Attention-based deep multiple-instance learning (MIL) has been applied to many machine-learning tasks with imprecise training labels. It is also appealing in hyperspectral target detection, which only requires the label of an area containing some targets, relaxing the effort of labeling the individual pixel in the scene. This article proposes an L1 sparsity-regularized attention multiple-instance neural network (L1-attention MINN) for hyperspectral target detection with imprecise labels that enforces the discrimination of false-positive instances from positively labeled bags. The sparsity constraint applied to the attention estimated for the positive training bags strictly complies with the definition of MIL and maintains better discriminative ability. The proposed algorithm has been evaluated on both simulated and real-field hyperspectral (subpixel) target detection tasks, where advanced performance has been achieved over the state-of-the-art comparisons, showing the effectiveness of the proposed method for target detection from imprecisely labeled hyperspectral data.