Extending the Reach of AND/OR Search for Optimization over Graphical ModelsLars Otten
This thesis presents substantial enhancements to the state of the art in combinatorial optimization over graphical models. Our contributions are relevant in the context of both exact and approximate reasoning over Bayesian and Markov networks, weighted constraint satisfaction problems, and other related queries. While the focus of this work is on probabilistic and constraint inference, we also draw from the areas of distributed computing and statistical learning. Relevant practical applications we consider include genetic linkage analysis, protein side-chain prediction, medical diagnosis, resource scheduling, and signal processing.
We extend AND/OR Branch-and-Bound (AOBB), a leading algorithm for optimization queries over graphical models. AOBB applies the principle of depth-first branch-and-bound to AND/OR search spaces, which exploit conditional independencies via problem decomposition and merge unifiable subproblems through caching of partial solutions. This thesis presents fundamental extensions to AOBB in three regards.
First, we significantly improve the applicability of AOBB as an approximation scheme. We analyze and demonstrate the inherent conflict between problem decomposition (through AND/OR search spaces) and the anytime behavior of AOBB and depth-first search in general. We introduce a new algorithm, Breadth-Rotating AND/OR Branch-and-Bound (BRAOBB), which drastically improves upon AOBB with respect to its anytime performance while maintaining desirable depth-first complexity guarantees. Comprehensive analysis and experimental evaluation demonstrate the scheme’s effectiveness. Furthermore, our entry based on BRAOBB placed first in all three optimization tracks of the PASCAL 2012 Probabilistic Inference Challenge.
Second, we investigate the instance-based run-time complexity of AOBB. The asymptotic worst-case bounds are both time and space exponential in the problem’s induced width, but often prove to be very loose due to the algorithm’s powerful pruning, as we show empirically. We identify a range of (sub)problem features and develop learning schemes to estimate runtime complexity based on statistical regression analysis. We conduct extensive experimental evaluation within and across various problem classes and demonstrate convincing predictive performance.
Third, we describe a parallel AND/OR Branch-and-Bound scheme that pushes the boundaries of feasibility for exact reasoning by orders of magnitude. We adapt the paradigm of parallel tree search to AND/OR search spaces; our implementation distributes conditioned subproblems on a grid of independent computers. In this context, we show how the pruning power of AOBB can cause large variance in subproblem complexity, which makes load balancing extremely elusive and impairs parallel performance. We thus propose load balancing based on the run-time estimation scheme presented earlier in the thesis, learning a complexity model offline from previously solved subproblems. Through experimental results using hundreds of computers on problem instances from a variety of classes we show convincing parallel performance with several orders of magnitude speedup over sequential AOBB, but we also highlight and analyze some inherent limitations.
Our implementations of AOBB, BRAOBB and parallel AOBB are available online under an open-source license.