Barrier Frank-Wolfe for Marginal Inference
We introduce a globally-convergent algorithm for optimizing the tree-reweighted (TRW) variational objective over the marginal polytope. The algorithm is based on the conditional gradient method (Frank-Wolfe) and moves pseudomarginals within the marginal polytope through repeated maximum a posteriori (MAP) calls. This modular structure enables us to leverage black-box MAP solvers (both exact and approximate) for variational inference, and obtains more accurate results than tree-reweighted algorithms that optimize over the local consistency relaxation. Theoretically, we bound the sub-optimality for the proposed algorithm despite the TRW objective having unbounded gradients at the boundary of the marginal polytope. Empirically, we demonstrate the increased quality of results found by tightening the relaxation over the marginal polytope as well as the spanning tree polytope on synthetic and real-world instances.
Overview: The present manuscript introduces an approach to approximate marginal inference in graphical models that is based on the Frank-Wolfe/conditional gradient method.The main objective is to optimize over the full marginal polytope (a constraint set that grows exponentially) instead of the local polytope (a relaxation of the marginal polytope that typical message passing algorithms optimize over).The presented approach works by repeatedly solving a linearization of the original problem, which in this specific case corresponds to MAP inference, with the natural/theta parameters determined by the current iterate.A specific problem that should be addressed in this context is that the gradient is unbounded at the boundary of the marginal polytope. A barrier-like approach, inspired by interior point methods, is introduced to handle this situation and aid convergence. The empirical results demonstrate improvements versus optimization over the local polytope.Positive points: + This paper provides interesting new perspectives on optimization-based/variational approximate marginal inference in graphical models.While a lot of recent work has focused on "smoothing" the MAP problem to achieve faster convergence (thereby effectively marginalizing), in this work, the smooth marginalization problem is solved by a sequence of non-smooth MAP problems. This highlights again the close connection between the two problems (MAP/marginalization) and provides interesting perspectives. + Related work is discussed in sufficient detail, and the work is well embedded into its scientific context. + The theory regarding the "barrier" approach is developed in great detail in the supplementary material.It is clear that a lot of effort went into development of the approach. + The material developed in the paper seems to be technically correct.Negative points: - I have some criticism regarding the approach in general (even though I find it interesting, from a pure curiosity-based research perspective).The use of the Frank-Wolfe method is indicated if optimization over a *non-linear* objective function is not plausible due to the difficulty of the constraint set, but optimization of a linear objective function over the same constraint set can be handled gracefully.However, this is not the case here.Optimization of a linear objective function over the exact marginal polytope is NP-hard as well.How much is really gained here?In fact, one could argue that it is even harder to solve the MAP problem, sine it is non-smooth. Many recent approaches have focused on smoothing the MAP problem, effectively turning it into a smooth marginalization problem, precisely to improve convergence.What must be done in practice, with the proposed approach, is to solve the inner MAP problem approximately, thereby voiding guarantees.One might argue that it is more elegant to establish a well-defined approximation in the first place, and to then solve the approximation exactly.- The manuscript is purely concerned about approximation quality, and not at all about computational cost. As an example, when evaluating the impact of using approximate MAP solutions in the inner loop, the authors run three solvers (QPBO, TRW-S, ICM) and use the result that realizes the highest energy.Clearly, this is not very practical outside of synthetic experiments.- In the discussion (section 6), the authors refer to their approach as performing "exact marginal inference".This is not the case: While the constraint set is exact, the objective function still uses an approximation of the exact entropy, the TRW bound.I also do not find it true that tightening over the spanning tree polytope has remained relatively unexplored in recent work.This was done both in the original work of Wainwright et al. (2005), as well as the more recent work of Jancsary and Matz (2011).- Figures 1 and 2 requires further polishing: The caption lines run over the space allocated to each specific sub-figure, making the captions very hard to read.In Figure 2 (a), the label of the Y coordinate states "Error in Marginals", but the caption states "Error in logZ".In general, the experiments section and the figures therein leave a rather disorganized impression. I assume that this is partially due to the lack of space, but it seems that further improvements could be achieved here.- A question regarding Figures 2 (c) and (d), and the statement in Section 5.1: How can the optimization of the edge occurrence probabilities rho possibly not have any effect on the tightness of the bound on on logZ if optimizing over the local polytope? Rho is expected to have a significant influence on the quality of the entropy approximation, both when optimizing over the marginal polytope *and* when optimizing over the local polytope.Both in Wainwright et al. (2005) and Jancsary and Matz (2011), the improvement of the bound was considerable in simulations.Is this an artifact of the specific type of interactions chosen, or does it perhaps point to a problem in the code/experimental setup?- A question regarding Figure 3: I assume that for TRBP, no optimization over rho was performed (this relates to my previous question - why should optimization over rho not help)?This is a different problem now than in Figure 2, so even if optimization over rho did not help there, one cannot in general conclude that it would not help for the Chinese characters problem.It seems somewhat unfair to compare the proposed method *with* optimization over rho to TRBP *without* such optimization.- Minor point: Footnote 2 is a fragment -"which converges which iterates differing by 0.002"