  We propose a new application of embedding techniques for problem retrieval in
adaptive tutoring. The objective is to retrieve problems whose mathematical
concepts are similar. There are two challenges: First, like sentences, problems
helpful to tutoring are never exactly the same in terms of the underlying
concepts. Instead, good problems mix concepts in innovative ways, while still
displaying continuity in their relationships. Second, it is difficult for
humans to determine a similarity score that is consistent across a large enough
training set. We propose a hierarchical problem embedding algorithm, called
Prob2Vec, that consists of abstraction and embedding steps. Prob2Vec achieves
96.88\% accuracy on a problem similarity test, in contrast to 75\% from
directly applying state-of-the-art sentence embedding methods. It is
interesting that Prob2Vec is able to distinguish very fine-grained differences
among problems, an ability humans need time and effort to acquire. In addition,
the sub-problem of concept labeling with imbalanced training data set is
interesting in its own right. It is a multi-label problem suffering from
dimensionality explosion, which we propose ways to ameliorate. We propose the
novel negative pre-training algorithm that dramatically reduces false negative
and positive ratios for classification, using an imbalanced training data set.
