A Discrete Optimization Approach to Supervised Ranking

In many ranking tasks in machine learning, the goal is to construct a scoring function f:X → R, where X⊂R, that can be used to rank a set of labeled examples {(x�,y�)}��� � , where x� ∈ X and y� ∈ {0,1}, that are randomly drawn from an unknown distribution on X × {0,1}. We present a mixed integer programming (MIP) method to generate this scoring function. In particular, the scoring function is chosen to be a linear combination of features, i.e., f(x�) = wx�, such that the coefficients w are optimal with respect to a ranking quality measure (area under the curve, discounted cumulative gain, P-Norm Push, etc.) on a training set. Other methods for ranking approximate the ranking quality measure by a convex function, whereas our MIP approach is exact. As a result, the proposed approach provides a benchmark against which other methods can be judged. We use datasets from various applications, and show that this novel method performs well on both training and test data, compared with traditional machine learning techniques.