Guarantees and Limits of Preprocessing in Constraint Satisfaction and Reasoning

We present a first theoretical analysis of the power of polynomial-time preprocessing for important combinatorial problems from various areas in AI. We consider problems from Constraint Satisfaction, Global Constraints, Satisfiability, Nonmonotonic and Bayesian Reasoning under structural restrictions. All these problems involve two tasks: (i) identifying the structure in the input as required by the restriction, and (ii) using the identified structure to solve the reasoning task efficiently. We show that for most of the considered problems, task (i) admits a polynomial-time preprocessing to a problem kernel whose size is polynomial in a structural problem parameter of the input, in contrast to task (ii) which does not admit such a reduction to a problem kernel of polynomial size, subject to a complexity theoretic assumption. As a notable exception we show that the consistency problem for the AtMost-NValue constraint admits a polynomial kernel consisting of a quadratic number of variables and domain values. Our results provide a firm worst-case guarantees and theoretical boundaries for the performance of polynomial-time preprocessing algorithms for the considered problems.Comment: arXiv admin note: substantial text overlap with arXiv:1104.2541, arXiv:1104.556

Narrowing the Gap: Random Forests In Theory and In Practice

Despite widespread interest and practical use, the theoretical properties of random forests are still not well understood. In this paper we contribute to this understanding in two ways. We present a new theoretically tractable variant of random regression forests and prove that our algorithm is consistent. We also provide an empirical evaluation, comparing our algorithm and other theoretically tractable random forest models to the random forest algorithm used in practice. Our experiments provide insight into the relative importance of different simplifications that theoreticians have made to obtain tractable models for analysis.Comment: Under review by the International Conference on Machine Learning (ICML) 201

Learning High-Dimensional Markov Forest Distributions: Analysis of Error Rates

The problem of learning forest-structured discrete graphical models from i.i.d. samples is considered. An algorithm based on pruning of the Chow-Liu tree through adaptive thresholding is proposed. It is shown that this algorithm is both structurally consistent and risk consistent and the error probability of structure learning decays faster than any polynomial in the number of samples under fixed model size. For the high-dimensional scenario where the size of the model d and the number of edges k scale with the number of samples n, sufficient conditions on (n,d,k) are given for the algorithm to satisfy structural and risk consistencies. In addition, the extremal structures for learning are identified; we prove that the independent (resp. tree) model is the hardest (resp. easiest) to learn using the proposed algorithm in terms of error rates for structure learning.Comment: Accepted to the Journal of Machine Learning Research (Feb 2011

Maximum Weight Matching via Max-Product Belief Propagation

Max-product "belief propagation" is an iterative, local, message-passing algorithm for finding the maximum a posteriori (MAP) assignment of a discrete probability distribution specified by a graphical model. Despite the spectacular success of the algorithm in many application areas such as iterative decoding, computer vision and combinatorial optimization which involve graphs with many cycles, theoretical results about both correctness and convergence of the algorithm are known in few cases (Weiss-Freeman Wainwright, Yeddidia-Weiss-Freeman, Richardson-Urbanke}. In this paper we consider the problem of finding the Maximum Weight Matching (MWM) in a weighted complete bipartite graph. We define a probability distribution on the bipartite graph whose MAP assignment corresponds to the MWM. We use the max-product algorithm for finding the MAP of this distribution or equivalently, the MWM on the bipartite graph. Even though the underlying bipartite graph has many short cycles, we find that surprisingly, the max-product algorithm always converges to the correct MAP assignment as long as the MAP assignment is unique. We provide a bound on the number of iterations required by the algorithm and evaluate the computational cost of the algorithm. We find that for a graph of size $n$, the computational cost of the algorithm scales as $O(n^3)$, which is the same as the computational cost of the best known algorithm. Finally, we establish the precise relation between the max-product algorithm and the celebrated {\em auction} algorithm proposed by Bertsekas. This suggests possible connections between dual algorithm and max-product algorithm for discrete optimization problems.Comment: In the proceedings of the 2005 IEEE International Symposium on Information Theor