Efficient Learning of Label Ranking by Soft Projections onto Polyhedra

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...

• Type: Journal
• Year: 2006
• Where: JMLR
• Authors: Shai Shalev-Shwartz, Yoram Singer
• Keywords: Feedback, Feedback Graph, JMLR 2006, Labels