Mixed inference such as the marginal MAP query (some variables marginalized by summation and others by maximization) is key to many prediction and decision models. It is known to be extremely hard; the problem is NPPP-complete while the decision problem for MAP is only NP-complete and the summation problem is #P-complete. Consequently, approximation anytime schemes are essential. In this paper, we show that the framework of heuristic AND/OR search, which exploits conditional independence in the graphical model, coupled with variational-based mini-bucket heuristics can be extended to this task and yield powerful state-of-the-art schemes. Specifically, we explore the complementary properties of best-first search for reducing the number of conditional sums and providing time-improving upper bounds, with depth-first search for rapidly generating and improving solutions and lower bounds. We show empirically that a class of solvers that interleaves depth-first with best-first schemes emerges as the most competitive anytime scheme.

This paper explores the anytime performance of search-based algorithms for solving the Marginal MAP task over graphical models. The current state of the art for solving this challenging task is based on best-first search exploring the AND/OR graph with the guidance of heuristics based on mini-bucket and variational cost-shifting principles. Yet, those schemes are uncompromising in that they solve the problem exactly, or not at all, and often suffer from memory problems. In this work, we explore the well known principle of weighted search for converting best-first search solvers into anytime schemes. The weighted best-first search schemes report a solution early in the process by using inadmissible heuristics, and subsequently improve the solution. While it was demonstrated recently that weighted schemes can yield effective anytime behavior for pure MAP tasks, Marginal MAP is far more challenging (e.g., a conditional sum must be evaluated for every solution). Yet, in an extensive empirical analysis we show that weighted schemes are indeed highly effective for Marginal MAP yielding the most competitive schemes to date for this task.

The paper explores the potential of weighted best-first search schemes as anytime optimization algorithms for solving graphical models tasks such as MPE (Most Probable Explanation) or MAP (Maximum a Posteriori) and WCSP (Weighted Constraint Satisfaction Problem). While such schemes were widely investigated for path-finding tasks, their application for graphical models was largely ignored, possibly due to their memory requirements. Compared to the depth-first branch and bound, which has long been the algorithm of choice for optimization in graphical models, a valuable virtue of weighted best-first search is that they are w-optimal, i.e. when terminated, they return a solution cost C and a weight w, such that C < = wC*, where C* is the optimal cost. We report on a significant empirical evaluation, demonstrating the usefulness of weighted best-first search as approximation anytime schemes (that have suboptimality bounds) and compare against one of the best depth-first branch and bound solver to date. We also investigate the impact of different heuristic functions on the behaviour of the algorithms.

One popular and efficient scheme for solving exactly combinatorial optimization problems over graphical models is depth-first Branch and Bound. However, when the algorithm exploits problem decomposition using AND/OR search spaces, its anytime behavior breaks down. This paper 1) analyzes and demonstrates this inherent conflict between effective exploitation of problem decomposition (through AND/OR search spaces) and the anytime behavior of depth-first search (DFS), 2) presents a first scheme to address this issue while maintaining desirable DFS memory properties, 3) analyzes and demonstrates its effectiveness. Our work is applicable to any problem that can be cast as search over an AND/OR search space.