Learning-based approaches to grasp planning are preferred over analytical methods due to their ability to better generalize to new, partially observed objects. However, data collection remains one of the biggest bottlenecks for grasp learning methods, particularly for multi-fingered hands. The relatively high dimensional configuration space of the hands coupled with the diversity of objects common in daily life requires a significant number of samples to produce robust and confident grasp success classifiers. In this paper, we present the first active deep learning approach to grasping that searches over the grasp configuration space and classifier confidence in a unified manner. We base our approach on recent success in planning multi-fingered grasps as probabilistic inference with a learned neural network likelihood function. We embed this within a multi-armed bandit formulation of sample selection. We show that our active grasp learning approach uses fewer training samples to produce grasp success rates comparable with the passive supervised learning method trained with grasping data generated by an analytical planner. We additionally show that grasps generated by the active learner have greater qualitative and quantitative diversity in shape.