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1. Submodel Decomposition Bounds for Influence Diagrams

   https://doi.org/10.1609/AAAI.V35I13.17442  

   Lee, Junkyu; Marinescu, Radu; Dechter, Rina (May 2021, Proceedings of the AAAI Conference on Artificial Intelligence)

   Influence diagrams (IDs) are graphical models for representing and reasoning with sequential decision-making problems under uncertainty. Limited memory influence diagrams (LIMIDs) model a decision-maker (DM) who forgets the history in the course of making a sequence of decisions. The standard inference task in IDs and LIMIDs is to compute the maximum expected utility (MEU), which is one of the most challenging tasks in graphical models. We present a model decomposition framework in both IDs and LIMIDs, which we call submodel decomposition that generates a tree of single-stage decision problems through a tree clustering scheme. We also develop a valuation algebra over the submodels that leads to a hierarchical message passing algorithm that propagates conditional expected utility functions over a submodel-tree as external messages. We show that the overall complexity is bounded by the maximum tree-width over the submodels, common in graphical model algorithms. Finally, we present a new method for computing upper bounds over a submodel-tree by first exponentiating the utility functions yielding a standard probabilistic graphical model as an upper bound and then applying standard variational upper bounds for the marginal MAP inference, yielding tighter upper bounds compared with state-of-the-art bounding schemes for the MEU task. 

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2. Heuristic AND/OR Search for Solving Influence Diagram

   https://doi.org/10.1609/SOCS.V11I1.18523  

   Lee, Junkyu; Marinescu, Radu; Dechter, Rina (May 2020, Proceedings of the International Symposium on Combinatorial Search)

   An influence diagram is a graphical representation of sequential decision-making under uncertainty, defining a structured decision problem by conditional probability functions and additive utility functions over discrete state and action variables. The task of finding the maximum expected utility of influence diagrams is closely related to the cost-optimal probabilistic planning, stochastic programmings, or model-based reinforcement learning. In this position paper, we address the heuristic search for solving influence diagram, where we generate admissible heuristic functions from graph decomposition schemes. Then, we demonstrate how such heuristics can guide an AND/OR branch and bound search. Finally, we briefly discuss the future directions for improving the quality of heuristic functions and search strategies. 

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3. A graphical global optimization framework for parameter estimation of statistical models with nonconvex regularization functions

   Davarnia, Danial; Kiaghadi, Mohammadreza (May 2025, International Conference on Artificial Intelligence and Statistics; Proceedings of Machine Learning Research)

   Li, Yingzhen; Mandt, Stephan; Agrawal, Shipra; Khan, Emtiyaz (Ed.)

   Optimization problems with norm-bounding constraints appear in various applications, from portfolio optimization to machine learning, feature selection, and beyond. A widely used variant of these problems relaxes the norm-bounding constraint through Lagrangian relaxation and moves it to the objective function as a form of penalty or regularization term. A challenging class of these models uses the zero-norm function to induce sparsity in statistical parameter estimation models. Most existing exact solution methods for these problems use additional binary variables together with artificial bounds on variables to formulate them as a mixed-integer program in a higher dimension, which is then solved by off-the-shelf solvers. Other exact methods utilize specific structural properties of the objective function to solve certain variants of these problems, making them non-generalizable to other problems with different structures. An alternative approach employs nonconvex penalties with desirable statistical properties, which are solved using heuristic or local methods due to the structural complexity of those terms. In this paper, we develop a novel graph-based method to globally solve optimization problems that contain a generalization of norm-bounding constraints. This includes standard ℓp-norms for p∈[0,∞) as well as nonconvex penalty terms, such as SCAD and MCP, as special cases. Our method uses decision diagrams to build strong convex relaxations for these constraints in the original space of variables without the need to introduce additional auxiliary variables or impose artificial variable bounds. We show that the resulting convexification method, when incorporated into a spatial branch-and-cut framework, converges to the global optimal value of the problem. To demonstrate the capabilities of the proposed framework, we conduct preliminary computational experiments on benchmark sparse linear regression problems with challenging nonconvex penalty terms that cannot be modeled or solved by existing global solvers. 

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4. Strategy Graphs for Influence Diagrams

   https://doi.org/10.1613/jair.1.13865  

   Hansen, Eric A.; Shi, Jinchuan; Kastrantas, James (September 2022, Journal of Artificial Intelligence Research)

   An influence diagram is a graphical model of a Bayesian decision problem that is solved by finding a strategy that maximizes expected utility. When an influence diagram is solved by variable elimination or a related dynamic programming algorithm, it is traditional to represent a strategy as a sequence of policies, one for each decision variable, where a policy maps the relevant history for a decision to an action. We propose an alternative representation of a strategy as a graph, called a strategy graph, and show how to modify a variable elimination algorithm so that it constructs a strategy graph. We consider both a classic variable elimination algorithm for influence diagrams and a recent extension of this algorithm that has more relaxed constraints on elimination order that allow improved performance. We consider the advantages of representing a strategy as a graph and, in particular, how to simplify a strategy graph so that it is easier to interpret and analyze. 

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5. Generating Extended Resolution Proofs with a BDD-Based SAT Solver

   https://doi.org/10.1007/978-3-030-72016-2_5  

   Bryant, Randal E.; Heule, Marijn J. (January 2021, Tools and Algorithms for the Construction and Analysis of Systems)

   Groote, J. F.; Larsen, K. G. (Ed.)

   In 2006, Biere, Jussila, and Sinz made the key observation that the underlying logic behind algorithms for constructing Reduced, Ordered Binary Decision Diagrams (BDDs) can be encoded as steps in a proof in the extended resolution logical framework. Through this, a BDD-based Boolean satisfiability (SAT) solver can generate a checkable proof of unsatisfiability. Such proofs indicate that the formula is truly unsatisfiable without requiring the user to trust the BDD package or the SAT solver built on top of it. We extend their work to enable arbitrary existential quantification of the formula variables, a critical capability for BDD-based SAT solvers. We demonstrate the utility of this approach by applying a prototype solver to obtain polynomially sized proofs on benchmarks for the mutilated chessboard and pigeonhole problems—ones that are very challenging for search-based SAT solvers. 

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