Sampling from Probabilistic Submodular Models
Submodular and supermodular functions have found wide applicability in machine learning, capturing notions such as diversity and regularity, respectively. These notions have deep consequences for optimization, and the problem of (approximately) optimizing submodular functions has received much attention. However, beyond optimization, these notions allow specifying expressive probabilistic models that can be used to quantify predictive uncertainty via marginal inference. Prominent, well-studied special cases include Ising models and determinantal point processes, but the general class of log-submodular and log-supermodular models is much richer and little studied. In this paper, we investigate the use of Markov chain Monte Carlo sampling to perform approximate inference in general log-submodular and log-supermodular models. In particular, we consider a simple Gibbs sampling procedure, and establish two sufficient conditions, the first guaranteeing polynomial-time, and the second fast (O(n log n)) mixing. We also evaluate the efficiency of the Gibbs sampler on three examples of such models, and compare against a recently proposed variational approach.
The authors provide a formal analysis of the mixing time of simple Gibbs sampling schemes on probabilistic submodular models. The main result is that the mixing time of Gibbs is polynomial or even less nlog(n) with the dimensionality on submodular models. To my knowledge, this work is novel and the result is interesting.Quality: The writing of this paper is clear and most of the paper is easy to follow. However, the author gives little introductions on some key results, like Therom 2, so it is not well self-contained. This makes it hard to understand for people not familiar the specific literature.Strength: The bound of mixing time is important for MCMC methods, but often difficult to establish. This work provides a novel result of the mixing time using submodularity.Questions:In addition, the proof of two main theorems on the bounds is not clear to me. More specific: Theorem 1: The idea of the set of canonical path is clear, but the definition C_{s,s'} seems simply the product space of two configurations, rather than a canonical path between them. This makes it difficult to understand the injective map eta_{s,s'}.Theorem 3: It is not clear to me what is the relationship between the chain (X_t, Y_t) and set (R, S). Is (R,S) the current sample from the pair of chains? In particular, what does it mean the two chain are identical if v = r?One more point is that it is good have more discussion on how realistic the strong condition for fast mixing. It seems very strong to have gamma_{f, beta} < 1 on most interesting models.I think the author should be careful about not over advertising the polynomial mixing time, because the bound is actually exponential to zeta, the notation of distance to modular function. It can be very large in many models, so the bounds may not be a good guid of real mixing time in practice.For the experiment, it is good to compare Gibbs with variational method, but it is not clear to me to see the bound of mixing time in the result.