arXiv subjects

Ali Jadbabaie

Publications and source records attributed to Ali Jadbabaie.

At least 127 records · Page 7Linked to original sources

arXiv2016

Near-Optimal Sensor Scheduling for Batch State Estimation: Complexity, Algorithms, and Limits

In this paper, we focus on batch state estimation for linear systems. This problem is important in applications such as environmental field estimation, robotic navigation, and target tracking. Its difficulty lies on that limited operational resources among the sensors, e.g., shared communication bandwidth or battery power, constrain the number of sensors that can be active at each measurement step. As a result, sensor scheduling algorithms must be employed. Notwithstanding, current sensor scheduling algorithms for batch state estimation scale poorly with the system size and the time horizon. In addition, current sensor scheduling algorithms for Kalman filtering, although they scale better, provide no performance guarantees or approximation bounds for the minimization of the batch state estimation error. In this paper, one of our main contributions is to provide an algorithm that enjoys both the estimation accuracy of the batch state scheduling algorithms and the low time complexity of the Kalman filtering scheduling algorithms. In particular: 1) our algorithm is near-optimal: it achieves a solution up to a multiplicative factor 1/2 from the optimal solution, and this factor is close to the best approximation factor 1/e one can achieve in polynomial time for this problem; 2) our algorithm has (polynomial) time complexity that is not only lower than that of the current algorithms for batch state estimation; it is also lower than, or similar to, that of the current algorithms for Kalman filtering. We achieve these results by proving two properties for our batch state estimation error metric, which quantifies the square error of the minimum variance linear estimator of the batch state vector: a) it is supermodular in the choice of the sensors; b) it has a sparsity pattern (it involves matrices that are block tri-diagonal) that facilitates its evaluation at each sensor set.

math.OC↗

arXiv2016

Distributed Online Optimization in Dynamic Environments Using Mirror Descent

This work addresses decentralized online optimization in non-stationary environments. A network of agents aim to track the minimizer of a global time-varying convex function. The minimizer evolves according to a known dynamics corrupted by an unknown, unstructured noise. At each time, the global function can be cast as a sum of a finite number of local functions, each of which is assigned to one agent in the network. Moreover, the local functions become available to agents sequentially, and agents do not have a prior knowledge of the future cost functions. Therefore, agents must communicate with each other to build an online approximation of the global function. We propose a decentralized variation of the celebrated Mirror Descent, developed by Nemirovksi and Yudin. Using the notion of Bregman divergence in lieu of Euclidean distance for projection, Mirror Descent has been shown to be a powerful tool in large-scale optimization. Our algorithm builds on Mirror Descent, while ensuring that agents perform a consensus step to follow the global function and take into account the dynamics of the global minimizer. To measure the performance of the proposed online algorithm, we compare it to its offline counterpart, where the global functions are available a priori. The gap between the two is called dynamic regret. We establish a regret bound that scales inversely in the spectral gap of the network, and more notably it represents the deviation of minimizer sequence with respect to the given dynamics. We then show that our results subsume a number of results in distributed optimization. We demonstrate the application of our method to decentralized tracking of dynamic parameters and verify the results via numerical experiments.

math.OC↗

arXiv2016

An Exact Distributed Newton Method for Reinforcement Learning

In this paper, we propose a distributed second- order method for reinforcement learning. Our approach is the fastest in literature so-far as it outperforms state-of-the-art methods, including ADMM, by significant margins. We achieve this by exploiting the sparsity pattern of the dual Hessian and transforming the problem of computing the Newton direction to one of solving a sequence of symmetric diagonally dominant system of equations. We validate the above claim both theoretically and empirically. On the theoretical side, we prove that similar to exact Newton, our algorithm exhibits super-linear convergence within a neighborhood of the optimal solution. Empirically, we demonstrate the superiority of this new method on a set of benchmark reinforcement learning tasks.

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arXiv2016

A Distributed Newton Method for Large Scale Consensus Optimization

In this paper, we propose a distributed Newton method for consensus optimization. Our approach outperforms state-of-the-art methods, including ADMM. The key idea is to exploit the sparsity of the dual Hessian and recast the computation of the Newton step as one of efficiently solving symmetric diagonally dominant linear equations. We validate our algorithm both theoretically and empirically. On the theory side, we demonstrate that our algorithm exhibits superlinear convergence within a neighborhood of optimality. Empirically, we show the superiority of this new method on a variety of machine learning problems. The proposed approach is scalable to very large problems and has a low communication overhead.

cs.DC↗

arXiv2016

Online Optimization in Dynamic Environments: Improved Regret Rates for Strongly Convex Problems

In this paper, we address tracking of a time-varying parameter with unknown dynamics. We formalize the problem as an instance of online optimization in a dynamic setting. Using online gradient descent, we propose a method that sequentially predicts the value of the parameter and in turn suffers a loss. The objective is to minimize the accumulation of losses over the time horizon, a notion that is termed dynamic regret. While existing methods focus on convex loss functions, we consider strongly convex functions so as to provide better guarantees of performance. We derive a regret bound that captures the path-length of the time-varying parameter, defined in terms of the distance between its consecutive values. In other words, the bound represents the natural connection of tracking quality to the rate of change of the parameter. We provide numerical experiments to complement our theoretical findings.

cs.LG↗

arXiv2016

Distributed Estimation of Dynamic Parameters : Regret Analysis

This paper addresses the estimation of a time- varying parameter in a network. A group of agents sequentially receive noisy signals about the parameter (or moving target), which does not follow any particular dynamics. The parameter is not observable to an individual agent, but it is globally identifiable for the whole network. Viewing the problem with an online optimization lens, we aim to provide the finite-time or non-asymptotic analysis of the problem. To this end, we use a notion of dynamic regret which suits the online, non-stationary nature of the problem. In our setting, dynamic regret can be recognized as a finite-time counterpart of stability in the mean- square sense. We develop a distributed, online algorithm for tracking the moving target. Defining the path-length as the consecutive differences between target locations, we express an upper bound on regret in terms of the path-length of the target and network errors. We further show the consistency of the result with static setting and noiseless observations.

math.OC↗

arXiv2015

Finite-time Analysis of the Distributed Detection Problem

This paper addresses the problem of distributed detection in fixed and switching networks. A network of agents observe partially informative signals about the unknown state of the world. Hence, they collaborate with each other to identify the true state. We propose an update rule building on distributed, stochastic optimization methods. Our main focus is on the finite-time analysis of the problem. For fixed networks, we bring forward the notion of Kullback-Leibler cost to measure the efficiency of the algorithm versus its centralized analog. We bound the cost in terms of the network size, spectral gap and relative entropy of agents' signal structures. We further consider the problem in random networks where the structure is realized according to a stationary distribution. We then prove that the convergence is exponentially fast (with high probability), and the non-asymptotic rate scales inversely in the spectral gap of the expected network.

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arXiv2015

Optimal distributed control for platooning via sparse coprime factorizations

We introduce a novel distributed control architecture for heterogeneous platoons of linear time--invariant autonomous vehicles. Our approach is based on a generalization of the concept of {\em leader--follower} controllers for which we provide a Youla--like parameterization while the sparsity constraints are imposed on the controller's left coprime factors, outlying a new concept of structural constraints in distributed control. The proposed scheme is amenable to optimal controller design via norm based costs, it guarantees string stability and eliminates the accordion effect from the behavior of the platoon. We also introduce a synchronization mechanism for the exact compensation of the time delays induced by the wireless broadcasting of information.

eess.SY↗

arXiv2015

Learning without Recall: A Case for Log-Linear Learning

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 beliefs of their neighboring agents at each time. Fully rational agents would successively apply Bayes rule to the entire history of observations. This leads to forebodingly complex inferences due to lack of knowledge about the global network structure that causes those observations. To address these complexities, we consider a Learning without Recall model, which in addition to providing a tractable framework for analyzing the behavior of rational agents in social networks, can also provide a behavioral foundation for the variety of non-Bayesian update rules in the literature. We present the implications of various choices for time-varying priors of such agents and how this choice affects learning and its rate.

cs.SI↗

arXiv2015

Learning without Recall by Random Walks on Directed Graphs

We consider a network of agents that aim to learn some unknown state of the world using private observations and exchange of beliefs. At each time, agents observe private signals generated based on the true unknown state. Each agent might not be able to distinguish the true state based only on her private observations. This occurs when some other states are observationally equivalent to the true state from the agent's perspective. To overcome this shortcoming, agents must communicate with each other to benefit from local observations. We propose a model where each agent selects one of her neighbors randomly at each time. Then, she refines her opinion using her private signal and the prior of that particular neighbor. The proposed rule can be thought of as a Bayesian agent who cannot recall the priors based on which other agents make inferences. This learning without recall approach preserves some aspects of the Bayesian inference while being computationally tractable. By establishing a correspondence with a random walk on the network graph, we prove that under the described protocol, agents learn the truth exponentially fast in the almost sure sense. The asymptotic rate is expressed as the sum of the relative entropies between the signal structures of every agent weighted by the stationary distribution of the random walk.

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arXiv2015

Distributed SDDM Solvers: Theory & Applications

In this paper, we propose distributed solvers for systems of linear equations given by symmetric diagonally dominant M-matrices based on the parallel solver of Spielman and Peng. We propose two versions of the solvers, where in the first, full communication in the network is required, while in the second communication is restricted to the R-Hop neighborhood between nodes for some $R \geq 1$. We rigorously analyze the convergence and convergence rates of our solvers, showing that our methods are capable of outperforming state-of-the-art techniques. Having developed such solvers, we then contribute by proposing an accurate distributed Newton method for network flow optimization. Exploiting the sparsity pattern of the dual Hessian, we propose a Newton method for network flow optimization that is both faster and more accurate than state-of-the-art techniques. Our method utilizes the distributed SDDM solvers for determining the Newton direction up to any arbitrary precision $ε>0$. We analyze the properties of our algorithm and show superlinear convergence within a neighborhood of the optimal. Finally, in a set of experiments conducted on randomly generated and barbell networks, we demonstrate that our approach is capable of significantly outperforming state-of-the-art techniques.

math.OC↗

arXiv2015

Fast, Accurate Second Order Methods for Network Optimization

Dual descent methods are commonly used to solve network flow optimization problems, since their implementation can be distributed over the network. These algorithms, however, often exhibit slow convergence rates. Approximate Newton methods which compute descent directions locally have been proposed as alternatives to accelerate the convergence rates of conventional dual descent. The effectiveness of these methods, is limited by the accuracy of such approximations. In this paper, we propose an efficient and accurate distributed second order method for network flow problems. The proposed approach utilizes the sparsity pattern of the dual Hessian to approximate the the Newton direction using a novel distributed solver for symmetric diagonally dominant linear equations. Our solver is based on a distributed implementation of a recent parallel solver of Spielman and Peng (2014). We analyze the properties of the proposed algorithm and show that, similar to conventional Newton methods, superlinear convergence within a neighbor- hood of the optimal value is attained. We finally demonstrate the effectiveness of the approach in a set of experiments on randomly generated networks.

math.OC↗

arXiv2015

Minimal Actuator Placement with Optimal Control Constraints

We introduce the problem of minimal actuator placement in a linear control system so that a bound on the minimum control effort for a given state transfer is satisfied while controllability is ensured. We first show that this is an NP-hard problem following the recent work of Olshevsky. Next, we prove that this problem has a supermodular structure. Afterwards, we provide an efficient algorithm that approximates up to a multiplicative factor of O(logn), where n is the size of the multi-agent network, any optimal actuator set that meets the specified energy criterion. Moreover, we show that this is the best approximation factor one can achieve in polynomial-time for the worst case. Finally, we test this algorithm over large Erdos-Renyi random networks to further demonstrate its efficiency.

eess.SY↗

arXiv2015

Switching to Learn

A network of agents attempt to learn some unknown state of the world drawn by nature from a finite set. Agents observe private signals conditioned on the true state, and form beliefs about the unknown state accordingly. Each agent may face an identification problem in the sense that she cannot distinguish the truth in isolation. However, by communicating with each other, agents are able to benefit from side observations to learn the truth collectively. Unlike many distributed algorithms which rely on all-time communication protocols, we propose an efficient method by switching between Bayesian and non-Bayesian regimes. In this model, agents exchange information only when their private signals are not informative enough; thence, by switching between the two regimes, agents efficiently learn the truth using only a few rounds of communications. The proposed algorithm preserves learnability while incurring a lower communication cost. We also verify our theoretical findings by simulation examples.

cs.LG↗

arXiv2015

Competitive Diffusion in Social Networks: Quality or Seeding?

In this paper, we study a strategic model of marketing and product consumption in social networks. We consider two firms in a market competing to maximize the consumption of their products. Firms have a limited budget which can be either invested on the quality of the product or spent on initial seeding in the network in order to better facilitate spread of the product. After the decision of firms, agents choose their consumptions following a myopic best response dynamics which results in a local, linear update for their consumption decision. We characterize the unique Nash equilibrium of the game between firms and study the effect of the budgets as well as the network structure on the optimal allocation. We show that at the equilibrium, firms invest more budget on quality when their budgets are close to each other. However, as the gap between budgets widens, competition in qualities becomes less effective and firms spend more of their budget on seeding. We also show that given equal budget of firms, if seeding budget is nonzero for a balanced graph, it will also be nonzero for any other graph, and if seeding budget is zero for a star graph it will be zero for any other graph as well. As a practical extension, we then consider a case where products have some preset qualities that can be only improved marginally. At some point in time, firms learn about the network structure and decide to utilize a limited budget to mount their market share by either improving the quality or new seeding some agents to incline consumers towards their products. We show that the optimal budget allocation in this case simplifies to a threshold strategy. Interestingly, we derive similar results to that of the original problem, in which preset qualities simulate the role that budgets had in the original setup.

cs.GT↗

arXiv2015

A Fast Distributed Solver for Symmetric Diagonally Dominant Linear Equations

In this paper, we propose a fast distributed solver for linear equations given by symmetric diagonally dominant M-Matrices. Our approach is based on a distributed implementation of the parallel solver of Spielman and Peng by considering a specific approximated inverse chain which can be computed efficiently in a distributed fashion. Representing the system of equations by a graph $\mathbb{G}$, the proposed distributed algorithm is capable of attaining $ε$-close solutions (for arbitrary $ε$) in time proportional to $n^{3}$ (number of nodes in $\mathbb{G}$), $α$ (upper bound on the size of the R-Hop neighborhood), and $\frac{{W}_{max}}{{W}_{min}}$ (maximum and minimum weight of edges in $\mathbb{G}$).

cs.DC↗

arXiv2015

Online Optimization : Competing with Dynamic Comparators

Recent literature on online learning has focused on developing adaptive algorithms that take advantage of a regularity of the sequence of observations, yet retain worst-case performance guarantees. A complementary direction is to develop prediction methods that perform well against complex benchmarks. In this paper, we address these two directions together. We present a fully adaptive method that competes with dynamic benchmarks in which regret guarantee scales with regularity of the sequence of cost functions and comparators. Notably, the regret bound adapts to the smaller complexity measure in the problem environment. Finally, we apply our results to drifting zero-sum, two-player games where both players achieve no regret guarantees against best sequences of actions in hindsight.

cs.LG↗

arXiv2015

Distributed Resource Allocation for Epidemic control

We present a distributed resource allocation strategy to control an epidemic outbreak in a networked population based on a Distributed Alternating Direction Method of Multipliers (D-ADMM) algorithm. We consider a linearized Susceptible- Infected-Susceptible (SIS) epidemic spreading model in which agents in the network are able to allocate vaccination resources (for prevention) and antidotes (for treatment) in the presence of a contagion. We express our epidemic control condition as a spectral constraint involving the Perron-Frobenius eigenvalue, and formulate the resource allocation problem as a Geometric Program (GP). Next, we separate the network-wide optimization problem into subproblems optimally solved by each agent in a fully distributed way. We conclude the paper by illustrating performance of our solution framework with numerical simulations.

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