On the optimality of solutions of the max-product belief-propagation algorithm in arbitrary graphs

Graphical models, such as Bayesian networks and Markov random elds, represent statistical dependencies of variables by a graph. The max-product \belief propagation" algorithm is a local-message passing algorithm on this graph that is known to converge to a unique xed point when the graph is a tree. Furthermore, when the graph is a tree, the assignment based on the xed-point yields the most probable a posteriori (MAP) values of the unobserved variables given the observed ones. Recently, good empirical performance has been obtained by running the max-product algorithm (or the equivalent min-sum algorithm) on graphs with loops, for applications including the decoding of \turbo" codes. Except for two simple graphs (cycle codes and single loop graphs) there has been little theoretical understanding of the max-product algorithm on graphs with loops. Here we prove a result on the xed points of max-product on a graph with arbitrary topology and with arbitrary probability distributions (discrete or continuous valued nodes). We show that the assignment based on a xed-point is a \neighborhood maximum" of the posterior probability: the posterior probability of the max-product assignment is guaranteed to be greater than all other assignments in a particular large region around that assignment. The region includes all assignments that di er from the max-product assignment in any subset of nodes that form no more than a single loop in the graph. In some graphs this neighborhood is exponentially large. We illustrate the analysis with examples. Problems involving probabilistic belief propagation arise in a wide variety of applications, including error correcting codes, speech recognition and image understanding. Typically, a probability distribution is assumed over a set of variables and the task is to infer the values of the unobserved variables given the observed ones. The assumed probability distribution is described using a graphical model [13] | the qualitative aspects of the distribution are speci ed by a graph structure. The graph may either be directed as in a Bayesian network [17], [11] or undirected as in a Markov Random Field [17], [9]. Here we focus on the problem of nding an assignment for the unobserved variables that is most probable given the observed ones. In general, this problem is NP hard [18] but if the graph is singly connected (i.e. there is only one path between any two given nodes) then there exist e cient local message{passing schemes to perform this task. Pearl [17] derived such a scheme for singly connected Bayesian networks. The algorithm, which he called \belief revision", is identical to his algorithm for nding posterior marginals over nodes except that the summation operator is replaced with a maximization. Aji et al. [2] have shown that both of Pearl's algorithms can be seen as special cases of generalized distributive laws over particular semirings. In particular, Pearl's algorithm for nding maximum a posteriori (MAP) assignments can be seen as a generalized distributive law over the max-product semiring. We will henceforth refer to it as the \max-product" algorithm. Pearl showed that for singly connected networks, the max-product algorithm is guaranteed to converge and that the assignment based on the messages at convergence is guaranteed to give the optimal assignment{values corresponding to the MAP solution. Several groups have recently reported excellent experimental results by running the max-product algorithm on graphs with loops [22], [6], [3], [19], [6], [10]. Benedetto et al. used the max-product algorithm to decode \turbo" codes and obtained excellent results that were slightly inferior to the original turbo decoding algorithm (which is equivalent to the sum-product algorithm). Weiss [19] compared the performance of sum-product and max-product on a \toy" turbo code problem while distinguishing between converged and unconverged cases. He found that if one considers only the convergent cases, the performance of max-product decoding is signi cantly better than sum-product decoding. However, the max-product algorithm converges less often so its overall performance (including both convergent and nonconvergent cases) is inferior. Progress in the analysis of the max-product algorithm has been made for two special topologies: single loop graphs, and \cycle codes". For graphs with a single loop [22], [19], [20], [5], [2], it can be shown that the algorithm converges to a stable xed point or a periodic oscillation. If it converges to a stable xed-point, then the assignment based on the xed-point messages is the optimal assignment. For graphs that correspond to cycle codes (low density parity check codes in which each bit is checked by exactly two check nodes), Wiberg [22] gave su cient conditions for max-product to converge to the transmitted codeword and Horn [10] gave su cient conditions for convergence to the MAP assignment. In this paper we analyze the max-product algorithm in graphs of arbitrary topology. We show that at a xed-point of the algorithm, the assignment is a \neighborhood maximum" of the posterior probability: the posterior probability of the max-product assignment is guaranteed to be greater than all other assignments in a particular large region around that assignment. These results motivate using this powerful algorithm in a broader class of networks. Y. Weiss is with Computer Science Division, 485 Soda Hall, UC Berkeley, Berkeley, CA 94720-1776. E-mail: [email protected]. Support by MURI-ARO-DAAH04-96-1-0341, MURI N00014-00-1-0637 and NSF IIS-9988642 is gratefully acknowledged. W. T. Freeman is with MERL, Mitsubishi Electric Research Labs., 201 Broadway, Cambridge, MA 02139. E-mail: [email protected].