arXiv subjects

Tamer Basar

Publications and source records attributed to Tamer Basar.

At least 55 records · Page 3Linked to original sources

arXiv2017

Exponential Convergence of a Distributed Algorithm for Solving Linear Algebraic Equations

In a recent paper, a distributed algorithm was proposed for solving linear algebraic equations of the form $Ax = b$ assuming that the equation has at least one solution. The equation is presumed to be solved by $m$ agents assuming that each agent knows a subset of the rows of the matrix $[A \; b]$, the current estimates of the equation's solution generated by each of its neighbors, and nothing more. Neighbor relationships are represented by a time-dependent directed graph $N(t)$ whose vertices correspond to agents and whose arcs characterize neighbor relationships. Sufficient conditions on $N(t)$ were derived under which the algorithm can cause all agents' estimates to converge exponentially fast to the same solution to $Ax = b$. These conditions were also shown to be necessary for exponential convergence, provided the data about $[A \; b]$ available to the agents is "non-redundant". The aim of this paper is to relax this "non-redundant" assumption. This is accomplished by establishing exponential convergence under conditions which are the weakest possible for the problem at hand; the conditions are based on a new notion of graph connectivity. An improved bound on the convergence rate is also derived.

math.OC↗

arXiv2016

Price of Transparency in Strategic Machine Learning

Based on the observation that the transparency of an algorithm comes with a cost for the algorithm designer when the users (data providers) are strategic, this paper studies the impact of strategic intent of the users on the design and performance of transparent ML algorithms. We quantitatively study the {\bf price of transparency} in the context of strategic classification algorithms, by modeling the problem as a nonzero-sum game between the users and the algorithm designer. The cost of having a transparent algorithm is measured by a quantity, named here as price of transparency which is the ratio of the designer cost at the Stackelberg equilibrium, when the algorithm is transparent (which allows users to be strategic) to that of the setting where the algorithm is not transparent.

cs.GT↗

arXiv2016

Optimal Capacity Allocation for Sampled Networked Systems

We consider the problem of estimating the states of weakly coupled linear systems from sampled measurements. We assume that the total capacity available to the sensors to transmit their samples to a network manager in charge of the estimation is bounded above, and that each sample requires the same amount of communication. Our goal is then to find an optimal allocation of the capacity to the sensors so that the average estimation error is minimized. We show that when the total available channel capacity is large, this resource allocation problem can be recast as a strictly convex optimization problem, and hence there exists a unique optimal allocation of the capacity. We further investigate how this optimal allocation varies as the available capacity increases. In particular, we show that if the coupling among the subsystems is weak, then the sampling rate allocated to each sensor is nondecreasing in the total sampling rate, and is strictly increasing if and only if the total sampling rate exceeds a certain threshold.

eess.SY↗

arXiv2016

Optimal Communication Scheduling and Remote Estimation over an Additive Noise Channel

This paper considers a sequential sensor scheduling and remote estimation problem with one sensor and one estimator. The sensor makes sequential observations about the state of an underlying memoryless stochastic process and makes a decision as to whether or not to send this measurement to the estimator. The sensor and the estimator have the common objective of minimizing expected distortion in the estimation of the state of the process, over a finite time horizon. The sensor is either charged a cost for each transmission or constrained on transmission times. As opposed to the prior work where communication between the sensor and the estimator was assumed to be perfect (noiseless), in this work an additive noise channel with fixed power constraint is considered; hence, the sensor has to encode its message before transmission. Under some technical assumptions, we obtain the optimal encoding and estimation policies in conjunction with the optimal transmission schedule. The impact of the presence of a noisy channel is analyzed numerically based on dynamic programming. This analysis yields some rather surprising results such as a phase transition phenomenon in the number of used transmission opportunities, which was not encountered in the noiseless communication setting.

math.OC↗

arXiv2016

Scalar Quadratic-Gaussian Soft Watermarking Games

We introduce the zero-sum game problem of soft watermarking: The hidden information (watermark) comes from a continuum and has a perceptual value; the receiver generates an estimate of the embedded watermark to minimize the expected estimation error (unlike the conventional watermarking schemes where both the hidden information and the receiver output are from a discrete finite set). Applications include embedding a multimedia content into another. We consider in this paper the scalar Gaussian case and use expected mean-squared distortion. We formulate the resulting problem as a zero-sum game between the encoder & receiver pair and the attacker. We show that for the lin- ear encoder, the optimal attacker is Gaussian-affine, derive the optimal system parameters in that case, and discuss the corresponding system behavior. We also provide numerical results to gain further insight and understanding of the system behavior at optimality.

cs.CR↗

arXiv2016

On the Structure of Equilibrium Strategies in Dynamic Gaussian Signaling Games

This paper analyzes a finite horizon dynamic signaling game motivated by the well-known strategic information transmission problems in economics. The mathematical model involves information transmission between two agents, a sender who observes two Gaussian processes, state and bias, and a receiver who takes an action based on the received message from the sender. The players incur quadratic instantaneous costs as functions of the state, bias and action variables. Our particular focus is on the Stackelberg equilibrium, which corresponds to information disclosure and Bayesian persuasion problems in economics. Prior work solved the static game, and showed that the Stackelberg equilibrium is achieved by pure strategies that are linear functions of the state and the bias variables. The main focus of this work is on the dynamic (multi-stage) setting, where we show that the existence of a pure strategy Stackelberg equilibrium, within the set of linear strategies, depends on the problem parameters. Surprisingly, for most problem parameters, a pure linear strategy does not achieve the Stackelberg equilibrium which implies the existence of a trade-off between exploiting and revealing information, which was also encountered in several other asymmetric information games.

eess.SY↗

arXiv2016

Information-Theoretic Approach to Strategic Communication as a Hierarchical Game

This paper analyzes the information disclosure problems originated in economics through the lens of information theory. Such problems are radically different from the conventional communication paradigms in information theory since they involve different objectives for the encoder and the decoder, which are aware of this mismatch and act accordingly. This leads, in our setting, to a hierarchical communication game, where the transmitter announces an encoding strategy with full commitment, and its distortion measure depends on a private information sequence whose realization is available at the transmitter. The receiver decides on its decoding strategy that minimizes its own distortion based on the announced encoding map and the statistics. Three problem settings are considered, focusing on the quadratic distortion measures, and jointly Gaussian source and private information: compression, communication, and the simple equilibrium conditions without any compression or communication. The equilibrium strategies and associated costs are characterized. The analysis is then extended to the receiver side information setting and the major changes in the structure of optimal strategies are identified. Finally, an extension of the results to the broader context of decentralized stochastic control is presented.

cs.IT↗

arXiv2016

Approximation Algorithm for the Binary-Preference Capacitated Selfish Replication Game and a Tight Bound on its Price of Anarchy

We consider the capacitated selfish replication (CSR) game with binary preferences, over general undirected networks. We first show that such games have an associated ordinary potential function, and hence always admit a pure-strategy Nash equilibrium (NE). Further, when the minimum degree of the network and the number of resources are of the same order, there exists an exact polynomial time algorithm which can find a NE. Following this, we study the price of anarchy of such games, and show that it is bounded above by 3; we further provide some instances for which the price of anarchy is at least 2. We develop a quasi-polynomial algorithm O(n^2D^{ln n}), where n is the number of players and D is the diameter of the network, which can find, in a distributed manner, an allocation profile that is within a constant factor of the optimal allocation, and hence of any pure-strategy NE of the game. Proof of this result uses a novel potential function.

cs.GT↗

arXiv2016

On a Modified DeGroot-Friedkin Model of Opinion Dynamics

This paper studies the opinion dynamics that result when individuals consecutively discuss a sequence of issues. Specifically, we study how individuals' self-confidence levels evolve via a reflected appraisal mechanism. Motivated by the DeGroot-Friedkin model, we propose a Modified DeGroot-Friedkin model which allows individuals to update their self-confidence levels by only interacting with their neighbors and in particular, the modified model allows the update of self-confidence levels to take place in finite time without waiting for the opinion process to reach a consensus on any particular issue. We study properties of this Modified DeGroot-Friedkin model and compare the associated equilibria and stability with those of the original DeGroot-Friedkin model. Specifically, for the case when the interaction matrix is doubly stochastic, we show that for the modified model, the vector of individuals' self-confidence levels asymptotically converges to a unique nontrivial equilibrium which for each individual is equal to 1/n, where n is the number of individuals. This implies that eventually, individuals reach a democratic state.

math.OC↗

arXiv2016

On the Role of Side Information In Strategic Communication

This paper analyzes the fundamental limits of strate- gic communication in network settings. Strategic communication differs from the conventional communication paradigms in in- formation theory since it involves different objectives for the encoder and the decoder, which are aware of this mismatch and act accordingly. This leads to a Stackelberg game where both agents commit to their mappings ex-ante. Building on our prior work on the point-to-point setting, this paper studies the compression and communication problems with the receiver and/or transmitter side information setting. The equilibrium strategies and associated costs are characterized for the Gaussian variables with quadratic cost functions. Several questions on the benefit of side information in source and joint source-channel coding in such strategic settings are analyzed. Our analysis has uncovered an interesting result on optimality of uncoded mappings in strategic source-channel coding in networks.

cs.IT↗

arXiv2016

Convergence Time of Quantized Metropolis Consensus Over Time-Varying Networks

We consider the quantized consensus problem on undirected time-varying connected graphs with n nodes, and devise a protocol with fast convergence time to the set of consensus points. Specifically, we show that when the edges of each network in a sequence of connected time-varying networks are activated based on Poisson processes with Metropolis rates, the expected convergence time to the set of consensus points is at most O(n^2 log^2 n), where each node performs a constant number of updates per unit time.

eess.SY↗

arXiv2016

Voltage Graphs and Cluster Consensus with Point Group Symmetries

A cluster consensus system is a multi-agent system in which the autonomous agents communicate to form multiple clusters, with each cluster of agents asymptotically converging to the same clustering point. We introduce in this paper a special class of cluster consensus dynamics, termed the $G$-clustering dynamics for $G$ a point group, whereby the autonomous agents can form as many as $|G|$ clusters, and moreover, the associated $|G|$ clustering points exhibit a geometric symmetry induced by the point group. The definition of a $G$-clustering dynamics relies on the use of the so-called voltage graph. We recall that a $G$-voltage graph is comprised of two elements---one is a directed graph (digraph), and the other is a map assigning elements of a group~$G$ to the edges of the digraph. For example, in the case when $G = \{1, -1\}$, i.e., a cyclic group of order~$2$, a voltage graph is nothing but a signed graph. A $G$-clustering dynamics can then be viewed as a generalization of the so-called Altafini's model, which was originally defined over a signed graph, by defining the dynamics over a voltage graph. One of the main contributions of this paper is to identify a necessary and sufficient condition for the exponential convergence of a $G$-clustering dynamics. Various properties of voltage graphs that are necessary for establishing the convergence result are also investigated, some of which might be of independent interest in topological graph theory.

math.DS↗

arXiv2015

Exponential Convergence of the Discrete-Time Altafini Model

This paper considers the discrete-time version of Altafini's model for opinion dynamics in which the interaction among a group of agents is described by a time-varying signed digraph. Prompted by an idea from [1], exponential convergence of the system is studied using a graphical approach. Necessary and sufficient conditions for exponential convergence with respect to each possible type of limit states are provided. Specifically, under the assumption of repeatedly jointly strong connectivity, it is shown that (1) a certain type of two-clustering will be reached exponentially fast for almost all initial conditions if, and only if, the sequence of signed digraphs is repeatedly jointly structurally balanced corresponding to that type of two-clustering; (2) the system will converge to zero exponentially fast for all initial conditions if, and only if, the sequence of signed digraphs is repeatedly jointly structurally unbalanced. An upper bound on the convergence rate is also provided.

math.OC↗

arXiv2015

Distributed Evaluation and Convergence of Self-Appraisals in Social Networks

We consider in this paper a networked system of opinion dynamics in continuous time, where the agents are able to evaluate their self-appraisals in a distributed way. In the model we formulate, the underlying network topology is described by a rooted digraph. For each ordered pair of agents $(i,j)$, we assign a function of self-appraisal to agent $i$, which measures the level of importance of agent $i$ to agent $j$. Thus, by communicating only with her neighbors, each agent is able to calculate the difference between her level of importance to others and others' level of importance to her. The dynamical system of self-appraisals is then designed to drive these differences to zero. We show that for almost all initial conditions, the trajectory generated by this dynamical system asymptotically converges to an equilibrium point which is exponentially stable.

eess.SY↗

arXiv2015

Privacy Constrained Information Processing

This paper studies communication scenarios where the transmitter and the receiver have different objectives due to privacy concerns, in the context of a variation of the strategic information transfer (SIT) model of Sobel and Crawford. We first formulate the problem as the minimization of a common distortion by the transmitter and the receiver subject to a privacy constrained transmitter. We show the equivalence of this formulation to a Stackelberg equilibrium of the SIT problem. Assuming an entropy based privacy measure, a quadratic distortion measure and jointly Gaussian variables, we characterize the Stackelberg equilibrium. Next, we consider asymptotically optimal compression at the transmitter which inherently provides some level of privacy, and study equilibrium conditions. We finally analyze the impact of the presence of an average power constrained Gaussian communication channel between the transmitter and the receiver on the equilibrium conditions.

cs.IT↗

arXiv2015

On Remote Estimation with Multiple Communication Channels

This paper considers a sequential estimation and sensor scheduling problem in the presence of multiple communication channels. As opposed to the classical remote estimation problem that involves one perfect (noiseless) channel and one extremely noisy channel (which corresponds to not transmitting the observed state), a more realistic additive noise channel with fixed power constraint along with a more costly perfect channel is considered. It is shown, via a counter-example, that the common folklore of applying symmetric threshold policy, which is well known to be optimal (for unimodal state densities) in the classical two-channel remote estimation problem, can be suboptimal for the setting considered. Next, in order to make the problem tractable, a side channel which signals the sign of the underlying state is considered. It is shown that, under some technical assumptions, threshold-in-threshold communication scheduling is optimal for this setting. The impact of the presence of a noisy channel is analyzed numerically based on dynamic programming. This numerical analysis uncovers some rather surprising results inheriting known properties from the noisy and noiseless settings.

cs.IT↗

arXiv2015

Optimal Sensor Scheduling and Remote Estimation over an Additive Noise Channel

We consider a sensor scheduling and remote estimation problem with one sensor and one estimator. At each time step, the sensor makes an observation on the state of a source, and then decides whether to transmit its observation to the estimator or not. The sensor is charged a cost for each transmission. The remote estimator generates a real-time estimate on the state of the source based on the messages received from the sensor. The estimator is charged for estimation error. As compared with previous works from the literature, we further assume that there is an additive communication channel noise. As a consequence, the sensor needs to encode the message before transmitting it to the estimator. For some specific distributions of the underlying random variables, we obtain the optimal solution to the problem of minimizing the expected value of the sum of communication cost and estimation cost over the time horizon.

eess.SY↗

arXiv2015

Optimal Estimation with Limited Measurements and Noisy Communication

This paper considers a sequential estimation and sensor scheduling problem with one sensor and one estimator. The sensor makes sequential observations about the state of an underlying memoryless stochastic process, and makes a decision as to whether or not to send this measurement to the estimator. The sensor and the estimator have the common objective of minimizing expected distortion in the estimation of the state of the process, over a finite time horizon, with the constraint that the sensor can transmit its observation only a limited number of times. As opposed to the prior work where communication between the sensor and the estimator was assumed to be perfect (noiseless), in this work an additive noise channel with fixed power constraint is considered; hence, the sensor has to encode its message before transmission. For some specific source and channel noise densities, we obtain the optimal encoding and estimation policies in conjunction with the optimal transmission schedule. The impact of the presence of a noisy channel is analyzed numerically based on dynamic programming. This analysis yields some rather surprising results such as a phase-transition phenomenon in the number of used transmission opportunities, which was not encountered in the noiseless communication setting.

eess.SY↗