On Sampling from the Gibbs Distribution with Random Maximum A-Posteriori Perturbations
In this paper we describe how MAP inference can be used to sample efficiently from Gibbs distributions. Specifically, we provide means for drawing either approximate or unbiased samples from Gibbs' distributions by introducing low dimensional perturbations and solving the corresponding MAP assignments. Our approach also leads to new ways to derive lower bounds on partition functions. We demonstrate empirically that our method excels in the typical "high signalhigh coupling" regime. The setting results in ragged energy landscapes that are challenging for alternative approaches to sampling and/or lower bounds.
This paper proposes several approaches to sample from 
      a Gibbs distribution over a discrete space by solving randomly perturbed 
      combinatorial optimization problems (MAP inference) over the same space. 
      The starting point is a known result [5] that allows to do sampling (in 
      principle, using high dimensional perturbations with exponential 
      complexity) by solving a single optimization problem. In this paper they 
      propose to 1) use more efficient low-dimensional random perturbations 
      to do approximate sampling (with probabilistic accuracy guarantees on tree 
      structured models) 2) estimate (conditional) marginals using ratios of 
      partition function estimates, and sequentially sample variables. They 
      propose a clever rejection strategy based on self reduction that 
      guarantees unbiasedness of the samples. Low-dimensional perturbations 
      are also used heuristically on loopy graphs, showing some promising 
      empirical results on Ising models and for an image segmentation problem. 
      New lower bounds on the partition function are also derived and evaluated 
      on Ising grids, showing improvements over variational bounds when 
      interactions are strong. Sampling is a fundamental problem with 
      lots of applications, so this paper is definitely very relevant for NIPS. 
      This direction of using random perturbations and MAP inference is as far 
      as I know novel and very different from traditional MCMC sampling. I think 
      there is a potential for this type of approach, especially for 
      probabilistic models with hard dependencies that are notoriously hard for 
      MCMC. I also like the fact that MAP inference is used as a black box, so 
      that future improvements on MAP solvers would directly translate into 
      better sampling methods. This is an interesting paper making a 
      step forward towards understanding MAP inference with random 
      perturbations, but I feel there is still a significant gap between theory 
      and practice. The theory on low dimensional perturbations (section 3) only 
      works for trees, so it's not really useful in practice (it can be done by 
      dynamic programming). Low dimensional perturbations are then used 
      heuristically on loopy models showing some good empirical results, but 
      without any guarantee (not even asymptotically), which is undesirable. On 
      the other hand the rejection sampler (section 4) is shown to work in 
      theory but not really evaluated in practice. The paper is 
      generally well written and easy to read. The intuitive idea is clear, and 
      the mathematical formalization is also good except for a few minor things 
      (see below) The idea of casting the averaging of multiple MAP 
      queries into a single one is clear. Is there an advantage (computational 
      or statistical) in using the extended model to do the averaging in a 
      single run over collecting multiple (independent) samples perturbing the 
      original model and then averaging them (this seems to be what is done in 
      the experiments)? Since the graphical model is extended creating 
      multiple copies of the variables, which “copy” do you propose to use as a 
      sample? I like the approach in section 4. The self-reduction and 
      rejection strategy is clever. One issue that is not addressed is how to 
      compute the expectations (line 1 in the pseudocode). Do you have any 
      concentration results on those expectations (like end of section 2)? can 
      you estimate them with high probability using a small number of samples? 
      Wouldn't sample averages introduce some bias (and some failure 
      probability)? I found the part on lower bounds (section 5) 
      interesting, but not so relevant to the sampling problem addressed in the 
      paper. Related to this, I would have liked to see more experiments 
      on sampling in place of the ones on the tightness of the bounds on the 
      partition function (a topic already addressed in [6]). It would have 
      been more convincing to actually run algorithm 1 and plot the runtime 
      (taking into account the time for MAP inference) in place of the second 
      experiment (figure 1, middle). This would also have answered the issue of 
      how the expectations are estimated, and the resulting quality of the 
      samples if the estimates are not exact. The last two experiments 
      on sampling using low dimensional perturbations are interesting, and show 
      the potential of this type of approach. I’m surprised to see that Gibbs 
      sampling performs so poorly, even for low couplings strength (I would 
      expect the performance of Gibbs sampling to be good for almost uniform 
      probability distributions, and then drop as the coupling strength 
      increases). Any intuition why it is not the case? Perhaps something like 
      Swendsen-Wang would also be a good (stronger) baseline, and a more fair 
      comparison with respect to variational methods. Other minor 
      comments: the frequent use of “efficiently” might be misleading 
      considering that it involves solving a potentially NP-hard optimization 
      problem is theta^hat on line 127-128 a function of all the copies 
      x_{alpha,j_alpha} (extended variables) or just x_alpha? line 
      140-141, it would be better to use a different symbol for the extended 
      variable vector (x is already used) i is not binded in the sum 
      defining theta^hat in line 156. maybe missing a sum? (same thing in 
      corollary 2) how are the m_i chosen in theorem 2? the restriction 
      m_r=m_s=1 in line 158 is not clear line 161,what does it mean that 
      x_r,x_s \in argmax? the argmax should be a set of solutions (assignments 
      to the extended variables), while it seems x_r and x_s are single 
      variables what is |dom(theta)| on line 268? Is it 2^n (for n 
      binary variables)? line 339, The sub-exponential runtime comment 
      should be made more precise, specifying that the tightness of the bounds 
      (hence rejection probability) is dependent on the graphical model (e.g. 
      coupling strength for Ising grids) what is the size of the grid 
      used for the approximate sampling experiment? (figure 1 right) If 
      possible, it would be good to use higher resolution figures as they 
      currently do not print well.