## Abstract

We discuss the problem of learning to rank labels from a real valued feedback associated with each label. We cast the feedback as a preferences graph where the nodes of the graph are the labels and edges express preferences over labels. We tackle the learning problem by defining a loss function for comparing a predicted graph with a feedback graph. This loss is materialized by decomposing the feedback graph into bipartite sub-graphs. We then adopt the maximum-margin framework which leads to a quadratic optimization problem with linear constraints. While the size of the problem grows quadratically with the number of the nodes in the feedback graph, we derive a problem of a significantly smaller size and prove that it attains the same minimum We then describe an efficient algorithm, called SOPOPO, for solving the reduced problem by employing a soft projection onto the polyhedron defined by a reduced set of constraints. We also describe and analyze a wrapper procedure for batch learning when multiple graphs are provided for training. We conclude with a set of experiments which show significant improvements in run time over a state of the art interior-point algorithm.

Original language | English (US) |
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Pages (from-to) | 1567-1599 |

Number of pages | 33 |

Journal | Journal of Machine Learning Research |

Volume | 7 |

State | Published - Jul 1 2006 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Software
- Control and Systems Engineering
- Statistics and Probability
- Artificial Intelligence