Partition strategies for incremental Mini-Bucket

Los modelos en grafo probabilísticos, tales como los campos aleatorios de Markov y las redes bayesianas, ofrecen poderosos marcos de trabajo para la representación de conocimiento y el razonamiento en modelos con gran número de variables. Sin embargo, los problemas de inferencia exacta en modelos de grafos son NP-hard en general, lo que ha causado que se produzca bastante interés en métodos de inferencia aproximados. El mini-bucket incremental es un marco de trabajo para inferencia aproximada que produce como resultado límites aproximados inferior y superior de la función de partición exacta, a base de -empezando a partir de un modelo con todos los constraints relajados, es decir, con las regiones más pequeñas posibleincrementalmente añadir regiones más grandes a la aproximación. Los métodos de inferencia aproximada que existen actualmente producen límites superiores ajustados de la función de partición, pero los límites inferiores suelen ser demasiado imprecisos o incluso triviales. El objetivo de este proyecto es investigar estrategias de partición que mejoren los límites inferiores obtenidos con el algoritmo de mini-bucket, trabajando dentro del marco de trabajo de mini-bucket incremental. Empezamos a partir de la idea de que creemos que debería ser beneficioso razonar conjuntamente con las variables de un modelo que tienen una alta correlación, y desarrollamos una estrategia para la selección de regiones basada en esa idea. Posteriormente, implementamos nuestra estrategia y exploramos formas de mejorarla, y finalmente medimos los resultados obtenidos usando nuestra estrategia y los comparamos con varios métodos de referencia. Nuestros resultados indican que nuestra estrategia obtiene límites inferiores más ajustados que nuestros dos métodos de referencia. También consideramos y descartamos dos posibles hipótesis que podrían explicar esta mejora.Els models en graf probabilístics, com bé els camps aleatoris de Markov i les xarxes bayesianes, ofereixen poderosos marcs de treball per la representació del coneixement i el raonament en models amb grans quantitats de variables. Tanmateix, els problemes d’inferència exacta en models de grafs son NP-hard en general, el qual ha provocat que es produeixi bastant d’interès en mètodes d’inferència aproximats. El mini-bucket incremental es un marc de treball per a l’inferència aproximada que produeix com a resultat límits aproximats inferior i superior de la funció de partició exacta que funciona començant a partir d’un model al qual se li han relaxat tots els constraints -és a dir, un model amb les regions més petites possibles- i anar afegint a l’aproximació regions incrementalment més grans. Els mètodes d’inferència aproximada que existeixen actualment produeixen límits superiors ajustats de la funció de partició. Tanmateix, els límits inferiors acostumen a ser massa imprecisos o fins aviat trivials. El objectiu d’aquest projecte es recercar estratègies de partició que millorin els límits inferiors obtinguts amb l’algorisme de mini-bucket, treballant dins del marc de treball del mini-bucket incremental. La nostra idea de partida pel projecte es que creiem que hauria de ser beneficiós per la qualitat de l’aproximació raonar conjuntament amb les variables del model que tenen una alta correlació entre elles, i desenvolupem una estratègia per a la selecció de regions basada en aquesta idea. Posteriorment, implementem la nostra estratègia i explorem formes de millorar-la, i finalment mesurem els resultats obtinguts amb la nostra estratègia i els comparem a diversos mètodes de referència. Els nostres resultats indiquen que la nostra estratègia obté límits inferiors més ajustats que els nostres dos mètodes de referència. També considerem i descartem dues possibles hipòtesis que podrien explicar aquesta millora.Probabilistic graphical models such as Markov random fields and Bayesian networks provide powerful frameworks for knowledge representation and reasoning over models with large numbers of variables. Unfortunately, exact inference problems on graphical models are generally NP-hard, which has led to signifi- cant interest in approximate inference algorithms. Incremental mini-bucket is a framework for approximate inference that provides upper and lower bounds on the exact partition function by, starting from a model with completely relaxed constraints, i.e. with the smallest possible regions, incrementally adding larger regions to the approximation. Current approximate inference algorithms provide tight upper bounds on the exact partition function but loose or trivial lower bounds. This project focuses on researching partitioning strategies that improve the lower bounds obtained with mini-bucket elimination, working within the framework of incremental mini-bucket. We start from the idea that variables that are highly correlated should be reasoned about together, and we develop a strategy for region selection based on that idea. We implement the strategy and explore ways to improve it, and finally we measure the results obtained using the strategy and compare them to several baselines. We find that our strategy performs better than both of our baselines. We also rule out several possible explanations for the improvement

Graphical Models for Optimal Power Flow

Optimal power flow (OPF) is the central optimization problem in electric power grids. Although solved routinely in the course of power grid operations, it is known to be strongly NP-hard in general, and weakly NP-hard over tree networks. In this paper, we formulate the optimal power flow problem over tree networks as an inference problem over a tree-structured graphical model where the nodal variables are low-dimensional vectors. We adapt the standard dynamic programming algorithm for inference over a tree-structured graphical model to the OPF problem. Combining this with an interval discretization of the nodal variables, we develop an approximation algorithm for the OPF problem. Further, we use techniques from constraint programming (CP) to perform interval computations and adaptive bound propagation to obtain practically efficient algorithms. Compared to previous algorithms that solve OPF with optimality guarantees using convex relaxations, our approach is able to work for arbitrary distribution networks and handle mixed-integer optimization problems. Further, it can be implemented in a distributed message-passing fashion that is scalable and is suitable for "smart grid" applications like control of distributed energy resources. We evaluate our technique numerically on several benchmark networks and show that practical OPF problems can be solved effectively using this approach.Comment: To appear in Proceedings of the 22nd International Conference on Principles and Practice of Constraint Programming (CP 2016

The Lazy Flipper: MAP Inference in Higher-Order Graphical Models by Depth-limited Exhaustive Search

This article presents a new search algorithm for the NP-hard problem of optimizing functions of binary variables that decompose according to a graphical model. It can be applied to models of any order and structure. The main novelty is a technique to constrain the search space based on the topology of the model. When pursued to the full search depth, the algorithm is guaranteed to converge to a global optimum, passing through a series of monotonously improving local optima that are guaranteed to be optimal within a given and increasing Hamming distance. For a search depth of 1, it specializes to Iterated Conditional Modes. Between these extremes, a useful tradeoff between approximation quality and runtime is established. Experiments on models derived from both illustrative and real problems show that approximations found with limited search depth match or improve those obtained by state-of-the-art methods based on message passing and linear programming.Comment: C++ Source Code available from http://hci.iwr.uni-heidelberg.de/software.ph

Efficient Localized Inference for Large Graphical Models

We propose a new localized inference algorithm for answering marginalization queries in large graphical models with the correlation decay property. Given a query variable and a large graphical model, we define a much smaller model in a local region around the query variable in the target model so that the marginal distribution of the query variable can be accurately approximated. We introduce two approximation error bounds based on the Dobrushin's comparison theorem and apply our bounds to derive a greedy expansion algorithm that efficiently guides the selection of neighbor nodes for localized inference. We verify our theoretical bounds on various datasets and demonstrate that our localized inference algorithm can provide fast and accurate approximation for large graphical models