Weighted Best First Search for Graphical Models

The paper considers Weighted Best First (WBF) search schemes, popular for path-finding domain, as approximations and as anytime schemes for the MAP task. We demonstrate empirically the ability of these schemes to effectively provide approximations with guaranteed suboptimality and also show that as anytime schemes they can be competitive on some benchmarks with one of the best state-of-the-art scheme, Depth-First Branchand-Bound. Introduction The most common search scheme for combinatorial optimization tasks, such as MAP/MPE or Weighted CSP, is Depth-First Branch-and-Bound. Its use for finding both exact and approximate solutions was extensively studied in recent years (Kask and Dechter 2001; Marinescu and Dechter 2009b; Otten and Dechter 2011; de Givry, Schiex, and Verfaillie 2006). Meanwhile, best-first search algorithms, though known to be more effective in bounding the search space (Dechter and Pearl 1985), are seldom considered for graphical models due to their inherently large memory requirements and their inability to provide any solution before termination. Furthermore, one of best-first’s most attractive features, avoiding the exploration of unbounded paths, seems irrelevant since solutions are of equal depth (i.e., the number of variables). In contrast, in path-finding domains, where solution length varies (e.g., planning), best-first search and especially its popular variant A* (Hart, Nilsson, and Raphael 1968) is clearly favored. However, A*’s exponential memory needs, coupled with its inability to provide a solution any time before termination, lead to extension into more flexible anytime schemes based on the Weighted A* (WA*) (Pohl 1970). The idea is to inflate the heuristic function guiding the search by a constant factor of w > 1, making the heuristic inadmissible, while still guaranteeing a solution cost within a factor of w from the optimal and typically yielding faster search. If the (non-optimal) solution is found quickly, the search for a better solution may resume. Several anytime weighted best-first search schemes were proposed in the context of path-finding in the past decade (Hansen and Zhou 2007; Likhachev, Gordon, and Thrun 2003; van den Berg et al. 2011; Richter, Thayer, and Ruml 2010). In our work we extended the above methods to graphical models and investigated their potential empirically. We used AND/OR Best First search (AOBF) (Marinescu and Dechter 2009b), a best-first algorithm developed for AND/OR search spaces over graphical models, as a basis. AOBF explores the context minimal AND/OR graph in a best-first manner, guided by the admissible and consistent mini-bucket heuristic (Dechter and Rish 2003; Kask and Dechter 2001; Dechter and Mateescu 2007). After exploring a variety of approaches and following extensive empirical analysis (including two non-parametric algorithms that interleave depthand best-first exploration), the two schemes that emerged as most promising were those running WA* iteratively while decreasing w. One (wAOBF) starts from scratch at each iteration and the other (wRAOBF) reuses search efforts from previous iterations, extending ideas of Anytime Repairing A* (ARA*) (Likhachev, Gordon, and Thrun 2003). We report on a comprehensive empirical evaluation of the two candidate algorithms (which ran multiple time with multiple heuristics) on over 100 instances from 4 different benchmarks, evaluating their performance both as approximation and anytime schemes. We compared against Breadth-Rotating AND/OR Branch-and-Bound (BRAOBB) (Otten and Dechter 2011), a state-of-the-art anytime DepthFirst Branch-and-Bound which won the 2011 Probabilistic Inference Challenge1 in all optimization categories. Our empirical analysis revealed that best-first search schemes can be used effectively for optimization over graphical models. Specifically: 1. Weighted BFS provides an effective scheme for generating approximate solutions with upfront guaranteed level of sub-optimality, 2. Weighted BFS can be made into effective anytime schemes and on some benchmarks even outperforms one of the best state of the art scheme which is based on depth-first search.