Margin-based structured prediction commonly uses a maximum loss over all possible structured outputs \cite{Altun03,Collins04b,Taskar03}. In natural language processing, recent work \cite{Zhang14,Zhang15} has proposed the use of the maximum loss over random structured outputs sampled independently from some proposal distribution. This method is linear-time in the number of random structured outputs and trivially parallelizable. We study this family of loss functions in the PAC-Bayes framework under Gaussian perturbations \cite{McAllester07}. Under some technical conditions and up to statistical accuracy, we show that this family of loss functions produces a tighter upper bound of the Gibbs decoder distortion than commonly used methods. Thus, using the maximum loss over random structured outputs is a principled way of learning the parameter of structured prediction models. Besides explaining the experimental success of \cite{Zhang14,Zhang15}, our theoretical results show that more general techniques are possible.
['Jean Honorio', 'Tommi S. Jaakkola']
Structured Prediction: From Gaussian Perturbations to Linear-Time Principled Algorithms