arXiv subjects

Asuman Ozdaglar

Publications and source records attributed to Asuman Ozdaglar.

At least 109 records · Page 6Linked to original sources

arXiv2015

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.

cs.SI↗

arXiv2015

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.

cs.SI↗

arXiv2015

On the Convergence Rate of Incremental Aggregated Gradient Algorithms

Motivated by applications to distributed optimization over networks and large-scale data processing in machine learning, we analyze the deterministic incremental aggregated gradient method for minimizing a finite sum of smooth functions where the sum is strongly convex. This method processes the functions one at a time in a deterministic order and incorporates a memory of previous gradient values to accelerate convergence. Empirically it performs well in practice; however, no theoretical analysis with explicit rate results was previously given in the literature to our knowledge, in particular most of the recent efforts concentrated on the randomized versions. In this paper, we show that this deterministic algorithm has global linear convergence and characterize the convergence rate. We also consider an aggregated method with momentum and demonstrate its linear convergence. Our proofs rely on a careful choice of a Lyapunov function that offers insight into the algorithm's behavior and simplifies the proofs considerably.

math.OC↗

arXiv2014

A globally convergent incremental Newton method

Motivated by machine learning problems over large data sets and distributed optimization over networks, we develop and analyze a new method called incremental Newton method for minimizing the sum of a large number of strongly convex functions. We show that our method is globally convergent for a variable stepsize rule. We further show that under a gradient growth condition, convergence rate is linear for both variable and constant stepsize rules. By means of an example, we show that without the gradient growth condition, incremental Newton method cannot achieve linear convergence. Our analysis can be extended to study other incremental methods: in particular, we obtain a linear convergence rate result for the incremental Gauss-Newton algorithm under a variable stepsize rule.

math.OC↗

arXiv2014

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 $Ω(W)$, where $W$ is the CutWidth of the graph, and also of order $Ω(\log n)$, 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 $r$.

cs.SI↗

arXiv2014

Exchangeable Equilibria, Part I: Symmetric Bimatrix Games

We introduce the notion of exchangeable equilibria of a symmetric bimatrix game, defined as those correlated equilibria in which players' strategy choices are conditionally independently and identically distributed given some hidden variable. We give several game-theoretic interpretations and a version of the "revelation principle". Geometrically, the set of exchangeable equilibria is convex and lies between the symmetric Nash equilibria and the symmetric correlated equilibria. Exchangeable equilibria can achieve higher expected utility than symmetric Nash equilibria.

cs.GT↗

arXiv2014

Message passing optimization of Harmonic Influence Centrality

This paper proposes a new measure of node centrality in social networks, the Harmonic Influence Centrality, which emerges naturally in the study of social influence over networks. Using an intuitive analogy between social and electrical networks, we introduce a distributed message passing algorithm to compute the Harmonic Influence Centrality of each node. Although its design is based on theoretical results which assume the network to have no cycle, the algorithm can also be successfully applied on general graphs.

math.OC↗

arXiv2013

On the O(1/k) Convergence of Asynchronous Distributed Alternating Direction Method of Multipliers

We consider a network of agents that are cooperatively solving a global optimization problem, where the objective function is the sum of privately known local objective functions of the agents and the decision variables are coupled via linear constraints. Recent literature focused on special cases of this formulation and studied their distributed solution through either subgradient based methods with O(1/sqrt(k)) rate of convergence (where k is the iteration number) or Alternating Direction Method of Multipliers (ADMM) based methods, which require a synchronous implementation and a globally known order on the agents. In this paper, we present a novel asynchronous ADMM based distributed method for the general formulation and show that it converges at the rate O(1/k).

math.OC↗

arXiv2013

On Threshold Models over Finite Networks

We study a model for cascade effects over finite networks based on a deterministic binary linear threshold model. Our starting point is a networked coordination game where each agent's payoff is the sum of the payoffs coming from pairwise interactions with each of the neighbors. We first establish that the best response dynamics in this networked game is equivalent to the linear threshold dynamics with heterogeneous thresholds over the agents. While the previous literature has studied such linear threshold models under the assumption that each agent may change actions at most once, a study of best response dynamics in such networked games necessitates an analysis that allows for multiple switches in actions. In this paper, we develop such an analysis and construct a combinatorial framework to understand the behavior of the model. To this end, we establish that the agents behavior cycles among different actions in the limit and provide three sets of results. We first characterize the limiting behavioral properties of the dynamics. We determine the length of the limit cycles and reveal bounds on the time steps required to reach such cycles for different network structures. We then study the complexity of decision/counting problems that arise within the context. Specifically, we consider the tractability of counting the number of limit cycles and fixed-points, and deciding the reachability of action profiles. We finally propose a measure of network resilience that captures the nature of the involved dynamics. We prove bounds and investigate the resilience of different network structures under this measure.

cs.DM↗

arXiv2012

Opinion fluctuations and disagreement in social networks

We study a tractable opinion dynamics model that generates long-run disagreements and persistent opinion fluctuations. Our model involves an inhomogeneous stochastic gossip process of continuous opinion dynamics in a society consisting of two types of agents: regular agents, who update their beliefs according to information that they receive from their social neighbors; and stubborn agents, who never update their opinions. When the society contains stubborn agents with different opinions, the belief dynamics never lead to a consensus (among the regular agents). Instead, beliefs in the society fail to converge almost surely, the belief profile keeps on fluctuating in an ergodic fashion, and it converges in law to a non-degenerate random vector. The structure of the network and the location of the stubborn agents within it shape the opinion dynamics. The expected belief vector evolves according to an ordinary differential equation coinciding with the Kolmogorov backward equation of a continuous-time Markov chain with absorbing states corresponding to the stubborn agents and converges to a harmonic vector, with every regular agent's value being the weighted average of its neighbors' values, and boundary conditions corresponding to the stubborn agents'. Expected cross-products of the agents' beliefs allow for a similar characterization in terms of coupled Markov chains on the network. We prove that, in large-scale societies which are highly fluid, meaning that the product of the mixing time of the Markov chain on the graph describing the social network and the relative size of the linkages to stubborn agents vanishes as the population size grows large, a condition of \emph{homogeneous influence} emerges, whereby the stationary beliefs' marginal distributions of most of the regular agents have approximately equal first and second moments.

cs.SI↗

arXiv2012

A Fast Distributed Proximal-Gradient Method

We present a distributed proximal-gradient method for optimizing the average of convex functions, each of which is the private local objective of an agent in a network with time-varying topology. The local objectives have distinct differentiable components, but they share a common nondifferentiable component, which has a favorable structure suitable for effective computation of the proximal operator. In our method, each agent iteratively updates its estimate of the global minimum by optimizing its local objective function, and exchanging estimates with others via communication in the network. Using Nesterov-type acceleration techniques and multiple communication steps per iteration, we show that this method converges at the rate 1/k (where k is the number of communication rounds between the agents), which is faster than the convergence rate of the existing distributed methods for solving this problem. The superior convergence rate of our method is also verified by numerical experiments.

cs.DC↗

arXiv2012

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.

cs.GT↗

arXiv2012

QoE-aware Media Streaming in Technology and Cost Heterogeneous Networks

We present a framework for studying the problem of media streaming in technology and cost heterogeneous environments. We first address the problem of efficient streaming in a technology-heterogeneous setting. We employ random linear network coding to simplify the packet selection strategies and alleviate issues such as duplicate packet reception. Then, we study the problem of media streaming from multiple cost-heterogeneous access networks. Our objective is to characterize analytically the trade-off between access cost and user experience. We model the Quality of user Experience (QoE) as the probability of interruption in playback as well as the initial waiting time. We design and characterize various control policies, and formulate the optimal control problem using a Markov Decision Process (MDP) with a probabilistic constraint. We present a characterization of the optimal policy using the Hamilton-Jacobi-Bellman (HJB) equation. For a fluid approximation model, we provide an exact and explicit characterization of a threshold policy and prove its optimality using the HJB equation. Our simulation results show that under properly designed control policy, the existence of alternative access technology as a complement for a primary access network can significantly improve the user experience without any bandwidth over-provisioning.

eess.SY↗

arXiv2011

The Reliability Value of Storage in a Volatile Environment

This paper examines the value of storage in securing reliability of a system with uncertain supply and demand, and supply friction. The storage is frictionless as a supply source, but once used, it cannot be filled up instantaneously. The focus application is a power supply network in which the base supply and demand are assumed to match perfectly, while deviations from the base are modeled as random shocks with stochastic arrivals. Due to friction, the random surge shocks cannot be tracked by the main supply sources. Storage, when available, can be used to compensate, fully or partially, for the surge in demand or loss of supply. The problem of optimal utilization of storage with the objective of maximizing system reliability is formulated as minimization of the expected discounted cost of blackouts over an infinite horizon. It is shown that when the stage cost is linear in the size of the blackout, the optimal policy is myopic in the sense that all shocks are compensated by storage up to the available level of storage. However, when the stage cost is strictly convex, it may be optimal to curtail some of the demand and allow a small current blackout in the interest of maintaining a higher level of reserve to avoid a large blackout in the future. The value of storage capacity in improving system's reliability, as well as the effects of the associated optimal policies under different stage costs on the probability distribution of blackouts are examined.

math.OC↗

arXiv2011

Dynamics in Near-Potential Games

Except for special classes of games, there is no systematic framework for analyzing the dynamical properties of multi-agent strategic interactions. Potential games are one such special but restrictive class of games that allow for tractable dynamic analysis. Intuitively, games that are "close" to a potential game should share similar properties. In this paper, we formalize and develop this idea by quantifying to what extent the dynamic features of potential games extend to "near-potential" games. We study convergence of three commonly studied classes of adaptive dynamics: discrete-time better/best response, logit response, and discrete-time fictitious play dynamics. For better/best response dynamics, we focus on the evolution of the sequence of pure strategy profiles and show that this sequence converges to a (pure) approximate equilibrium set, whose size is a function of the "distance" from a close potential game. We then study logit response dynamics and provide a characterization of the stationary distribution of this update rule in terms of the distance of the game from a close potential game and the corresponding potential function. We further show that the stochastically stable strategy profiles are pure approximate equilibria. Finally, we turn attention to fictitious play, and establish that the sequence of empirical frequencies of player actions converges to a neighborhood of (mixed) equilibria of the game, where the size of the neighborhood increases with distance of the game to a potential game. Thus, our results suggest that games that are close to a potential game inherit the dynamical properties of potential games. Since a close potential game to a given game can be found by solving a convex optimization problem, our approach also provides a systematic framework for studying convergence behavior of adaptive learning dynamics in arbitrary finite strategic form games.

cs.GT↗

arXiv2011

A Distributed Newton Method for Network Utility Maximization

Most existing work uses dual decomposition and subgradient methods to solve Network Utility Maximization (NUM) problems in a distributed manner, which suffer from slow rate of convergence properties. This work develops an alternative distributed Newton-type fast converging algorithm for solving network utility maximization problems with self-concordant utility functions. By using novel matrix splitting techniques, both primal and dual updates for the Newton step can be computed using iterative schemes in a decentralized manner with limited information exchange. Similarly, the stepsize can be obtained via an iterative consensus-based averaging scheme. We show that even when the Newton direction and the stepsize in our method are computed within some error (due to finite truncation of the iterative schemes), the resulting objective function value still converges superlinearly to an explicitly characterized error neighborhood. Simulation results demonstrate significant convergence rate improvement of our algorithm relative to the existing subgradient methods based on dual decomposition.

math.OC↗

arXiv2011

Completion Time Minimization and Robust Power Control in Wireless Packet Networks

A wireless packet network is considered in which each user transmits a stream of packets to its destination. The transmit power of each user interferes with the transmission of all other users. A convex cost function of the completion times of the user packets is minimized by optimally allocating the users' transmission power subject to their respective power constraints. At all ranges of SINR, completion time minimization can be formulated as a convex optimization problem and hence can be efficiently solved. In particular, although the feasible rate region of the wireless network is non-convex, its corresponding completion time region is shown to be convex. When channel knowledge is imperfect, robust power control is considered based on the channel fading distribution subject to outage probability constraints. The problem is shown to be convex when the fading distribution is log-concave in exponentiated channel power gains; e.g., when each user is under independent Rayleigh, Nakagami, or log-normal fading. Applying the optimization frameworks in a wireless cellular network, the average completion time is significantly reduced as compared to full power transmission.

cs.IT↗

arXiv2011

Optimal Pricing in Networks with Externalities

We study the optimal pricing strategies of a monopolist selling a divisible good (service) to consumers that are embedded in a social network. A key feature of our model is that consumers experience a (positive) local network effect. In particular, each consumer's usage level depends directly on the usage of her neighbors in the social network structure. Thus, the monopolist's optimal pricing strategy may involve offering discounts to certain agents, who have a central position in the underlying network. First, we consider a setting where the monopolist can offer individualized prices and derive an explicit characterization of the optimal price for each consumer as a function of her network position. In particular, we show that it is optimal for the monopolist to charge each agent a price that is proportional to her Bonacich centrality in the social network. In the second part of the paper, we discuss the optimal strategy of a monopolist that can only choose a single uniform price for the good and derive an algorithm polynomial in the number of agents to compute such a price. Thirdly, we assume that the monopolist can offer the good in two prices, full and discounted, and study the problem of determining which set of consumers should be given the discount. We show that the problem is NP-hard, however we provide an explicit characterization of the set of agents that should be offered the discounted price. Next, we describe an approximation algorithm for finding the optimal set of agents. We show that if the profit is nonnegative under any feasible price allocation, the algorithm guarantees at least 88% of the optimal profit. Finally, we highlight the value of network information by comparing the profits of a monopolist that does not take into account the network effects when choosing her pricing policy to those of a monopolist that uses this information optimally.

cs.GT↗