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We propose and analyze a novel theoretical and algorithmic framework forstructured prediction. While so far the term has referred to discrete outputspaces, here we consider more general settings, such as manifolds or spaces ofprobability measures. We define structured prediction as a problem where theoutput space lacks a vectorial structure. We identify and study a large classof loss functions that implicitly defines a suitable geometry on the problem.The latter is the key to develop an algorithmic framework amenable to a sharpstatistical analysis and yielding efficient computations. When dealing withoutput spaces with infinite cardinality, a suitable implicit formulation of theestimator is shown to be crucial.

A General Framework for Consistent Structured Prediction with Implicit Loss Embeddings