In many real-world applications, one object (e.g., image) can be represented or described by multiple instances (e.g., image patches) and simultaneously associated with multiple labels. Such applications can be formulated as multi-instance multi-label learning (MIML) problems and have been extensively studied during the past few years. Existing MIML methods are useful in many applications but most of which suffer from relatively low accuracy and training efficiency due to several issues: i) the inter-label correlations (i.e., the probabilistic correlations between the multiple labels corresponding to an object) are neglected; ii) the inter-instance correlations cannot be learned directly (or jointly) with other types of correlations due to the missing instance labels; iii) diverse inter-correlations (e.g., inter-label correlations, inter-instance correlations) can only be learned in multiple stages. To resolve these issues, a new single-stage framework called broad multi-instance multi-label learning (BMIML) is proposed. In BMIML, there are three innovative modules: i) an auto-weighted label enhancement learning (AWLEL) based on broad learning system (BLS); ii) A specific MIML neural network called scalable multi-instance probabilistic regression (SMIPR); iii) Finally, an interactive decision optimization (IDO). As a result, BMIML can achieve simultaneous learning of diverse inter-correlations between whole images, instances, and labels in single stage for higher classification accuracy and much faster training time. Experiments show that BMIML is highly competitive to (or even better than) existing methods in accuracy and much faster than most MIML methods even for large medical image data sets (> 90K images).