  Semi-supervised classification, one of the most prominent fields in machine
learning, studies how to combine the statistical knowledge of the often
abundant unlabeled data with the often limited labeled data in order to
maximize overall classification accuracy. In this context, the process of
actively choosing the data to be labeled is referred to as active learning. In
this paper, we initiate the non-asymptotic analysis of the optimal policy for
semi-supervised classification with actively obtained labeled data. Considering
a general Bayesian classification model, we provide the first characterization
of the jointly optimal active learning and semi-supervised classification
policy, in terms of the cost-performance tradeoff driven by the label query
budget (number of data items to be labeled) and overall classification
accuracy. Leveraging recent results on the R\'enyi Entropy, we derive tight
information-theoretic bounds on such active learning cost-performance tradeoff.
