Monte Carlo methods are powerful tools for solving problems involving complex probability distributions. Despite their versatility, these methods often suffer from inefficiencies, especially when dealing with rare events. As such, importance sampling emerged as a prominent technique for alleviating these challenges. Recently, a new scheme called Abstraction Sampling was developed that incorporated stratification to importance sampling over graphical models. However, existing work only explored a limited set of abstraction functions that guide stratification. This study introduces three new classes of abstraction functions combined with seven distinct partitioning schemes, resulting in twenty-one new abstraction functions, each motivated by theory and intuition from both search and sampling domains. An extensive empirical analysis on over 400 problems compares these new schemes highlighting several well-performing candidates.

A major limiting factor in graphical model inference is the complexity of computing the partition function. Exact message-passing algorithms such as Bucket Elimination (BE) require exponential memory to compute the partition function; therefore, approximations are necessary. In this paper, we build upon a recently introduced methodology called Deep Bucket Elimination (DBE) that uses classical Neural Networks to approximate messages generated by BE for large buckets. The main feature of our new scheme, renamed NeuroBE, is that it customizes the architecture of the neural networks, their learning process and in particular, adapts the loss function to the internal form or distribution of messages. Our experiments demonstrate significant improvements in accuracy and time compared with the earlier DBE scheme.

Bucket Elimination (BE) is a universal inference scheme that can solve most tasks over probabilistic and deterministic graphical models exactly. However, it often requires exponentially high levels of memory (in the induced-width) preventing its execution. In the spirit of exploiting Deep Learning for inference tasks, in this paper, we will use neural networks to approximate BE. The resulting Deep Bucket Elimination (DBE) algorithm is developed for computing the partition function. We provide a proof-of-concept empirically using instances from several different benchmarks, showing that DBE can be a more accurate approximation than current state-of-the-art approaches for approximating BE (e.g. the mini-bucket schemes), especially when problems are sufficiently hard.

Abstraction Sampling (AS) is a recently introduced enhancement of Importance Sampling that exploits stratification by using a notion of abstractions: groupings of similar nodes into abstract states. It was previously shown that AS performs particularly well when sampling over an AND/OR search space; however, existing schemes were limited to ``proper'' abstractions in order to ensure unbiasedness, severely hindering scalability. In this paper, we introduce AOAS, a new Abstraction Sampling scheme on AND/OR search spaces that allow more flexible use of abstractions by circumventing the properness requirement. We analyze the properties of this new algorithm and, in an extensive empirical evaluation on five benchmarks, over 480 problems, and comparing against other state of the art algorithms, illustrate AOAS's properties and show that it provides a far more powerful and competitive Abstraction Sampling framework.

We introduce the concept of local bucket error for the mini-bucket heuristics and show how it can be used to improve the power of AND/OR search for combinatorial optimization tasks in graphical models (e.g. MAP/MPE or weighted CSPs). The local bucket error illuminates how the heuristic errors are distributed in the search space, guided by the mini-bucket heuristic. We present and analyze methods for compiling the local bucket-errors (exactly and approximately) and show that they can be used to yield an effective tool for balancing look-ahead overhead during search. This can be especially instrumental when memory is restricted, accommodating the generation of only weak compiled heuristics. We illustrate the impact of the proposed schemes in an extensive empirical evaluation for both finding exact solutions and anytime suboptimal solutions.

Best-first search can be regarded as anytime scheme for producing lower bounds on the optimal solution, a characteristic that is mostly overlooked. We explore this topic in the context of AND/OR best-first search, guided by the MBE heuristic, when solving graphical models. In that context, the impact of the secondary heuristic for subproblem ordering may be significant, especially in the anytime context. Indeed, our paper illustrates this, showing that the new concept of bucket errors can advise in providing effective subproblem orderings in AND/OR search.

The paper explores the potential of look-ahead methods within the context of AND/OR search in graphical models using the Mini-Bucket heuristic for combinatorial optimization tasks (e.g., weighted CSPS or MAP inference). We study how these methods can be used to compensate for the approximation error of the initially generated Mini-Bucket heuristics, within the context of anytime Branch-And-Bound search.

We explore the use of iterative cost-shifting as a dynamic heuristic generator for solving MPE in graphical models via Branch and Bound. When mini-bucket elimination is limited by its memory budget, it may not provide good heuristics. This can happen often when the graphical model has a very high induced width with large variable domain sizes. In addition, we explore a hybrid setup where both MBE and the iterative cost-shifting bound are used in a combined heuristic. We compare these approaches with the most advanced statically generated heuristics.

Many algorithms for performing inference in graphical models have complexity that is exponential in the treewidth — a parameter of the underlying graph structure. Computing the (minimal) treewidth is NPcomplete, so stochastic algorithms are sometimes used to find low width tree decompositions. A common approach for finding good decompositions is iteratively executing a greedy triangulation algorithm (e.g. minfill) with randomized tie-breaking. However, utilizing a stochastic algorithm as part of the inference task introduces a new problem — namely, deciding how long the stochastic algorithm should be allowed to execute before performing inference on the best tree decomposition found so far. We refer to this dilemma as the Stopping Problem and formalize it in terms of the total time needed to answer a probabilistic query. We propose a rule for discontinuing the search for improved decompositions and demonstrate the benefit (in terms of time saved) of applying this rule to Bayes and Markov network instances.

We study iterative randomized greedy algorithms for generating (elimination) orderings with small induced width and state space size — two parameters known to bound the complexity of inference in graphical models. We propose and implement the Iterative Greedy Variable Ordering (IGVO) algorithm, a new variant within this algorithm class. An empirical evaluation using different ranking functions and conditions of randomness, demonstrates that IGVO finds significantly better orderings than standard greedy ordering implementations when evaluated within an anytime framework. Additional order of magnitude improvements are demonstrated on a multi-core system, thus further expanding the set of solvable graphical models. The experiments also confirm the superiority of the MinFill heuristic within the iterative scheme.

The paper investigates parameterized approximate message-passing schemes that are based on bounded inference and are inspired by Pearl's belief propagation algorithm (BP). We start with the bounded inference mini-clustering algorithm and then move to the iterative scheme called Iterative Join-Graph Propagation (IJGP), that combines both iteration and bounded inference. Algorithm IJGP belongs to the class of Generalized Belief Propagation algorithms, a framework that allowed connections with approximate algorithms from statistical physics and is shown empirically to surpass the performance of mini-clustering and belief propagation, as well as a number of other state-of-the-art algorithms on several classes of networks. We also provide insight into the accuracy of iterative BP and IJGP by relating these algorithms to well known classes of constraint propagation schemes.