Neural Document-level Multi-aspect Sentiment Classification (DMSC) usually requires a lot of manual aspect-level sentiment annotations, which is time-consuming and laborious. As document-level sentiment labeled data are widely available from online service, it is valuable to perform DMSC with such free document-level annotations. To this end, we propose a novel Diversified Multiple Instance Learning Network (D-MILN), which is able to achieve aspect-level sentiment classification with only document-level weak supervision. Specifically, we connect aspect-level and document-level sentiment by formulating this problem as multiple instance learning, providing a way to learn aspect-level classifier from the back propagation of document-level supervision. Two diversified regularizations are further introduced in order to avoid the overfitting on document-level signals during training. Diversified textual regularization encourages the classifier to select aspect-relevant snippets, and diversified sentimental regularization prevents the aspect-level sentiments from being overly consistent with document-level sentiment. Experimental results on TripAdvisor and BeerAdvocate datasets show that D-MILN remarkably outperforms recent weakly-supervised baselines, and is also comparable to the supervised method.