A New Class of Upper Bounds on the Log Partition Function (1301.0610v1)

Abstract: Bounds on the log partition function are important in a variety of contexts, including approximate inference, model fitting, decision theory, and large deviations analysis. We introduce a new class of upper bounds on the log partition function, based on convex combinations of distributions in the exponential domain, that is applicable to an arbitrary undirected graphical model. In the special case of convex combinations of tree-structured distributions, we obtain a family of variational problems, similar to the Bethe free energy, but distinguished by the following desirable properties: i. they are cnvex, and have a unique global minimum; and ii. the global minimum gives an upper bound on the log partition function. The global minimum is defined by stationary conditions very similar to those defining fixed points of belief propagation or tree-based reparameterization Wainwright et al., 2001. As with BP fixed points, the elements of the minimizing argument can be used as approximations to the marginals of the original model. The analysis described here can be extended to structures of higher treewidth e.g., hypertrees, thereby making connections with more advanced approximations e.g., Kikuchi and variants Yedidia et al., 2001; Minka, 2001.

• The paper presents a novel convex optimization framework based on convex combinations in the exponential family to bound the log partition function.
• It leverages tree-structured distributions to formulate a tractable variational problem that overcomes local minima issues of traditional mean field methods.
• Empirical evaluations on grid and complete graphs show that the optimized bounds outperform naive approaches, offering robust and reliable inference.

Upper Bounds on the Log Partition Function

The paper by Wainwright et al. presents a novel approach to deriving upper bounds on the log partition function, a quantity of fundamental significance in graphical models. The authors introduce an innovative class of bounds that utilize convex combinations of distributions defined in the exponential domain, applicable across arbitrary undirected graphical models.

Key Contributions and Methodology

1. Exponential Family and Convex Combinations: The methodology presented is rooted in the exponential family framework. By leveraging convex combinations of distributions represented by exponential parameters, the authors construct bounds that are more tractable. Specifically, the focus is on tree-structured distributions, leading to a variational formulation akin to the Bethe free energy.
2. Convex Optimization: Central to the paper is the formulation of an optimization problem that is both convex and characterized by a unique global minimum. This optimization problem is defined over the parameters of tractable substructures, specifically spanning trees within the graph. The solution to this problem provides an upper bound to the log partition function.
3. Theoretical Insights and Practical Implications: The convexity of the proposed bounds is contrasted with the non-convex nature of traditional methods like mean field theory. This convexity ensures that these bounds are less susceptible to local minima issues, providing robustness and reliability in the estimations. The derivations show a close relationship to belief propagation fixed points, hinting at connections with tree-based reparameterization.
4. Empirical Evaluation: Through experiments on grid structures and complete graphs, the paper demonstrates the efficacy of the proposed bounds. The results illustrate that the optimized upper bounds outperform naive mean field bounds under various conditions, particularly with increasing edge strengths and complexity of connections.

Implications and Future Directions

The implications of this research are twofold: theoretical and practical. Theoretically, the work extends the understanding of the log partition function bounding by incorporating concepts from convex optimization and variational approaches. Practically, it furnishes approximate inference algorithms with a robust tool for marginal approximations in complex graphical models.

Future research might focus on extending these bounds to more complex structures like hypertrees, potentially offering tighter bounds. Another avenue is exploring efficient algorithms for the distribution optimization over these structures, as challenges increase with graph complexity.

In conclusion, the work of Wainwright et al. adds a valuable perspective to the approximation of a critical statistical function, offering both novel theoretical insights and practical tools for computational inference within probabilistic graphical models.