Learning Efficient Random Maximum A-Posteriori Predictors with Non-Decomposable Loss Functions
In this work we develop efficient methods for learning random MAP predictors for structured label problems. In particular, we construct posterior distributions over perturbations that can be adjusted via stochastic gradient methods. We show that any smooth posterior distribution would suffice to define a smooth PAC-Bayesian risk bound suitable for gradient methods. In addition, we relate the posterior distributions to computational properties of the MAP predictors. We suggest multiplicative posteriors to learn super-modular potential functions that accompany specialized MAP predictors such as graph-cuts. We also describe label-augmented posterior models that can use efficient MAP approximations, such as those arising from linear program relaxations.
The paper adapts a random MAP predictor model for 
      structured prediction, and proposes to learn the model by minimizing the 
      PAC-bayesian bound. They discussed different possible choices of the 
      posterior distributions, including multiplicative models of non-negative 
      parameters for sub-modular models, and posterior distributions for 
      approximate MAP solutions (but without much details) . Quality: 
      median. 1. The paper compares with structured SVM. But more 
      natural comparisons may be methods in [6,22] which also learns random MAP 
      predictions. It would be nice to provide experimental comparison to these 
      methods. 2. It would be nice to present results on more than one 
      datasets. Clarity. The paper is well written overall. But some 
      part of Section 4 confuses me a bit. In particular, 1). Corollary 
      1 assumes additive posterior distributions, but right after it, the 
      statement that "every smooth posterior distribution remains quadratic 
      form" seems to over-generalize the result to any smooth posterior 
      distribution. Since this is considered as one of the important 
      contribution to the paper. Please you explain this point more clearly and 
      explicitly in the paper. 2). Section 4.1 proposes to build 
      posterior distributions directly for approximate MAP predictions. The idea 
      is interesting, but it seems that no solid example of this kind is 
      outlined, and that is no related experiments provided (am i correct?). 
      This makes me feel that this section is more a "future-looking" 
      discussion. If this true, please state this fact explicitly in the paper, 
      since otherwise it may confuse the readers. Also, is it necessary to state 
      Theorem 3 as a theorem? This is really a quite obvious result; stating it 
      as a theorem may make readers feel that this is an important important 
      contribution of the paper (especially in contrast to the other two 
      corollaries you used). Give that no solid result is provided in section 
      4.1, I suggest to significantly shorten this section and possibly 
      rearrange its position (e.g., putting it after Section 4.2). 
      Originality: median or low. The paper learns a random MAP 
      predictor model for structured prediction by directly minimizing the 
      PAC-bayesian bound. The random MAP framework has been proposed in [6,22, 
      8] and the loss minimization method is basically the same as the method in 
      [8] (probably the main difference is that [8] restricts on additive and 
      Gaussian posterior, but the technical difference seems very minor). I 
      think the real contribution of this paper is on proposing to use more 
      general posterior distributions, in particular, a multiplicative models of 
      non-negative parameters for sub-modular models, and (potentially) 
      posterior distributions for approximate MAP solutions. But note that the 
      paper only shows results on multiplicative posterior models; although 
      section 4.1 talks about building posterior models for approximate MAP, the 
      paper does not show any concrete example of this kind. 
      Significance. The method proposed in the paper may be useful to 
      improve accuracy of structured prediction problems.