Learning Preferences Graphs by Soft Projections onto Polyhedra ## AbstractWe discuss the problem of learning to predict the order of nodes in a graph from a real valued feedback associated with each node. This setting includes as special cases binary classification, multiclass categorization, and multilabel ordering. In the preference graph problem the nodes are labels and edges express preferences over labels. We approach the problem by defining a loss function for comparing a predicted graph with a feedback graph. This loss is defined 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 much smaller size and prove that it attains the same minimum. We then describe an efficient algorithm, called SOPOPO, for solving the reduced problem which employs a soft projection onto a polyhedron defined by a reduced set of constraints. We also describe and analyze a wrapper procedure for batch preference graph 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.
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