Ilan Lobel

Distributed Subgradient Methods for Convex Optimization Over Random Networks

We consider the problem of cooperatively minimizing the sum of convex functions, where the functions represent local objective functions of the agents. We assume that each agent has information about his local function, and communicate with the other agents over a time-varying network topology. For this problem, we propose a distributed subgradient method that uses averaging algorithms for locally sharing information among the agents. In contrast to previous works on multi-agent optimization that make worst-case assumptions about the connectivity of the agents (such as bounded communication intervals between nodes), we assume that links fail according to a given stochastic process. Under the assumption that the link failures are independent and identically distributed over time (possibly correlated across links), we provide almost sure convergence results for our subgradient algorithm.

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. 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.

Optimal Dynamic Mechanism Design and the Virtual Pivot Mechanism

We consider the problem of designing optimal mechanisms for settings where agents have dynamic private information. We present the virtual-pivot mechanism, which is optimal in a large class of environments that satisfy a separability condition. The mechanism satisfies a rather strong equilibrium notion (it is periodic ex post incentive compatible and individually rational). We provide both necessary and sufficient conditions for immediate incentive compatibility for mechanisms that satisfy periodic ex post incentive compatibility in future periods. The result also yields a strikingly simple mechanism for selling a sequence of items to a single buyer. We also show that the allocation rule of the virtual-pivot mechanism has a very simple structure (a virtual index) in multiarmed bandit settings. Finally, we show through examples that the relaxation technique we use does not produce optimal dynamic mechanisms in general nonseparable environments.

Optimizing Product Launches in the Presence of Strategic Consumers

A technology firm launches newer generations of a given product over time. At any moment, the firm decides whether to release a new version of the product that captures the current technology level at the expense of a fixed launch cost. Consumers are forward-looking and purchase newer models only when it maximizes their own future discounted surpluses. We start by assuming that consumers have a common valuation for the product and consider two product launch settings. In the first setting, the firm does not announce future release technologies and the equilibrium of the game is to release new versions cyclically with a constant level of technology improvement that is optimal for the firm. In the second setting, the firm is able to precommit to a schedule of technology releases and the optimal policy generally consists of alternating minor and major technology launch cycles. We verify that the difference in profits between the commitment and no-commitment scenarios can be significant, varying from 4% to 12%. Finally, we generalize our model to allow for multiple customer classes with different valuations for the product, demonstrating how to compute equilibria in this case and numerically deriving insights for different market compositions.

Convergence analysis of distributed subgradient methods over random networks

We consider the problem of cooperatively minimizing the sum of convex functions, where the functions represent local objective functions of the agents. We assume that each agent has information about his local function, and communicate with the other agents over a time-varying network topology. For this problem, we propose a distributed subgradient method that uses averaging algorithms for locally sharing information among the agents. In contrast to previous works that make worst-case assumptions about the connectivity of the agents (such as bounded communication intervals between nodes), we assume that links fail according to a given stochastic process. Under the assumption that the link failures are independent and identically distributed over time (possibly correlated across links), we provide convergence results and convergence rate estimates for our subgradient algorithm.

Optimal Multiperiod Pricing with Service Guarantees

We study the multiperiod pricing problem of a service firm with capacity levels that vary over time. Customers are heterogeneous in their arrival and departure periods as well as valuations, and are fully strategic with respect to their purchasing decisions. The firms problem is to set a sequence of prices that maximizes its revenue while guaranteeing service to all paying customers. We provide a dynamic programming based algorithm that computes the optimal sequence of prices for this problem in polynomial time. We show that due to the presence of strategic customers, available service capacity at a time period may bind the price offered at another time period. This phenomenon leads the firm to utilize the same price in multiple periods, in effect limiting the number of different prices that the service firm utilizes in optimal price policies. Also, when customers become more strategic patient for service, the firm offers higher prices. This may lead to the underutilization of capacity, lower revenues, and reduced customer welfare. We observe that the firm can combat this problem if it has an ability, beyond posted prices, to direct customers to different service periods. This paper was accepted by Dimitris Bertsimas, optimization.

Preferences, Homophily, and Social Learning

We study a sequential model of Bayesian social learning in networks in which agents have heterogeneous preferences, and neighbors tend to have similar preferences—a phenomenon known as homophily. We find that the density of network connections determines the impact of preference diversity and homophily on learning. When connections are sparse, diverse preferences are harmful to learning, and homophily may lead to substantial improvements. In contrast, in a dense network, preference diversity is beneficial. Intuitively, diverse ties introduce more independence between observations while providing less information individually. Homophilous connections individually carry more useful information, but multiple observations become redundant.

Optimal Long-Term Supply Contracts with Asymmetric Demand Information

We consider a manufacturer selling to a retailer with private demand information arising dynamically over an innite time horizon. Under a backlogging model, we show that the manufacturer’s optimal dynamic long-term contract takes a simple form: in the rst period, based on her private demand forecast, the retailer selects a wholesale price and pays an associated upfront fee, and, from then on, the two parties stick to a simple wholesale price contract with the retailer’s chosen price. We also show the structure of the optimal long-term contract under lost sales combines wholesale prices with options that coordinate the supply chain.

Lower bounds on the rate of learning in social networks

We study the rate of convergence of Bayesian learning in social networks. Each individual receives a signal about the underlying state of the world, observes a subset of past actions and chooses one of two possible actions. Our previous work [1] established that when signals generate unbounded likelihood ratios, there will be asymptotic learning under mild conditions on the social network topology-in the sense that beliefs and decisions converge (in probability) to the correct beliefs and action. The question of the speed of learning has not been investigated, however. In this paper, we provide estimates of the speed of learning (the rate at which the probability of the incorrect action converges to zero). We focus on a special class of topologies in which individuals observe either a random action from the past or the most recent action. We show that convergence to the correct action is faster than a polynomial rate when individuals observe the most recent action and is at a logarithmic rate when they sample a random action from the past. This suggests that communication in social networks that lead to repeated sampling of the same individuals lead to slower aggregation of information.

Feature-based Dynamic Pricing

We consider the problem faced by a firm that receives highly differentiated products in an online fashion and needs to price them in order to sell them to its customer base. Products are described by vectors of features and the market value of each product is linear in the values of the features. The firm does not initially know the values of the different features, but it can learn the values of the features based on whether products were sold at the posted prices in the past. This model is motivated by a question in online advertising, where impressions arrive over time and can be described by vectors of features. We first consider a multi-dimensional version of binary search over polyhedral sets, and show that it has exponential worst-case regret. We then propose a modification of the prior algorithm where uncertainty sets are replaced by their Lowner-John ellipsoids. We show that this algorithm has a worst-case regret that is quadratic in the dimensionality of the feature space and logarithmic in the time horizon.