In MIML (multi-instance multi-label learning) tasks, labels are often correlated with each other, and DAG (directed acyclic graph) is a common hierarchically structure which often occurs in the prediction of gene ontology biological functions of proteins. Considering the labels with directed acyclic graph structures in MIML, a novel algorithm named MIMLDAG (multi-instance multi-label directed acyclic graph) was proposed. MIMLDAG trained a low-dimensional subspace of shared labels from the feature space of original datasets, minimized the rank loss by a stochastic gradient descent method, and then incorporated the inner DAG hierarchical structure of labels for optimizing the output labels. MIMLDAG was applied to predict the protein functions in multiple datasets, and the results show that MIMLDAG possesses higher efficiency and predictive performance.