On Learning with Finite Memory

We consider an infinite collection of agents who make decisions, sequentially, about an unknown underlying binary state of the world. Each agent, prior to making a decision, receives an independent private signal whose distribution depends on the state of the world. Moreover, each agent also observes the decisions of its last K immediate predecessors. We study conditions under which the agent decisions converge to the correct value of the underlying state. We focus on the case where the private signals have bounded information content and investigate whether learning is possible, that is, whether there exist decision rules for the different agents that result in the convergence of their sequence of individual decisions to the correct state of the world. We first consider learning in the almost sure sense and show that it is impossible, for any value of K. We then explore the possibility of convergence in probability of the decisions to the correct state. Here, a distinction arises: if K equals 1, learning in probability is impossible under any decision rule, while for K greater or equal to 2, we design a decision rule that achieves it. We finally consider a new model, involving forward looking strategic agents, each of which maximizes the discounted sum (over all agents) of the probabilities of a correct decision. (The case, studied in previous literature, of myopic agents who maximize the probability of their own decision being correct is an extreme special case.) We show that for any value of K, for any equilibrium of the associated Bayesian game, and under the assumption that each private signal has bounded information content, learning in probability fails to obtain

When is a network epidemic hard to eliminate?

We consider the propagation of a contagion process (epidemic) on a network and study the problem of dynamically allocating a fixed curing budget to the nodes of the graph, at each time instant. For bounded degree graphs, we provide a lower bound on the expected time to extinction under any such dynamic allocation policy, in terms of a combinatorial quantity that we call the resistance of the set of initially infected nodes, the available budget, and the number of nodes n. Specifically, we consider the case of bounded degree graphs, with the resistance growing linearly in n. We show that if the curing budget is less than a certain multiple of the resistance, then the expected time to extinction grows exponentially with n. As a corollary, if all nodes are initially infected and the CutWidth of the graph grows linearly, while the curing budget is less than a certain multiple of the CutWidth, then the expected time to extinction grows exponentially in n. The combination of the latter with our prior work establishes a fairly sharp phase transition on the expected time to extinction (sub-linear versus exponential) based on the relation between the CutWidth and the curing budget

A lower bound on the performance of dynamic curing policies for epidemics on graphs

We consider an SIS-type epidemic process that evolves on a known graph. We assume that a fixed curing budget can be allocated at each instant to the nodes of the graph, towards the objective of minimizing the expected extinction time of the epidemic. We provide a lower bound on the optimal expected extinction time as a function of the available budget, the epidemic parameters, the maximum degree, and the CutWidth of the graph. For graphs with large CutWidth (close to the largest possible), and under a budget which is sublinear in the number of nodes, our lower bound scales exponentially with the size of the graph

An efficient curing policy for epidemics on graphs

We provide a dynamic policy for the rapid containment of a contagion process modeled as an SIS epidemic on a bounded degree undirected graph with n nodes. We show that if the budget $r$ of curing resources available at each time is ${\Omega}(W)$, where $W$ is the CutWidth of the graph, and also of order ${\Omega}(\log n)$, then the expected time until the extinction of the epidemic is of order $O(n/r)$, which is within a constant factor from optimal, as well as sublinear in the number of nodes. Furthermore, if the CutWidth increases only sublinearly with n, a sublinear expected time to extinction is possible with a sublinearly increasing budget $r$

Observational learning with finite memory

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 113-114).We study a model of sequential decision making under uncertainty by a population of agents. Each agent prior to making a decision receives a private signal regarding a binary underlying state of the world. Moreover she observes the actions of her last K immediate predecessors. We discriminate between the cases of bounded and unbounded informativeness of private signals. In contrast to the literature that typically assumes myopic agents who choose the action that maximizes the probability of making the correct decision (the decision that identifies correctly the underlying state), in our model we assume that agents are forward looking, maximizing the discounted sum of the probabilities of a correct decision from all the future agents including theirs. Therefore, an agent when making a decision takes into account the impact that this decision will have on the subsequent agents. We investigate whether in a Perfect Bayesian Equilibrium of this model individual's decisions converge to the correct state of the world, in probability, and we show that this cannot happen for any K and any discount factor if private signals' informativeness is bounded. As a benchmark, we analyze the design limits associated with this problem, which entail constructing decision profiles that dictate each agent's action as a function of her information set, given by her private signal and the last K decisions. We investigate the case of bounded informativeness of the private signals. We answer the question whether there exists a decision profile that results in agents' actions converging to the correct state of the world, a property that we call learning. We first study almost sure learning and prove that it is impossible under any decision rule. We then explore learning in probability, where a dichotomy arises. Specifically, if K = 1 we show that learning in probability is impossible under any decision rule, while for K > 2 we design a decision rule that achieves it.by Kimon Drakopoulos.S.M

Analysis and control of contagion processes on networks

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2016.Cataloged from PDF version of thesis.Includes bibliographical references (pages 137-141).We consider the propagation of a contagion process ("epidemic") on a network and study the problem of dynamically allocating a fixed curing budget to the nodes of the graph, at each time instant. We provide a dynamic policy for the rapid containment of a contagion process modeled as an SIS epidemic on a bounded degree undirected graph with n nodes. We show that if the budget r of curing resources available at each time is Q(W), where W is the CutWidth of the graph, and also of order [omega](log n), then the expected time until the extinction of the epidemic is of order O(n/r), which is within a constant factor from optimal, as well as sublinear in the number of nodes. Furthermore, if the CutWidth increases sublinearly with n, a sublinear expected time to extinction is possible with only a sublinearly increasing budget r. In contrast, we provide a lower bound on the expected time to extinction under any such dynamic allocation policy, for bounded degree graphs, in terms of a combinatorial quantity that we call the resistance of the set of initially infected nodes, the available budget, and the number of nodes n. Specifically, we consider the case of bounded degree graphs, with the resistance growing linearly in n. We show that if the curing budget is less than a certain multiple of the resistance, then the expected time to extinction grows exponentially with n. As a corollary, if all nodes are initially infected and the CutWidth of the graph grows linearly in n, while the curing budget is less than a certain multiple of the CutWidth, then the expected time to extinction grows exponentially in n. The combination of these two results establishes a fairly sharp phase transition on the expected time to extinction (sublinear versus exponential) based on the relation between the CutWidth and the curing budget. Finally, in the empirical part of the thesis, we analyze data on the evolution and propagation of influenza across the United States and discover that compartmental epidemic models enriched with environment dependent terms have fair prediction accuracy, and that the effect of inter-state traveling is negligible compared to the effect of intra-state contacts.by Kimon Drakopoulos.Ph. D

Theoretical Analysis of Active Contours on Graphs

Active contour models based on partial differential equations have proved successful in image segmentation, yet the study of their formulation on arbitrary geometric graphs, which place no restrictions in the spatial configuration of samples, is still at an early stage. In this paper, we introduce geometric approximations of gradient and curvature on arbitrary graphs, which enable a straightforward extension of active contour models that are formulated through level sets to such general inputs. We prove convergence in probability of our gradient approximation to the true gradient value and derive an asymptotic upper bound for the error of this approximation for the class of random geometric graphs. Two different approaches for the approximation of curvature are presented, and both are also proved to converge in probability in the case of random geometric graphs. We propose neighborhood-based filtering on graphs to improve the accuracy of the aforementioned approximations and define two variants of Gaussian smoothing on graphs which include normalization in order to adapt to graph nonuniformities. The performance of our active contour framework on graphs is demonstrated in the segmentation of regular images and geographical data defined on arbitrary graphs, using geodesic active contours and active contours without edges as representative models in our experiments.ISSN:1936-495

A lower bound on the performance of dynamic curing policies for epidemics on graphs

We consider an SIS-type epidemic process that evolves on a known graph. We assume that a fixed curing budget can be allocated at each instant to the nodes of the graph, towards the objective of minimizing the expected extinction time of the epidemic. We provide a lower bound on the optimal expected extinction time as a function of the available budget, the epidemic parameters, the maximum degree, and the CutWidth of the graph. For graphs with large CutWidth (close to the largest possible), and under a budget which is sublinear in the number of nodes, our lower bound scales exponentially with the size of the graph.United States. Army Research Office (Grant W911NF-12-1-0509)National Science Foundation (U.S.) (Grant CMMI-1234062