Variational Inference in Mixed Probabilistic Submodular Models
We consider the problem of variational inference in probabilistic models with both log-submodular and log-supermodular higher-order potentials. These models can represent arbitrary distributions over binary variables, and thus generalize the commonly used pairwise Markov random fields and models with log-supermodular potentials only, for which efficient approximate inference algorithms are known. While inference in the considered models is #P-hard in general, we present efficient approximate algorithms exploiting recent advances in the field of discrete optimization. We demonstrate the effectiveness of our approach in a large set of experiments, where our model allows reasoning about preferences over sets of items with complements and substitutes.
The authors consider the problem of inference in mixed submodular models, where the probability of a set "A" is proportional to exp(F(A) + G(A)), where F and G are submodular (i.e. -G is supermodular). Previous work was only able to handle a single submodular or supermodular function in the exponent.  They then proceed to upper bound the partition function of the model by using a modular function that upper bounds F(A) - G(A). It is easy to calculate the partition function of a modular function. They then optimize over upper bounds to find an estimate of the partition function that is reasonably tight. Their optimization procedure involves an inner bound on the set of possible modular functions, then a block coordinate descent procedure using the Frank-Wolfe algorithm. The authors then empirically show that their approach achieves reasonable results in a product recommendation experiment.
The authors consider a useful generalization to inference in submodular models, which has been receiving a good amount of attention in recent conferences.  Their paper is well written and synthesizes many results in the literature and can provide an example of how to use previously developed tools.  However, the approach they take is not entirely novel compared to past work. With limited new theoretical results, to make the work a solid contribution it should have a more in-depth experimental analysis. Their experiment does not sufficiently compare to approaches outside the variational inference for submodular model framework. Specifically, they should compare to other approaches for estimating marginals, such as MCMC, and other approaches for estimation (e.g. logistic regression, SVM, boosting etc.) or mention why other approaches would not work.  == post-rebuttal answer==  Thank you for your rebuttal. After reading the other reviews I would agree that this is somewhat incremental, with many of the core ideas coming from previous work. However I found the problem to be interesting and believe that there are some nice algorithmic and modeling contributions. I would have liked to see a more extensive experimentation section with details on runtimes and other accuracy baselines to better judge the impact of this approach, especially since the algorithms are more heuristical in nature.