Slack and margin rescaling are variants of the structured output SVM. They define convex surrogates to task specific loss functions, which, when specialized to non-additive loss functions for multi-label problems, yield extensions to increasing set functions. We demonstrate in this paper that we may use these concepts to define polynomial time convex extensions of arbitrary supermodular functions. We further show that slack and margin rescaling can be interpreted as dominating convex extensions over multiplicative and additive families, and that margin rescaling is strictly dominated by slack rescaling. However, we also demonstrate that, while the function value and gradient for margin rescaling can be computed in polynomial time, the same for slack rescaling corresponds to a non-supermodular maximization problem.
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