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The problem of bipartite ranking, where instances are labeled positive ornegative and the goal is to learn a scoring function that minimizes theprobability of mis-ranking a pair of positive and negative instances (orequivalently, that maximizes the area under the ROC curve), has been widelystudied in recent years. A dominant theoretical and algorithmic framework forthe problem has been to reduce bipartite ranking to pairwise classification; inparticular, it is well known that the bipartite ranking regret can beformulated as a pairwise classification regret, which in turn can be upperbounded using usual regret bounds for classification problems. Recently,Kotlowski et al. (2011) showed regret bounds for bipartite ranking in terms ofthe regret associated with balanced versions of the standard (non-pairwise)logistic and exponential losses. In this paper, we show that such(non-pairwise) surrogate regret bounds for bipartite ranking can be obtained interms of a broad class of proper (composite) losses that we term as stronglyproper. Our proof technique is much simpler than that of Kotlowski et al.(2011), and relies on properties of proper (composite) losses as elucidatedrecently by Reid and Williamson (2010, 2011) and others. Our result yieldsexplicit surrogate bounds (with no hidden balancing terms) in terms of avariety of strongly proper losses, including for example logistic, exponential,squared and squared hinge losses as special cases. We also obtain tightersurrogate bounds under certain low-noise conditions via a recent result ofClemencon and Robbiano (2011).

Surrogate Regret Bounds for Bipartite Ranking via Strongly Proper Losses