Structural magnetic resonance imaging (sMRI) is widely used for the brain neurological disease diagnosis, which could reflect the variations of brain. However, due to the local brain atrophy, only a few regions in sMRI scans have obvious structural changes, which are highly correlative with pathological features. Hence, the key challenge of sMRI-based brain disease diagnosis is to enhance the identification of discriminative features. To address this issue, we propose a dual attention multi-instance deep learning network (DA-MIDL) for the early diagnosis of Alzheimer's disease (AD) and its prodromal stage mild cognitive impairment (MCI). Specifically, DA-MIDL consists of three primary components: 1) the Patch-Nets with spatial attention blocks for extracting discriminative features within each sMRI patch whilst enhancing the features of abnormally changed micro-structures in the cerebrum, 2) an attention multi-instance learning (MIL) pooling operation for balancing the relative contribution of each patch and yield a global different weighted representation for the whole brain structure, and 3) an attention-aware global classifier for further learning the integral features and making the AD-related classification decisions. Our proposed DA-MIDL model is evaluated on the baseline sMRI scans of 1689 subjects from two independent datasets (i.e., ADNI and AIBL). The experimental results show that our DA-MIDL model can identify discriminative pathological locations and achieve better classification performance in terms of accuracy and generalizability, compared with several state-of-the-art methods.