We consider the question of computing Maximum A Posteriori (MAP) assignment in an arbitrary pair-wise Markov Random Field (MRF). We present a randomized iterative algorithm based on simple local updates. The algorithm, starting with an arbitrary initial assignment, updates it in each iteration by first, picking a random node, then selecting an (appropriately chosen) random local neighborhood and optimizing over this local neighborhood. Somewhat surprisingly, we show that this algorithm finds a near optimal assignment within $2n\ln n$ iterations on average and with high probability for {\em any} $n$ node pair-wise MRF with {\em geometry} (i.e. MRF graph with polynomial growth) with the approximation error depending on (in a reasonable manner) the geometric growth rate of the graph and the average radius of the local neighborhood -- this allows for a graceful tradeoff between the complexity of the algorithm and the approximation error. Through extensive simulations, we show that our algorithm finds extremely good approximate solutions for various kinds of MRFs with geometry.

We present a new local approximation algorithm for computing MAP and logpartition functionfor arbitrary exponential family distribution represented by a finite-valued pairwise Markov random field (MRF), say G. Our algorithm is based on decomposing G into appropriately chosen small components; computing estimates locally in each of these components and then producing a good global solution. We prove that the algorithm can provide approximate solution within arbitrary accuracy when G excludes some finite sized graph as its minor and G has bounded degree: all Planar graphs with bounded degree are examples of such graphs. The running time of the algorithm is Θ(n) (n is the number of nodes in G), with constant dependent on accuracy, degree of graph and size of the graph that is excluded as a minor (constant for Planar graphs). Our algorithm for minor-excluded graphs uses the decomposition scheme of Klein, Plotkin and Rao (1993). In general, our algorithm works with any decomposition schemeand provides quantifiable approximation guarantee that depends on the decomposition scheme.

We investigate the use of message-passing algorithms for the problem of finding the max-weight independent set (MWIS) in a graph. First, we study the performance ofloopy max-product belief propagation. We show that, if it converges, the quality of the estimate is closely related to the tightness of an LP relaxation of the MWIS problem. We use this relationship to obtain sufficient conditions for correctness of the estimate. We then develop a modification of max-product - one that converges to an optimal solution of the dual of the MWIS problem. We also develop a simple iterative algorithm for estimating the max-weight independent set from this dual solution. We show that the MWIS estimate obtained using these two algorithms in conjunction is correct when the graph is bipartite and the MWIS is unique. Finally, we show that any problem of MAP estimation for probability distributions over finite domains can be reduced to an MWIS problem. We believe this reduction will yield new insights and algorithms for MAP estimation.

We investigate the use of message-passing algorithms for the problem of finding the max-weight independent set (MWIS) in a graph. First, we study the performance of the classical loopy max-product belief propagation. We show that each fixed point estimate of max-product can be mapped in a natural way to an extreme point of the LP polytope associated with the MWIS problem. However, this extreme point may not be the one that maximizes the value of node weights; the particular extreme point at final convergence depends on the initialization of max-product. We then show that if max-product is started from the natural initialization of uninformative messages, it always solves the correct LP -- if it converges. This result is obtained via a direct analysis of the iterative algorithm, and cannot be obtained by looking only at fixed points. The tightness of the LP relaxation is thus necessary for max-product optimality, but it is not sufficient. Motivated by this observation, we show that a simple modification of max-product becomes gradient descent on (a convexified version of) the dual of the LP, and converges to the dual optimum. We also develop a message-passing algorithm that recovers the primal MWIS solution from the output of the descent algorithm. We show that the MWIS estimate obtained using these two algorithms in conjunction is correct when the graph is bipartite and the MWIS is unique. Finally, we show that any problem of MAP estimation for probability distributions over finite domains can be reduced to an MWIS problem. We believe this reduction will yield new insights and algorithms for MAP estimation.

We present a new local approximation algorithm for computing Maximum a Posteriori (MAP) and log-partition function for arbitrary exponential family distribution represented by a finite-valued pair-wise Markov random field (MRF), say $G$. Our algorithm is based on decomposition of $G$ into {\em appropriately} chosen small components; then computing estimates locally in each of these components and then producing a {\em good} global solution. We show that if the underlying graph $G$ either excludes some finite-sized graph as its minor (e.g. Planar graph) or has low doubling dimension (e.g. any graph with {\em geometry}), then our algorithm will produce solution for both questions within {\em arbitrary accuracy}. We present a message-passing implementation of our algorithm for MAP computation using self-avoiding walk of graph. In order to evaluate the computational cost of this implementation, we derive novel tight bounds on the size of self-avoiding walk tree for arbitrary graph. As a consequence of our algorithmic result, we show that the normalized log-partition function (also known as free-energy) for a class of {\em regular} MRFs will converge to a limit, that is computable to an arbitrary accuracy.