Slopey quantizers are locally optimal for Witsenhausen's counterexample

We study the perfect Bayesian equilibria of a leader-follower game of incomplete information. The follower makes a noisy observation of the leader's action (who moves first) and chooses an action minimizing her expected deviation from the leader's action. Knowing this, leader who observes the realization of the state, chooses an action that minimizes her distance to the state of the world and the ex-ante expected deviation from the follower's action. We show the existence of what we call “near piecewise-linear equilibria” when there is strong complementarity between the leader and the follower and the precision of the prior is poor. As a major consequence of this result, we prove local optimality of a class of slopey quantization strategies which had been suspected of being the optimal solution in the past, based on numerical evidence for Witsenhausen's counterexample

Complexity of Bayesian Belief Exchange over a Network

Many important real-world decision making prob- lems involve group interactions among individuals with purely informational externalities, such situations arise for example in jury deliberations, expert committees, medical diagnosis, etc. In this paper, we will use the framework of iterated eliminations to model the decision problem as well as the thinking process of a Bayesian agent in a group decision/discussion scenario. We model the purely informational interactions of rational agents in a group, where they receive private information and act based upon that information while also observing other people’s beliefs. As the Bayesian agent attempts to infer the true state of the world from her sequence of observations which include her neighbors’ beliefs as well as her own private signal, she recursively refines her belief about the signals that other players could have observed and beliefs that they would have hold given the assumption that other players are also rational. We further analyze the computational complexity of the Bayesian belief formation in groups and show that it is NP -hard. We also investigate the factors underlying this computational complexity and show how belief calculations simplify in special network structures or cases with strong inherent symmetries. We finally give insights about the statistical efficiency (optimality) of the beliefs and its relations to computational efficiency.United States. Army Research Office (grant MURI W911NF-12- 1-0509)National Science Foundation (U.S.). Computing and Communication Foundation (grant CCF 1665252)United States. Department of Defense (ONR grant N00014-17-1-2598)National Science Foundation (U.S.) (grant DMS-1737944

Bayesian Learning Without Recall

We analyze a model of learning and belief formation in networks in which agents follow Bayes rule yet they do not recall their history of past observations and cannot reason about how other agents’ beliefs are formed. They do so by making rational inferences about their observations which include a sequence of independent and identically distributed private signals as well as the actions of their neighboring agents at each time. Successive applications of Bayes rule to the entire history of past observations lead to forebodingly complex inferences: due to lack of knowledge about the global network structure, and unavailability of private observations, as well as third party interactions preceding every decision. Such difficulties make Bayesian updating of beliefs an implausible mechanism for social learning. To address these complexities, we consider a Bayesian without Recall model of inference. On the one hand, this model provides a tractable framework for analyzing the behavior of rational agents in social networks. On the other hand, this model also provides a behavioral foundation for the variety of non-Bayesian update rules in the literature. We present the implications of various choices for the structure of the action space and utility functions for such agents and investigate the properties of learning, convergence, and consensus in special cases.United States. Army Research Office. Multidisciplinary University Research Initiative (W911NF-12-1-0509

Equilibrium analysis for a leader-follower game with noisy observations: A pace forward in Witsenhausen's counterexample conjecture

In this paper, we view Witsenhausen's problem as a leader-follower coordination game in which the action of the leader is corrupted by an additive normal noise, before reaching the follower. The leader who observes the realization of the state, chooses an action that minimizes her distance to the state of the world and the ex-ante expected deviation from the follower's action. The follower then makes a noisy observation of the leader's action and chooses an action minimizing her expected deviation from the leader's action. To this end, we analyze the perfect Bayesian equilibria of this game and show that strong complimentarily between the leader and the follower combined with a prior with poor enough precision can give rise to what we call “near piecewise-linear equilibria”. As a major consequence, we prove local optimality of a class of slopey quantization strategies which had been suspected of being the optimal solution in the past, based on numerical evidence for Witsenhausen's counterexample

An online optimization approach for multi-agent tracking of dynamic parameters in the presence of adversarial noise

This paper addresses tracking of a moving target in a multi-agent network. The target follows a linear dynamics corrupted by an adversarial noise, i.e., the noise is not generated from a statistical distribution. The location of the target at each time induces a global time-varying loss function, and the global loss is a sum of local losses, each of which is associated to one agent. Agents noisy observations could be nonlinear. We for- mulate this problem as a distributed online optimization where agents communicate with each other to track the minimizer of the global loss. We then propose a decentralized version of the Mirror Descent algorithm and provide the non-asymptotic analysis of the problem. Using the notion of dynamic regret, we measure the performance of our algorithm versus its offline counterpart in the centralized setting. We prove that the bound on dynamic regret scales inversely in the network spectral gap, and it represents the adversarial noise causing deviation with respect to the linear dynamics. Our result subsumes a number of results in the distributed optimization literature. Finally, in a numerical experiment, we verify that our algorithm can be simply implemented for multi-agent tracking with nonlinear observations.United States. Office of Naval Research. Basic Research Challenge. Program of Decentralized Online Optimizatio

Dynamic Pricing in Social Networks: The Word-of-Mouth Effect

We study the problem of optimal dynamic pricing for a monopolist selling a product to consumers in a social network. In the proposed model, the only means of spread of information about the product is via word-of-mouth communication; consumers’ knowledge of the product is only through friends who already know about the product’s existence. Both buyers and nonbuyers contribute to information diffusion, while buyers are more likely to spread the news about the product. By analyzing the structure of the underlying endogenous dynamic process, we show that the optimal dynamic pricing policy for durable products with zero or negligible marginal cost drops the price to zero infinitely often. The firm uses free offers to attract low-valuation agents and to get them more engaged in the spread. As a result, the firm can reach potential high-valuation consumers in parts of the network that would otherwise remain untouched without the price drops. We provide evidence for this behavior from the smartphone app market, where price histories indicate frequent zero-price sales. Moreover, we show that despite dropping the price to zero infinitely often, the optimal price trajectory does not get trapped near zero. We demonstrate the validity of our results in the face of forward-looking consumers and homophily in word-of-mouth engagement. We further unravel the key role of the product type in the optimality of zero-price sales by showing that the price fluctuations disappear after a finite time for a nondurable product. ©2016 INFORMS.ARO MURI (Grant W911NF-12-1-050

Distributed estimation and learning over heterogeneous networks

We consider several estimation and learning problems that networked agents face when making decisions given their uncertainty about an unknown variable. Our methods are designed to efficiently deal with heterogeneity in both size and quality of the observed data, as well as heterogeneity over time (intermittence). The goal of the studied aggregation schemes is to efficiently combine the observed data that is spread over time and across several network nodes, accounting for all the network heterogeneities. Moreover, we require no form of coordination beyond the local neighborhood of every network agent or sensor node. The three problems that we consider are (i) maximum likelihood estimation of the unknown given initial data sets, (ii) learning the true model parameter from streams of data that the agents receive intermittently over time, and (iii) minimum variance estimation of a complete sufficient statistic from several data points that the networked agents collect over time. In each case, we rely on an aggregation scheme to combine the observations of all agents; moreover, when the agents receive streams of data over time, we modify the update rules to accommodate the most recent observations. In every case, we demonstrate the efficiency of our algorithms by proving convergence to the globally efficient estimators given the observations of all agents. We supplement these results by investigating the rate of convergence and providing finite-time performance guarantees

Resilient monotone submodular function maximization

In this paper, we focus on applications in machine learning, optimization, and control that call for the resilient selection of a few elements, e.g. features, sensors, or leaders, against a number of adversarial denial-of-service attacks or failures. In general, such resilient optimization problems are hard, and cannot be solved exactly in polynomial time, even though they often involve objective functions that are monotone and submodular. Notwithstanding, in this paper we provide the first scalable algorithm for their approximate solution, that is valid for any number of attacks or failures, and which, for functions with low curvature, guarantees superior approximation performance. Notably, the curvature has been known to tighten approximations for several non-resilient maximization problems, yet its effect on resilient maximization had hitherto been unknown. We complement our theoretical analyses with supporting empirical evaluations

Are deep ResNets provably better than linear predictors?

© 2019 Neural information processing systems foundation. All rights reserved. Recent results in the literature indicate that a residual network (ResNet) composed of a single residual block outperforms linear predictors, in the sense that all local minima in its optimization landscape are at least as good as the best linear predictor. However, these results are limited to a single residual block (i.e., shallow ResNets), instead of the deep ResNets composed of multiple residual blocks. We take a step towards extending this result to deep ResNets. We start by two motivating examples. First, we show that there exist datasets for which all local minima of a fully-connected ReLU network are no better than the best linear predictor, whereas a ResNet has strictly better local minima. Second, we show that even at the global minimum, the representation obtained from the residual block outputs of a 2-block ResNet do not necessarily improve monotonically over subsequent blocks, which highlights a fundamental difficulty in analyzing deep ResNets. Our main theorem on deep ResNets shows under simple geometric conditions that, any critical point in the optimization landscape is either (i) at least as good as the best linear predictor; or (ii) the Hessian at this critical point has a strictly negative eigenvalue. Notably, our theorem shows that a chain of multiple skip-connections can improve the optimization landscape, whereas existing results study direct skip-connections to the last hidden layer or output layer. Finally, we complement our results by showing benign properties of the “near-identity regions” of deep ResNets, showing depth-independent upper bounds for the risk attained at critical points as well as the Rademacher complexity

Escaping saddle points in constrained optimization

In this paper, we study the problem of escaping from saddle points in smooth nonconvex optimization problems subject to a convex set C. We propose a generic framework that yields convergence to a second-order stationary point of the problem, if the convex set C is simple for a quadratic objective function. Specifically, our results hold if one can find a ρ-approximate solution of a quadratic program subject to C in polynomial time, where ρ < 1 is a positive constant that depends on the structure of the set C. Under this condition, we show that the sequence of iterates generated by the proposed framework reaches an (ε, γ)-second order stationary point (SOSP) in at most O(max{ε- 2 , ρ -3 γ -3 }) iterations. We further characterize the overall complexity of reaching an SOSP when the convex set C can be written as a set of quadratic constraints and the objective function Hessian has a specific structure over the convex set C. Finally, we extend our results to the stochastic setting and characterize the number of stochastic gradient and Hessian evaluations to reach an (ε, γ)-SOSP.United States. Defense Advanced Research Projects Agency. Lagrange ProgramUnited States. Office of Naval Research. Basic Research Challeng