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.