In this paper, we study the problem of active learning for cost-sensitive multiclass classification. We propose selective sampling algorithms, which process the data in a streaming fashion,querying only a subset of the labels. For these algorithms, we analyze the regret and label complexity when the labels are generated according to a generalized linear model. We establish that the gains of active learning over passive learning can range from none to exponentially large, based on a natural notion of margin. We also present a safety guarantee to guard against model mismatch. Numerical simulations show that our algorithms indeed obtain a low regret with a small number of queries.