The value of information in shortest path optimization/

Information about a random event (termed the source) is typically treated as a (possibly noisy) function of that event. Information has a destination, an agent, that uses the information to make a decision. In traditional communication systems design, it is usually assumed that the agent uses the information to produce an estimate of the source, and that estimate is in turn used to make the decision. Consequently, the typical objective of communication-systems design is to construct the communication system so that the joint distribution between the source and the information is "optimal" in the sense that it minimizes the average error of the estimate. Due to resource limitations such as cost, power, or time, estimation quality is constrained in the sense that the set of allowable joint distribution is bounded in mutual information. In the context of an agent using information to make decisions, however, such metrics may not be appropriate. In particular, the true value of information is determined by how it impacts the average payoff of the agent's decisions, not its estimation accuracy. To this end, mutual information may not the most convenient measure of information quantity since its relationship to decision quality may be very complicated, making it difficult to develop algorithms for information optimization. In this thesis, we study the value of information in an instance of an uncertain decision framework: shortest path optimization on a graph with random edge weights. Specifically, we consider an agent that seeks to traverse the shortest path of a graph subject to some side information it receives about the edge weights in advance of and during its travel. In this setting, decision quality is determined by the average length of the paths the agent chooses, not how often the agent decodes the optimal path. For this application, we define and quantify a notion of information that is compatible with this problem, bound the performance of the agent subject to a bound on the amount of information available to it, study the impact of spreading information sequentially over partial decisions, and provide algorithms for information optimization. Meaningful, analytic performance bounds and practical algorithms for information optimization are obtained by leveraging a new type of geometric graph reduction for shortest path optimization as well as an abstraction of the geometry of sequential decision making. Thesis Supervisor: Munther A. Dahleh Title: Professor Acknowledgments First and foremost among those I owe my gratitude is my adviser Prof. Munther Dahleh. I want to first thank him for giving me the opportunity to join his research group and LIDS, the reasons for which of course go without saying. I also wish to thank him for the many lessons I learned from him while a student in his research group, and specifically for what I regard to be the most important of these lessons: how to seek out interesting questions and new landscapes for exploration. While I still have a few more years to go in refining this skill, I am fortunate to start from a such a strong base. Finally, I want to thank him for his excellent teaching and patience, which have contributed significantly to the very positive experience I had while a member of his group. Within LIDS, I would like to thank some of the current and former students who have made my time here so enjoyable, specifically AmirAli Ahmadi, Aliaa Atwi, Ola Ayaso, Rose Faghih, Sleiman Itani, Sertac Karaman, Yola Katsargyri, Georgios Kotsalis, Patrick Kreidl, Nuno Martins, Sidhant Misra, Paul Njoroge, Mesrob Ohannessian, Mitra Osqui, Mardavij Roozbehani, Keith Santarelli, Sri Sarma, Pari Shah, Noah Stein, Danielle Tarraf, and Ermin Wei. Among these individuals, I want to specifically thank Yola for the many fun conversations, interesting stories, and good times; Paul for the many conversations and good jokes over free CSAIL tea and Muddy Charles beers; and my office mates (Yola, Mitra, and Rose) in part for their willingness to listen to my stories and far-out ideas, but mostly for just for making the office such a fun place to work. Some friends outside of LIDS that I wish to thank are Al Avestruz, Jose Bohorquez, Wilfried Hofstetter, Deborah Howell, Behnam Jafarpour, Kelly Klima, Bill Nadir, and Crystal Ng. In particular, I would like to thank Al for countless enlightening conversations, good ideas, and just plain good times; Wilfried for many great discussions of geopolitics over many great beers; and Kelly for all the fun times while hanging out at the Thirsty. I also want to thank Lisa Vora and Jason Wampler for the good times and their hospitality whenever I visited Maryland, and I want to thank the many fun and interesting people I met while working with the Thirsty Ear Pub, including the staff (current and former), the executive committee, and the management. My time working with the Thirsty Ear holds a special place in my memories, and everyone involved with the pub is a part of those memories. Finally, I wish to offer a special set of acknowledgments to my family and my girlfriend. To my sister Kristin and my brothers Jason, Scott, and Tim, you make going back home more fun than any other destination that I can imagine, and you are collectively funnier than any other crowd I've met. You make traveling back to New Jersey a delight and traveling away a chore. To my girlfriend Leila, thank you for your love, friendship, personality, sense of humor, support, unending patience, support, and unending patience. I have no doubt that you knew right away that wasn't a typo. There is also no doubt that I would not be here today if it was not for your support of my academic work and aspirations. At the same time, you keep me grounded and well balanced, and you keep me level when stress would otherwise have its own way. You are truly a rock for me as well as my best friend. And to my parents, I want to thank you for your unending support of my goals at every step of the way. In particular, I want to thank my father for fostering and supporting the development of my interests, skills, and (by example) strong work ethic. I want to thank my mother for her constant belief in me (you were right! ) and for her seeming unending wisdom that has had such a profound effect on me. For these reasons and more, I dedicate this dissertation to you both. Michael Rinehart November 24, 2009 This work was sponsored by the Army Research Office (W911NF-07-1-0568).