Anytime Anyspace AND/OR Search for Bounding the Partition Function

Bounding the partition function is a key inference task in many graphical models. In this paper, we develop an anytime anyspace search algorithm taking advantage of AND/OR tree structure and optimized variational heuristics to tighten deterministic bounds on the partition function. We study how our priority-driven best-first search scheme can improve on stateof-the-art variational bounds in an anytime way within limited memory resources, as well as the effect of the AND/OR framework to exploit conditional independence structure within the search process within the context of summation. We compare our resulting bounds to a number of existing methods, and show that our approach offers a number of advantages on realworld problem instances taken from recent UAI competitions. Introduction Probabilistic graphical models, including Bayesian networks and Markov random fields, provide a framework for representing and reasoning with probabilistic and deterministic information (Dechter, Geffner, and Halpern, 2010; Darwiche, 2009; Dechter, 2013). Reasoning in a probabilistic graphical model often requires computing the partition function, i.e., the normalizing constant of the underlying distribution. In general, exact computation of the partition function is known to be #P-hard (Valiant, 1979), leading to the development of a broad array of approximate schemes. Particularly useful are schemes that provide guarantees, such as a confidence interval (upper and lower bounds), that can also improve them in an anytime and anyspace manner. Approximate elimination methods (Dechter and Rish, 2003; Liu and Ihler, 2011) and closely related variational bounds (Wainwright and Jordan, 2008) provide deterministic guarantees on the partition function. However, these bounds are not anytime; their quality often depends on the amount of memory available, and do not improve without additional memory. On the other hand, Monte Carlo methods, such as those based on importance sampling (Liu, Fisher, and Ihler, 2015), or approximate hash-based counting for weighted SAT (e.g., Chakraborty, Meel, and Vardi, 2016) can smoothly trade time for quality, but provide only probabilistic bounds (e.g., they hold with probability 1− δ for some confidence parameter δ), and can be slow to provide tight intervals. Copyright c � 2017, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. In this work, we explore AND/OR search algorithms for providing anytime, deterministic bounds on the partition function. Historically, search techniques are well studied in graphical models for optimization (e.g., MAP, or maximum a posteriori and weighted CSP tasks), (Shimony and Charniak, 1991; Santos, 1991; Kask and Dechter, 2001; F. Bacchus and Piassi, 2003b) and exact summation (Darwiche, 2001; Chavira and Darwiche, 2008; F. Bacchus and Piassi, 2003a; Sang, Beame, and Kautz, 2005; Darwiche, 2009). However, there have been relatively little recent study for approximate summation problems such as the partition function via search. One exception is Viricel et al. (2016), which adapts a depth-first branchand-bound search scheme to provide deterministic upper and lower bounds on the partition function. AND/OR search spaces (Dechter and Mateescu, 2007) provide an elegant framework that exploits conditional, possibly context-specific independence structure during search. In contrast to methods such as recursive conditioning or “clamping” (Darwiche, 2001; Weller and Domke, 2015; Dechter and Mateescu, 2007) and recent work on knowledge compilation (Kisa et al., 2014), that can also explore the AND/OR search space, most explicit AND/OR search algorithms were used for optimization, employing a fixed search order that restricts the search but enables the use of strong, pre-compiled heuristic functions and can lead to faster exploration and better early pruning (Marinescu and Dechter, 2009a,b; Otten et al., 2011; Otten and Dechter, 2012; Marinescu, Dechter, and Ihler, 2014, 2015). Other, related approaches for summation queries include cutset-conditioning for exact solutions (Pearl, 1988; Dechter, 2013) or approximation with sampling (Bidyuk and Dechter, 2007). Bidyuk, Dechter, and Rollon (2010) used conditioning to combine bound intervals on marginal probabilities. Our contributions In this paper, we develop an anytime anyspace AND/OR best-first search algorithm to improve deterministic bounds for the partition function. Our prioritydriven best-first search scheme takes advantage of both AND/OR tree search and optimized variational heuristics, to efficiently reduce the bound gap on the partition function. Empirical results demonstrate that our approach with heuristics extracted from weighted mini-bucket (Liu and Ihler, 2011) is almost always superior to the baselines on various benchmark-memory settings. Background Let X = (X1, . . . , XN ) be a vector of random variables, where each Xi takes values in a discrete domain Xi; we use lower case letters, e.g. xi ∈ Xi, to indicate a value of Xi. A graphical model over X consists of a set of factors F = {fα(Xα) | α ∈ I}, where each factor fα is defined on a subset Xα = {Xi | i ∈ α} ofX, called its scope. We associate an undirected graph G = (V,E) with F, where each node i ∈ V corresponds to a variable Xi and we connect two nodes, (i, j) ∈ E, iff {i, j} ⊆ α for some α. The set I then corresponds to cliques of G. We can interpret F as an unnormalized probability measure, so that