We study learning in a setting where agents receive independent noisy signals about the true value of a variable and then communi cate in a network. They naively update beliefs by repeatedly taking weighted averages of neighbors' opinions. We show that all opinions in a large society converge to the truth if and only if the influence of the most influential agent vanishes as the society grows. We also identify obstructions to this, including prominent groups, and pro vide structural conditions on the network ensuring efficient learn ing. Whether agents converge to the truth is unrelated to how quickly consensus is approached. (JEL D83, D85, Z13)

/pdf/naive-learning-in-social-networks-and-the-wisdom-of-crowds-15d43bhpge.pdf

Naïve Learning in Social Networks and the Wisdom of Crowds

We consider distributed optimization by a collection of nodes, each having access to its own convex function, whose collective goal is to minimize the sum of the functions. The communications between nodes are described by a time-varying sequence of directed graphs, which is uniformly strongly connected. For such communications, assuming that every node knows its out-degree, we develop a broadcast-based algorithm, termed the subgradient-push, which steers every node to an optimal value under a standard assumption of subgradient boundedness. The subgradient-push requires no knowledge of either the number of agents or the graph sequence to implement. Our analysis shows that the subgradient-push algorithm converges at a rate of O (ln t/√t), where the constant depends on the initial values at the nodes, the subgradient norms, and, more interestingly, on both the consensus speed and the imbalances of influence among the nodes.

Distributed optimization over time-varying directed graphs

We study distributed optimization problems when N nodes minimize the sum of their individual costs subject to a common vector variable. The costs are convex, have Lipschitz continuous gradient (with constant L), and bounded gradient. We propose two fast distributed gradient algorithms based on the centralized Nesterov gradient algorithm and establish their convergence rates in terms of the per-node communications K and the per-node gradient evaluations k. Our first method, Distributed Nesterov Gradient, achieves rates O( logK/K) and O(logk/k). Our second method, Distributed Nesterov gradient with Consensus iterations, assumes at all nodes knowledge of L and μ(W) - the second largest singular value of the N ×N doubly stochastic weight matrix W. It achieves rates O( 1/ K2-ξ) and O( 1/k2) ( ξ > 0 arbitrarily small). Further, we give for both methods explicit dependence of the convergence constants on N and W. Simulation examples illustrate our findings.

/pdf/fast-distributed-gradient-methods-40j6e4y7jy.pdf

Fast Distributed Gradient Methods

We provide an overview of recent research on belief and opinion dynamics in social networks. We discuss both Bayesian and non-Bayesian models of social learning and focus on the implications of the form of learning (e.g., Bayesian vs. non-Bayesian), the sources of information (e.g., observation vs. communication), and the structure of social networks in which individuals are situated on three key questions: (1) whether social learning will lead to consensus, i.e., to agreement among individuals starting with different views; (2) whether social learning will effectively aggregate dispersed information and thus weed out incorrect beliefs; (3) whether media sources, prominent agents, politicians and the state will be able to manipulate beliefs and spread misinformation in a society.

/pdf/opinion-dynamics-and-learning-in-social-networks-1y846zw0bs.pdf

Opinion Dynamics and Learning in Social Networks

/pdf/opinion-dynamics-and-learning-in-social-networks-3kf8bg71tx.pdf

We study distributed algorithms for solving global optimization problems in which the objective function is the sum of local objective functions of agents and the constraint set is given by the intersection of local constraint sets of agents. We assume that each agent knows only his own local objective function and constraint set, and exchanges information with the other agents over a randomly varying network topology to update his information state. We assume a state-dependent communication model over this topology: communication is Markovian with respect to the states of the agents and the probability with which the links are available depends on the states of the agents. We study a projected multi-agent subgradient algorithm under state-dependent communication. The state-dependence of the communication introduces significant challenges and couples the study of information exchange with the analysis of subgradient steps and projection errors. We first show that the multi-agent subgradient algorithm when used with a constant stepsize may result in the agent estimates to diverge with probability one. Under some assumptions on the stepsize sequence, we provide convergence rate bounds on a “disagreement metric” between the agent estimates. Our bounds are time-nonhomogeneous in the sense that they depend on the initial starting time. Despite this, we show that agent estimates reach an almost sure consensus and converge to the same optimal solution of the global optimization problem with probability one under different assumptions on the local constraint sets and the stepsize sequence.

https://pages.stern.nyu.edu/~ilobel/state-dep-optimization.pdf

Distributed multi-agent optimization with state-dependent communication

We study distributed algorithms for solving global optimization problems in which the objective function is the sum of local objective functions of agents and the constraint set is given by the intersection of local constraint sets of agents. We assume that each agent knows only his own local objective function and constraint set, and exchanges information with the other agents over a randomly varying network topology to update his information state. We assume a state-dependent communication model over this topology: communication is Markovian with respect to the states of the agents and the probability with which the links are available depends on the states of the agents. In this paper, we study a projected multi-agent subgradient algorithm under state-dependent communication. The algorithm involves each agent performing a local averaging to combine his estimate with the other agents' estimates, taking a subgradient step along his local objective function, and projecting the estimates on his local constraint set. The state-dependence of the communication introduces significant challenges and couples the study of information exchange with the analysis of subgradient steps and projection errors. We first show that the multi-agent subgradient algorithm when used with a constant stepsize may result in the agent estimates to diverge with probability one. Under some assumptions on the stepsize sequence, we provide convergence rate bounds on a "disagreement metric" between the agent estimates. Our bounds are time-nonhomogeneous in the sense that they depend on the initial starting time. Despite this, we show that agent estimates reach an almost sure consensus and converge to the same optimal solution of the global optimization problem with probability one under different assumptions on the local constraint sets and the stepsize sequence.

Distributed Multi-Agent Optimization with State-Dependent Communication

We study a zero-sum game formulation of a dynamic vehicle routing problem: a system planner seeks to design dynamic routing policies for a team of vehicles to minimize the average waiting time of demands that are strategically placed in a region by an adversarial agent with unitary capacity operating from a depot. We characterize an equilibrium in the limiting case where vehicles travel arbitrarily slower than the agent (heavy load). We show that such an equilibrium consists of a routing policy based on performing successive TSP tours through outstanding demands and a unique power-law spatial density centered at the depot location.

http://dspace.mit.edu/bitstream/handle/1721.1/68498/770668096-MIT.pdf;sequence=2

Strategic dynamic vehicle routing with spatio-temporal dependent demands

This paper studies endogenous liquidity crises as the result of information panics. Collective ignorance is welfare maximizing but it is fragile, susceptible to self-fulfilling fears about asymmetric information. When investors become worried about the potential of adverse selection, they raise interest rates. In turn, fearing unfairly high rates, borrowers have incentives to acquire information about their probability of repayment. More information worsens the average credit risk of borrowers in the market, and thus justifies investors' initial concerns. Adverse selection may obtain in equilibrium even though it is not justified by the fundamentals in the economy. Information panics amplify small aggregate shocks to asset qualities causing large detrimental effects to real economic activity.

Diego Feijer

Papers

Distributed multi-agent optimization with state-dependent communication

Distributed Multi-Agent Optimization with State-Dependent Communication

Strategic dynamic vehicle routing with spatio-temporal dependent demands

Information Panics and Liquidity Crises