A Consistent Regularization Approach for Structured Prediction
We propose and analyze a regularization approach for structured prediction problems. We characterize a large class of loss functions that allows to naturally embed structured outputs in a linear space. We exploit this fact to design learning algorithms using a surrogate loss approach and regularization techniques. We prove universal consistency and finite sample bounds characterizing the generalization properties of the proposed method. Experimental results are provided to demonstrate the practical usefulness of the proposed approach.
The paper presents a framework for stuctured prediction that allows for a wide range of loss functions through a "loss trick" that allows the losses to be represented through embeddings in RKHS for the outputs. They present an elegant framework for optimising the models, essentially boiling down to solving a kernel ridge regression type problem and then computing a pre-image (decoding) to arrive at the prediction. The authors provide analysis of consistency and present a generalization bound for the framework. The method is experimentally compared to alternative approaches in different structured prediction tasks with good results   
 In my view, this is a beautiful paper that will advance the field of structured prediction significantly and provides a platform for further development.   Nevertheless, the paper should be better related to existing work on vector-valued regression for structured output. A recent related work is but there are others: C ́eline Brouard, Florence D’Alch ́e-Buc, Marie Szafranski. Input Output Kernel Regression: Supervised and Semi-Supervised Structured Output Prediction with Operator-Valued Kernels. 2015.  The paper is generally well written, I have only few remarks: - line 70-72: you might note already here that this amounts to a ridge regression problem  in the output Hilbert space. - line 74: (7) is generally known as the pre-image (or decoding) problem. Good to mention it already here. - line 129: closing paranthesis missing: (\psi(y, - line 171: extra full stop  . [3]. - line 186: somehow convoluted grammar here: "of the same order of the generalization bounds...". Please rephrase. - line 211:  T => \ell in the dimension of the label space, I think. - line 212: i does not appear inside the argument of argmax or argmin - line 223-230: I found this subsection somehow cryptically written. Please write more to the point. What is the relationship of your results to SVMstruct. Can you cover the max-margin models such as SVMstruct and others (e.g. those in Bakir et al. 2007) as well or are those models outside your theory? - Figure 1: Please use only one of the terms, KRR or KRLS.