Projecting Ising Model Parameters for Fast Mixing
Inference in general Ising models is difficult, due to high treewidth making treebased algorithms intractable. Moreover, when interactions are strong, Gibbs sampling may take exponential time to converge to the stationary distribution. We present an algorithm to project Ising model parameters onto a parameter set that is guaranteed to be fast mixing, under several divergences. We find that Gibbs sampling using the projected parameters is more accurate than with the original parameters when interaction strengths are strong and when limited time is available for sampling.
Aims to improve the mixing rate of Gibbs sampling in 
      pairwise Ising models with strong interactions, which are known to be 
      "slow-mixing". Several projections to "fast-mixing" models are proposed; 
      essentially, parameters are identified which are as close as possible to 
      the original model, but weak enough to satisfy a spectral bound 
      establishing rapid mixing. Experiments show some regimes where this leads 
      to improved marginal estimates for a given computation time, compared to 
      variational methods (mean field and belief propagation variants) and Gibbs 
      sampling in the original model. The technical content builds 
      heavily on prior results establishing conditions for rapid mixing [4,8]. 
      But I haven't seen the idea of projecting onto the nearest rapidly mixing 
      model, as an approximate inference method, explored before. On the 
      domain of "toy tiny Ising models", results offer some nice improvements 
      over a good set of baselines (standard sampling and contemporary 
      variational methods). My main concern is that I think there should be more 
      than the usual amount of skepticism as to whether these results will scale 
      to larger models. Experiments involve comparisons of mixing times of 
      samplers on different models, and it is hard to judge how these will scale 
      with problem size. Also there is a (from all appearances) computationally 
      expensive projection step required to build the fast-mixing model, the 
      cost of which seems not to be accounted for. Finally, it is not at all 
      clear that generalization beyond binary states will be possible, since 
      establishing convergence bounds more generally is far more challenging. 
      CLARITY: In general the presentation is clear and reasonably 
      accessible, given the technical content. But there are some places where 
      reorganization and clarification is needed: * The paper refers several 
      times to "standard results" and even states these in the form of a theorem 
      (e.g. Theorem 6) without reference or proof sketch. Lemma 5 is left 
      unproven, the reference proves for a special case of zero-field. * 
      Discuss properties of the dependency matrix R. For instance, it does not 
      appear to be tractable due to the maximization, please state whether this 
      is the case. (I presume this is why lemma 5 is invoked.) * The second 
      half of Sec. 4 is nearly impossible to follow. Before Theorem 7 the text 
      references g, M, and Lambda before they are introduced. Then the statement 
      of Theorem 4 includes notation that is not really explained. If this 
      optimization is going to be discussed, more explanation is needed. 
      (Perhaps less time could be spent restating results from [8] which are not 
      really used.) * In Sec. 1, KL(q||p) is used for both directions of KL 
      divergence when the notation needs to be shifted for one. In the first 
      paragraph, q is used for the true distribution and p for the tractable 
      approximation; this is the opposite of almost all related literature. 
      EXPERIMENTS: It appears that the time comparison in Figure 2 
      does *not* include the computation required for projection. A somewhat 
      ambiguous statement to this effect appears in Sec 6.1 but is unclear; 
      please clarify and if it is the case, show results with projection time 
      included. As it stands, the proposed methods essentially get to use the 
      output of a sophisticated variational optimization without penalty, which 
      certainly makes the improvement over standard Gibbs less convincing. 
      It was disappointing not to see experiments on larger models, 
      given that Gibbs mixing times often depend on problem size. There are 
      certainly options for running experiments in regimes where junction tree 
      is intractable, like using a model where symmetries let the true marginals 
      be computed, or taking the output of a very long sampling run as truth. 
      I understand that KL(\theta||\psi) is intractable in general, but 
      it would still be interesting to explore here as a potential "best case" 
      for how sampling in an approximate model would perform. (Junction tree 
      could be used for the toy models used in the submission.) Mean 
      field is a degenerate case of the reverse KL projection, as the paper 
      points out, yet there is a large difference between mean field error and 
      the error from reverse KL projection. This deserves discussion.