Generalization Bounds for Online Learning Algorithms with Pairwise Loss Functions

Efficient online learning with pairwise loss functions is a crucial component in building largescale learning system that maximizes the area under the Receiver Operator Characteristic (ROC) curve. In this paper we investigate the generalization performance of online learning algorithms with pairwise loss functions. We show that the existing proof techniques for generalization bounds of online algorithms with a pointwise loss can not be directly applied to pairwise losses. Using the Hoeffding-Azuma inequality and various proof techniques for the risk bounds in the batch learning, we derive data-dependent bounds for the average risk of the sequence of hypotheses generated by an arbitrary online learner in terms of an easily computable statistic, and show how to extract a low risk hypothesis from the sequence. In addition, we analyze a natural extension of the perceptron algorithm for the bipartite ranking problem providing a bound on the empirical pairwise loss. Combining these results we get a complete risk analysis of the proposed algorithm.