Multiclass-Multilabel Learning when the Label Set Grows with the Number of Examples

We discuss multiclass-multilabel classification problems in which the set of candidate labels is extremely large. Most existing multiclass-multilabel learning algorithms expect to observe a statistically significant sample from each class, and fail if they receive only a handful of examples per class. We propose and analyze the following two-stage approach: first use an arbitrary (perhaps heuristic) classification algorithm to construct an initial classifier, then apply a simple but principled method to augment this classifier by removing harmful labels from the label set. A careful theoretical analysis allows us to justify our approach under some reasonable conditions (such as label sparsity and power-law distribution of label frequencies), even when the training set does not provide a statistically significant representation of most classes. Surprisingly, our theoretical analysis continues to hold even when the number of labels exceeds the sample size. We demonstrate the merits of our approach on the ambitious task of categorizing the entire web using the 1.5 million categories defined on Wikipedia.