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Margin-based structured prediction commonly uses a maximum loss over allpossible structured outputs \cite{Altun03,Collins04b,Taskar03}. In naturallanguage processing, recent work \cite{Zhang14,Zhang15} has proposed the use ofthe maximum loss over random structured outputs sampled independently from someproposal distribution. This method is linear-time in the number of randomstructured outputs and trivially parallelizable. We study this family of lossfunctions in the PAC-Bayes framework under Gaussian perturbations\cite{McAllester07}. Under some technical conditions and up to statisticalaccuracy, we show that this family of loss functions produces a tighter upperbound of the Gibbs decoder distortion than commonly used methods. Thus, usingthe maximum loss over random structured outputs is a principled way of learningthe parameter of structured prediction models. Besides explaining theexperimental success of \cite{Zhang14,Zhang15}, our theoretical results showthat more general techniques are possible.

Structured Prediction: From Gaussian Perturbations to Linear-Time Principled Algorithms