This paper considers a problem where multiple users make repeated decisions based on their own observed events. The events and decisions at each time step determine the values of a utility function and a collection of penalty functions. The goal is to make distributed decisions over time to maximize time average utility subject to time average constraints on the penalties. An example is a collection of power constrained sensor nodes that repeatedly report their own observations to a fusion center. Maximum time average utility is fundamentally reduced because users do not know the events observed by others. Optimality is characterized for this distributed context. It is shown that optimality is achieved by correlating user decisions through a commonly known pseudorandom sequence. An optimal algorithm is developed that chooses pure strategies at each time step based on a set of time-varying weights.
Much prior work on network optimization treats scenarios where it is possible to find distributed solutions with no loss of optimality. For example, network flow problems that are described by linear or separable convex programs can be optimally solved in a distributed manner @cite_10 @cite_22 @cite_8 @cite_20 . Problems where network nodes want to average sensor data @cite_3 or compute convex programs @cite_1 have distributed solutions. Work in @cite_18 solves for an optimal vector of parameters associated with an infinite horizon Markov decision problem using distributed agents. Work in @cite_21 @cite_11 @cite_6 develops distributed multiple access methods that converge to optimality. However, the above problems do not have random events that create a fundamental gap between centralized and distributed performance.
@cite_18 @cite_22 @cite_8 @cite_21 @cite_1 @cite_3 @cite_6 @cite_10 @cite_20 @cite_11
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The focus of this paper is the development and analysis of distributed algorithms for solving convex optimization problems that are defined over networks. Such network-structured optimization problems arise in a variety of application domains within the information sciences and engineering. For instance, problems such as multi-agent coordination, distributed tracking and localization, estimation problems in sensor networks and packet routing are all naturally cast as distributed convex minimization [2], [11], [10], [16], [20]. Common to these problems is the necessity for completely decentralized computation that is locally light — so as to avoid overburdening small sensors or flooding busy networks — and robust to periodic link or node failures. As a second example, data sets that are too large to be processed quickly by any single processor present related challenges. A canonical example that arises in statistical machine learning is the problem of minimizing a loss function averaged over a large dataset (e.g., optimization in support vector machines [5]). With terabytes of data, it is desirable to assign smaller subsets of the data to different processors, and the processors must communicate to find parameters that minimize the loss over the entire dataset. However, the communication should be efficient enough that network latencies do not offset computational gains. We study combinatorial optimization problems in which a set of distributed agents must achieve a global objective using only local information. Papadimitriou and Yannakakis [Proceedings of the 25th ACM Symposium on Theory of Computing, 1993, pp. 121--129] initiated the study of such problems in a framework where distributed decision-makers must generate feasible solutions to positive linear programs with information only about local constraints. We extend their model by allowing these distributed decision-makers to perform local communication to acquire information over time and then explore the tradeoff between the amount of communication and the quality of the solution to the linear program that the decision-makers can obtain. Our main result is a distributed algorithm that obtains a @math approximation to the optimal linear programming solution while using only a polylogarithmic number of rounds of local communication. This algorithm offers a significant improvement over the logarithmic approximation ratio previously obtained by Awerbuch and Azar [Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science, 1994, pp. 240--249] for this problem while providing a comparable running time. Our results apply directly to the application of network flow control, an application in which distributed routers must quickly choose how to allocate bandwidth to connections using only local information to achieve global objectives. The sequential version of our algorithm is faster and considerably simpler than the best known approximation algorithms capable of achieving a @math approximation ratio for positive linear programming. This thesis addresses problems of optimization to be solved collectively (distributely, we say) by groups of networked agents having limited communication and computation capabilities. The focus is on two paradigmatic problems, average consensus and coverage control. The former consists in computing the average of values which are priory known to the agents, while the latter consists in optimally deploying robotic agents in a given environment. Their solution is possible thanks to iterative algorithms which exploit the available communications among agents, described by a graph. Research has been very intense in latest years on these problems, due to strong applicative motivations: these algorithms are crucial for the design of networks of sensors, and for the coordinated motion of groups of unmanned vehicles. Namely, the ultimate goal of these algorithms is to make such artificial networks able to self-organize and perform specific tasks without a centralized supervision. In the thesis networks have been considered, which allow only communication in pairs, or limited to messages belonging to a discrete alphabet. Several novel algorithms for the solution of the above optimization problems are proposed: they are proved to lead the agents to the desired solution, and their performance is studied. The mathematical tools required for the analysis are novel, or need to be used in a non-standard way. They are drawn from very different branches of pure and applied mathematics: spectral graph theory, Markov chains, combinatorics, linear algebra, set-valued analysis, convex and discrete geometry, topology of hyper-spaces of sets. Flow control in high speed networks requires distributed routers to make fast decisions based only on local information in allocating bandwidth to connections. While most previous work on this problem focuses on achieving local objective functions, in many cases it may be necessary to achieve global objectives such as maximizing the total flow. This problem illustrates one of the basic aspects of distributed computing: achieving global objectives using local information. Papadimitriou and Yannakakis (1993) initiated the study of such problems in a framework of solving positive linear programs by distributed agents. We take their model further, by allowing the distributed agents to acquire more information over time. We therefore turn attention to the tradeoff between the running time and the quality of the solution to the linear program. We give a distributed algorithm that obtains a (1+ spl epsiv ) approximation to the global optimum solution and runs in a polylogarithmic number of distributed rounds. While comparable in running time, our results exhibit a significant improvement on the logarithmic ratio previously obtained by Awerbuch and Azar (1994). Our algorithm, which draws from techniques developed by Luby and Nisan (1993) is considerably simpler than previous approximation algorithms for positive linear programs, and thus may have practical value in both centralized and distributed settings. We study nonconvex distributed optimization in multiagent networks with time-varying (nonsymmetric) connectivity. We introduce the first algorithmic framework for the distributed minimization of the sum of a smooth (possibly nonconvex and nonseparable) function—the agents’ sum-utility—plus a convex (possibly nonsmooth and nonseparable) regularizer. The latter is usually employed to enforce some structure in the solution, typically sparsity. The proposed method hinges on successive convex approximation techniques while leveraging dynamic consensus as a mechanism to distribute the computation among the agents: each agent first solves (possibly inexactly) a local convex approximation of the nonconvex original problem, and then performs local averaging operations. Asymptotic convergence to (stationary) solutions of the nonconvex problem is established. Our algorithmic framework is then customized to a variety of convex and nonconvex problems in several fields, including signal processing, communications, networking, and machine learning. Numerical results show that the new method compares favorably to existing distributed algorithms on both convex and nonconvex problems. In Stolyar (Queueing Systems 50 (2005) 401---457) a dynamic control strategy, called greedy primal-dual (GPD) algorithm, was introduced for the problem of maximizing queueing network utility subject to stability of the queues, and was proved to be (asymptotically) optimal. (The network utility is a concave function of the average rates at which the network generates several "commodities.") Underlying the control problem of Stolyar (Queueing Systems 50 (2005) 401---457) is a convex optimization problem subject to a set of linear constraints. In this paper we introduce a generalized GPD algorithm, which applies to the network control problem with additional convex (possibly non-linear) constraints on the average commodity rates. The underlying optimization problem in this case is a convex problem subject to convex constraints. We prove asymptotic optimality of the generalized GPD algorithm. We illustrate key features and applications of the algorithm on simple examples. In this paper, we investigate the problem of maximizing the throughput over a finite-horizon time period for a sensor network with energy replenishment. The finite-horizon problem is important and challenging because it necessitates optimizing metrics over the short term rather than metrics that are averaged over a long period of time. Unlike the infinite-horizon problem, the fact that inefficiencies cannot be made to vanish to infinitesimally small values, means that the finite-horizon problem requires more delicate control. The finite-horizon throughput optimization problem can be formulated as a convex optimization problem, but turns out to be highly complex. The complexity is brought about by the “time coupling property,” which implies that current decisions can influence future performance. To address this problem, we employ a three-step approach. First, we focus on the throughput maximization problem for a single node with renewable energy assuming that the replenishment rate profile for the entire finite-horizon period is known in advance. An energy allocation scheme that is equivalent to computing a shortest path in a simply-connected space is developed and proven to be optimal. We then relax the assumption that the future replenishment profile is known and develop an online algorithm. The online algorithm guarantees a fraction of the optimal throughput. Motivated by these results, we propose a low-complexity heuristic distributed scheme, called NetOnline, in a rechargeable sensor network. We prove that this heuristic scheme is optimal under homogeneous replenishment profiles. Further, in more general settings, we show via simulations that NetOnline significantly outperforms a state-of-the-art infinite-horizon based scheme, and for certain configurations using data collected from a testbed sensor network, it achieves empirical performance close to optimal. We propose a new distributed algorithm based on alternating direction method of multipliers (ADMM) to minimize sum of locally known convex functions using communication over a network. This optimization problem emerges in many applications in distributed machine learning and statistical estimation. Our algorithm allows for a general choice of the communication weight matrix, which is used to combine the iterates at different nodes. We show that when functions are convex, both the objective function values and the feasibility violation converge with rate @math , where @math is the number of iterations. We then show that when functions are strongly convex and have Lipschitz continuous gradients, the sequence generated by our algorithm converges linearly to the optimal solution. In particular, an @math -optimal solution can be computed with @math iterations, where @math is the condition number of the problem. Our analysis highlights the effect of network and communication weights on the convergence rate through degrees of the nodes, the smallest nonzero eigenvalue, and operator norm of the communication matrix. This work deals with trajectory optimization for a robotic sensor network sampling a spatio-temporal random field. We examine the optimal sampling problem of minimizing the maximum predictive variance of the estimator over the space of network trajectories. This is a high-dimensional, multi-modal, nonsmooth optimization problem, known to be NP-hard even for static fields and discrete design spaces. Under an asymptotic regime of near-independence between distinct sample locations, we show that the solutions to a novel generalized disk-covering problem are solutions to the optimal sampling problem. This result effectively transforms the search for the optimal trajectories into a geometric optimization problem. Constrained versions of the latter are also of interest as they can accommodate trajectories that satisfy a maximum velocity restriction on the robots. We characterize the solution for the unconstrained and constrained versions of the geometric optimization problem as generalized multicircumcenter trajectories, and provide algorithms which enable the network to find them in a distributed fashion. Several simulations illustrate our results. In a heterogeneous wireless cellular network, each user may be covered by multiple access points such as macro pico relay femto base stations (BS). An effective approach to maximize the sum utility (e.g., system throughput) in such a network is to jointly optimize users' linear procoders as well as their BS associations. In this paper, we first show that this joint optimization problem is NP-hard and thus is difficult to solve to global optimality. To find a locally optimal solution, we formulate the problem as a noncooperative game in which the users and the BSs both act as players. We introduce a set of new utility functions for the players and show that every Nash equilibrium (NE) of the resulting game is a stationary solution of the original sum utility maximization problem. Moreover, we develop a best-response type algorithm that allows the players to distributedly reach a NE of the game. Simulation results show that the proposed distributed algorithm can effectively relieve local BS congestion and simultaneously achieve high throughput and load balancing in a heterogeneous network.