time and achieves a rate close to that obtained by scheduling schemes that require dedicated error-free feedback from all the network users.