The value of information in shortest path optimization/

Abstract

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.

(cont.) 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.