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Error bounds for computing the expectation by Markov chain Monte Carlo
, 2009
"... We study the error of reversible Markov chain Monte Carlo methods for approximating the expectation of a function. Explicit error bounds with respect to the l2, l4 and l∞norm of the function are proven. By the estimation the well known asymptotical limit of the error is attained, i.e. there is n ..."
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Cited by 116 (2 self)
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We study the error of reversible Markov chain Monte Carlo methods for approximating the expectation of a function. Explicit error bounds with respect to the l2, l4 and l∞norm of the function are proven. By the estimation the well known asymptotical limit of the error is attained, i.e. there is no gap between the estimate and the asymptotical behavior. We discuss the dependence of the error on a burnin of the Markov chain. Furthermore we suggest and justify a specific burnin for optimizing the algorithm.
Dual Averaging for Distributed Optimization: Convergence Analysis and Network Scaling
 IEEE TRANSACTIONS ON AUTOMATIC CONTROL
, 2010
"... The goal of decentralized optimization over a network is to optimize a global objective formed by a sum of local (possibly nonsmooth) convex functions using only local computation and communication. It arises in various application domains, including distributed tracking and localization, multiagen ..."
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Cited by 93 (12 self)
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The goal of decentralized optimization over a network is to optimize a global objective formed by a sum of local (possibly nonsmooth) convex functions using only local computation and communication. It arises in various application domains, including distributed tracking and localization, multiagent coordination, estimation in sensor networks, and largescale machine learning. We develop and analyze distributed algorithms based on dual subgradient averaging, and we provide sharp bounds on their convergence rates as a function of the network size and topology. Our analysis allows us to clearly separate the convergence of the optimization algorithm itself and the effects of communication dependent on the network structure. We show that the number of iterations required by our algorithm scales inversely in the spectral gap of the network and confirm this prediction’s sharpness both by theoretical lower bounds and simulations for various networks. Our approach includes the cases of deterministic optimization and communication as well as problems with stochastic optimization and/or communication.
Cover times, blanket times, and majorizing measures
 In ACM Symposium on Theory of Computing
, 2011
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Curvature, concentration, and error estimates for Markov chain
"... We provide explicit nonasymptotic estimates for the rate of convergence of empirical means of Markov chains, together with a Gaussian or exponential control on the deviations of empirical means. These estimates hold under a “positive curvature ” assumption expressing a kind of metric ergodicity, wh ..."
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Cited by 30 (3 self)
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We provide explicit nonasymptotic estimates for the rate of convergence of empirical means of Markov chains, together with a Gaussian or exponential control on the deviations of empirical means. These estimates hold under a “positive curvature ” assumption expressing a kind of metric ergodicity, which generalizes the Ricci curvature from differential geometry and, on finite graphs, amounts to contraction under path coupling. The goal of the Markov chain Monte Carlo method is to provide an efficient way to approximate the integral π(f): = ∫ f(x)π(dx) of a function f under a finite measure π on some space X. This approach, which has been very successful, consists in constructing a hopefully easytosimulate Markov chain (X1, X2,...,Xk,...) on X with stationary distribution π, waiting for a time T0 (the burnin) so that the chain gets close to its stationary distribution, and then estimating π(f) by the empirical mean on the next T steps of the trajectory, with T large enough:
OPINION FLUCTUATIONS AND DISAGREEMENT IN SOCIAL NETWORKS
 SUBMITTED TO THE ANNALS OF APPLIED PROBABILITY
, 2010
"... We study a stochastic gossip model 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 and might represent ..."
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Cited by 26 (5 self)
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We study a stochastic gossip model 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 and might represent leaders, political parties or media sources attempting to influence the beliefs in the rest of the society. When the society contains stubborn agents with different opinions, opinion dynamics never lead to a consensus (among the regular agents). Instead, beliefs in the society almost surely fail to converge, and the belief of each regular agent converges in law to a nondegenerate random variable. The model thus generates longrun disagreement and continuous opinion fluctuations. The structure of the social network and the location of stubborn agents within it shape opinion dynamics. When the society is “highly fluid”, meaning that the mixing time of the random walk on the graph describing the social network is small relative to (the inverse of) the relative size of the linkages to stubborn agents, the ergodic beliefs of most of the agents concentrate around a certain common value. We also show that under additional conditions, the ergodic beliefs distribution becomes “approximately chaotic”, meaning that the variance of the aggregate belief of the society vanishes in the large population limit while individual opinions still fluctuate significantly.
Nazarov: Perfect matchings as IID factors on nonamenable groups
 Europ. J. Combin
, 2011
"... ar ..."
The mixing time evolution of Glauber dynamics for the meanfield Ising model
 Comm. Math. Phys
"... Abstract. We consider Glauber dynamics for the Ising model on the complete graph on n vertices, known as the CurieWeiss model. It is wellknown that the mixingtime in the high temperature regime (β < 1) has order n log n, whereas the mixingtime in the case β> 1 is exponential in n. Recently ..."
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Cited by 24 (12 self)
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Abstract. We consider Glauber dynamics for the Ising model on the complete graph on n vertices, known as the CurieWeiss model. It is wellknown that the mixingtime in the high temperature regime (β < 1) has order n log n, whereas the mixingtime in the case β> 1 is exponential in n. Recently, Levin, Luczak and Peres proved that for any fixed β < 1 there is cutoff at time 1 2(1−β) n log n with a window of order n, whereas the mixingtime at the critical temperature β = 1 is Θ(n 3/2). It is natural to ask how the mixingtime transitions from Θ(n log n) to Θ(n 3/2) and finally to exp (Θ(n)). That is, how does the mixingtime behave when β = β(n) is allowed to tend to 1 as n → ∞. In this work, we obtain a complete characterization of the mixingtime of the dynamics as a function of the temperature, as it approaches its critical point βc = 1. In particular, we find a scaling window of order 1 / √ n around the critical temperature. In the high temperature regime, β = 1 − δ for some 0 < δ < 1 so that δ 2 n → ∞ with n, the mixingtime has order (n/δ) log(δ 2 n), and exhibits cutoff with constant 1 2 and window size n/δ. In the critical window, β = 1 ± δ where δ2n is O(1), there is no cutoff, and the mixingtime has order n 3/2. At low temperature, β = 1 + δ for δ> 0 with δ 2 n → ∞ and δ = o(1), there is no cutoff, and the mixing time has order n δ exp ( ( 3 4 + o(1))δ2n). 1.
Systemic Risk and Stability in Financial Networks ∗
, 2013
"... We provide a framework for studying the relationship between the financial network architecture and the likelihood of systemic failures due to contagion of counterparty risk. We show that financial contagion exhibits a form of phase transition as interbank connections increase: as long as the magnit ..."
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Cited by 22 (0 self)
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We provide a framework for studying the relationship between the financial network architecture and the likelihood of systemic failures due to contagion of counterparty risk. We show that financial contagion exhibits a form of phase transition as interbank connections increase: as long as the magnitude and the number of negative shocks affecting financial institutions are sufficiently small, more “complete ” interbank claims enhance the stability of the system. However, beyond a certain point, such interconnections start to serve as a mechanism for propagation of shocks and lead to a more fragile financial system. We also show that, under natural contracting assumptions, financial networks that emerge in equilibrium may be socially inefficient due to the presence of a network externality: even though banks take the effects of their lending, risktaking and failure on their immediate creditors into account, they do not internalize the consequences of their actions on the rest of the network.