Correlation Decay in Random Decision Networks

We consider a decision network on an undirected graph in which each node corresponds to a decision variable, and each node and edge of the graph is associated with a reward function whose value depends only on the variables of the corresponding nodes. The goal is to construct a decision vector which maximizes the total reward. This decision problem encompasses a variety of models, including maximum-likelihood inference in graphical models (Markov Random Fields), combinatorial optimization on graphs, economic team theory and statistical physics. The network is endowed with a probabilistic structure in which costs are sampled from a distribution. Our aim is to identify sufficient conditions to guarantee average-case polynomiality of the underlying optimization problem. We construct a new decentralized algorithm called Cavity Expansion and establish its theoretical performance for a variety of models. Specifically, for certain classes of models we prove that our algorithm is able to find near optimal solutions with high probability in a decentralized way. The success of the algorithm is based on the network exhibiting a correlation decay (long-range independence) property. Our results have the following surprising implications in the area of average case complexity of algorithms. Finding the largest independent (stable) set of a graph is a well known NP-hard optimization problem for which no polynomial time approximation scheme is possible even for graphs with largest connectivity equal to three, unless P=NP. We show that the closely related maximum weighted independent set problem for the same class of graphs admits a PTAS when the weights are i.i.d. with the exponential distribution. Namely, randomization of the reward function turns an NP-hard problem into a tractable one

Conditional dynamics of non-Markovian, infinite-server queues

Thesis (S.M.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2005.Includes bibliographical references (p. 75-79).We study the transient dynamics of a partially observed, infinite server queue fed with a Poisson arrival process whose controlled rate is changed at discrete points in time. More specifically, we define a state that incorporates partial information from the history of the process and write analytical formula for the dynamics of the system (state transition probabilities). Moreover, we develop an approximation method that makes the state finite-dimensional, and introduce techniques to further reduce the dimension of the state. This method could thus enable the formulation of tractable DPs in the future.by Theophane Weber.S.M

Correlation decay and decentralized optimization in graphical models

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 213-229) and index.Many models of optimization, statistics, social organizations and machine learning capture local dependencies by means of a network that describes the interconnections and interactions of different components. However, in most cases, optimization or inference on these models is hard due to the dimensionality of the networks. This is so even when using algorithms that take advantage of the underlying graphical structure. Approximate methods are therefore needed. The aim of this thesis is to study such large-scale systems, focusing on the question of how randomness affects the complexity of optimizing in a graph; of particular interest is the study of a phenomenon known as correlation decay, namely, the phenomenon where the influence of a node on another node of the network decreases quickly as the distance between them grows. In the first part of this thesis, we develop a new message-passing algorithm for optimization in graphical models. We formally prove a connection between the correlation decay property and (i) the near-optimality of this algorithm, as well as (ii) the decentralized nature of optimal solutions. In the context of discrete optimization with random costs, we develop a technique for establishing that a system exhibits correlation decay. We illustrate the applicability of the method by giving concrete results for the cases of uniform and Gaussian distributed cost coefficients in networks with bounded connectivity. In the second part, we pursue similar questions in a combinatorial optimization setting: we consider the problem of finding a maximum weight independent set in a bounded degree graph, when the node weights are i.i.d. random variables.(cont.) Surprisingly, we discover that the problem becomes tractable for certain distributions. Specifically, we construct a PTAS for the case of exponentially distributed weights and arbitrary graphs with degree at most 3, and obtain generalizations for higher degrees and different distributions. At the same time we prove that no PTAS exists for the case of exponentially distributed weights for graphs with sufficiently large but bounded degree, unless P=NP. Next, we shift our focus to graphical games, which are a game-theoretic analog of graphical models. We establish a connection between the problem of finding an approximate Nash equilibrium in a graphical game and the problem of optimization in graphical models. We use this connection to re-derive NashProp, a message-passing algorithm which computes Nash equilibria for graphical games on trees; we also suggest several new search algorithms for graphical games in general networks. Finally, we propose a definition of correlation decay in graphical games, and establish that the property holds in a restricted family of graphical games. The last part of the thesis is devoted to a particular application of graphical models and message-passing algorithms to the problem of early prediction of Alzheimer's disease. To this end, we develop a new measure of synchronicity between different parts of the brain, and apply it to electroencephalogram data. We show that the resulting prediction method outperforms a vast number of other EEG-based measures in the task of predicting the onset of Alzheimer's disease.by Théophane Weber.Ph.D

Correlation Decay in Random Decision Networks

We consider a decision network on an undirected graph in which each node corresponds to a decision variable, and each node and edge of the graph is associated with a reward function whose value depends only on the variables of the corresponding nodes. The goal is to construct a decision vector that maximizes the total reward. This decision problem encompasses a variety of models, including maximum-likelihood inference in graphical models (Markov Random Fields), combinatorial optimization on graphs, economic team theory, and statistical physics. The network is endowed with a probabilistic structure in which rewards are sampled from a distribution. Our aim is to identify sufficient conditions on the network structure and rewards distributions to guarantee average-case polynomiality of the underlying optimization problem. Additionally, we wish to characterize the efficiency of a decentralized solution generated on the basis of local information. We construct a new decentralized algorithm called Cavity Expansion and establish its theoretical performance for a variety of graph models and reward function distributions. Specifically, for certain classes of models we prove that our algorithm is able to find a near-optimal solution with high probability in a decentralized way. The success of the algorithm is based on the network exhibiting a certain correlation decay (long-range independence) property, and we prove that this property is indeed exhibited by the models of interest. Our results have the following surprising implications in the area of average-case complexity of algorithms. Finding the largest independent (stable) set of a graph is a well known NP-hard optimization problem for which no polynomial time approximation scheme is possible even for graphs with largest connectivity equal to three unless P D NP. Yet we show that the closely related Maximum Weight Independent Set problem for the same class of graphs admits a PTAS when the weights are independently and identically distributed with the exponential distribution. Namely, randomization of the reward function turns an NP-hard problem into a tractable one. Keywords: optimization; NP-hardness; long-range independenceNational Science Foundation (U.S.) (Grant CMMI-0726733