This paper explores the fundamental properties of distributed minimization of a sum of functions with each function only known to one node, and a pre-specified level of node knowledge and computational capacity. We define the optimization information each node receives from its objective function, the neighboring information each node receives from its neighbors, and the computational capacity each node can take advantage of in controlling its state. It is proven that there exist a neighboring information way and a control law that guarantee global optimal consensus if and only if the solution sets of the local objective functions admit a nonempty intersection set for fixed strongly connected graphs. Then we show that for any tolerated error, we can find a control law that guarantees global optimal consensus within this error for fixed, bidirectional, and connected graphs under mild conditions. For time-varying graphs, we show that optimal consensus can always be achieved as long as the graph is uniformly jointly strongly connected and the nonempty intersection condition holds. The results illustrate that nonempty intersection for the local optimal solution sets is a critical condition for successful distributed optimization for a large class of algorithms.
Distributed optimization is a classical topic in applied mathematics with several excellent textbooks, e.g., @cite_15 @cite_1 @cite_34 .
@cite_15 @cite_34 @cite_1
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We consider a network of agents that are cooperatively solving a global unconstrained optimization problem, where the objective function is the sum of privately known local objective functions of the agents. Recent literature on distributed optimization methods for solving this problem focused on subgradient based methods, which typically converge at the rate equation, where k is the number of iterations. In this paper, we introduce a new distributed optimization algorithm based on Alternating Direction Method of Multipliers (ADMM), which is a classical method for sequentially decomposing optimization problems with coupled constraints. We show that this algorithm converges at the rate equation. As the size of modern data sets exceeds the disk and memory capacities of a single computer, machine learning practitioners have resorted to parallel and distributed computing. Given that optimization is one of the pillars of machine learning and predictive modeling, distributed optimization methods have recently garnered ample attention in the literature. Although previous research has mostly focused on settings where either the observations, or features of the problem at hand are stored in distributed fashion, the situation where both are partitioned across the nodes of a computer cluster (doubly distributed) has barely been studied. In this work we propose two doubly distributed optimization algorithms. The first one falls under the umbrella of distributed dual coordinate ascent methods, while the second one belongs to the class of stochastic gradient coordinate descent hybrid methods. We conduct numerical experiments in Spark using real-world and simulated data sets and study the scaling properties of our methods. Our empirical evaluation of the proposed algorithms demonstrates the out-performance of a block distributed ADMM method, which, to the best of our knowledge is the only other existing doubly distributed optimization algorithm. We describe distributed algorithms for two widely-used topic models, namely the Latent Dirichlet Allocation (LDA) model, and the Hierarchical Dirichet Process (HDP) model. In our distributed algorithms the data is partitioned across separate processors and inference is done in a parallel, distributed fashion. We propose two distributed algorithms for LDA. The first algorithm is a straightforward mapping of LDA to a distributed processor setting. In this algorithm processors concurrently perform Gibbs sampling over local data followed by a global update of topic counts. The algorithm is simple to implement and can be viewed as an approximation to Gibbs-sampled LDA. The second version is a model that uses a hierarchical Bayesian extension of LDA to directly account for distributed data. This model has a theoretical guarantee of convergence but is more complex to implement than the first algorithm. Our distributed algorithm for HDP takes the straightforward mapping approach, and merges newly-created topics either by matching or by topic-id. Using five real-world text corpora we show that distributed learning works well in practice. For both LDA and HDP, we show that the converged test-data log probability for distributed learning is indistinguishable from that obtained with single-processor learning. Our extensive experimental results include learning topic models for two multi-million document collections using a 1024-processor parallel computer. 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. Distributed optimization has gained popularity in many areas because it can to cope with the increasing volume of data, and the corresponding demand for computational resources. A popular distributed solver is the alternating direction method of multipliers (ADMM). Fully decentralized (D) ADMM arises when node-to-node communications have to abide by the underlying network connectivity. DADMM’s convergence however, slows down as the network diameter grows large. To deal with this challenge, the recently proposed hybrid (H) ADMM provides considerable performance boost over DADMM by exploiting local topology information. But HADMM only applies to unweighted graphs. The present contribution develops a weighted hybrid (WH) consensus-based ADMM approach that can deal with weighted graphs. The resultant scheme further improves the performance of HADMM through graph-aware weight tuning. Theoretical analysis offers convergence guarantees and establishes linear convergence rate, while numerical tests on various graphs demonstrate the WHADMM merits. We study the problem of efficiently optimizing submodular functions under cardinality constraints in distributed setting. Recently, several distributed algorithms for this problem have been introduced which either achieve a sub-optimal solution or they run in super-constant number of rounds of computation. Unlike previous work, we aim to design distributed algorithms in multiple rounds with almost optimal approximation guarantees at the cost of outputting a larger number of elements. Toward this goal, we present a distributed algorithm that, for any e > 0 and any constant r, outputs a set S of O(rk e1 r) items in r rounds, and achieves a (1-e)-approximation of the value of the optimum set with k items. This is the first distributed algorithm that achieves an approximation factor of (1-e) running in less than log 1 e number of rounds. We also prove a hardness result showing that the output of any 1-e approximation distributed algorithm limited to one distributed round should have at least Ω(k e) items. In light of this hardness result, our distributed algorithm in one round, r = 1, is asymptotically tight in terms of the output size. We support the theoretical guarantees with an extensive empirical study of our algorithm showing that achieving almost optimum solutions is indeed possible in a few rounds for large-scale real datasets. 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. We present a novel Newton-type method for distributed optimization, which is particularly well suited for stochastic optimization and learning problems. For quadratic objectives, the method enjoys a linear rate of convergence which provably improves with the data size, requiring an essentially constant number of iterations under reasonable assumptions. We provide theoretical and empirical evidence of the advantages of our method compared to other approaches, such as one-shot parameter averaging and ADMM. Distributed Computing Through Combinatorial Topology describes techniques for analyzing distributed algorithms based on award winning combinatorial topology research. The authors present a solid theoretical foundation relevant to many real systems reliant on parallelism with unpredictable delays, such as multicore microprocessors, wireless networks, distributed systems, and Internet protocols. Today, a new student or researcher must assemble a collection of scattered conference publications, which are typically terse and commonly use different notations and terminologies. This book provides a self-contained explanation of the mathematics to readers with computer science backgrounds, as well as explaining computer science concepts to readers with backgrounds in applied mathematics. The first section presents mathematical notions and models, including message passing and shared-memory systems, failures, and timing models. The next section presents core concepts in two chapters each: first, proving a simple result that lends itself to examples and pictures that will build up readers' intuition; then generalizing the concept to prove a more sophisticated result. The overall result weaves together and develops the basic concepts of the field, presenting them in a gradual and intuitively appealing way. The book's final section discusses advanced topics typically found in a graduate-level course for those who wish to explore further. Gathers knowledge otherwise spread across research and conference papers using consistent notations and a standard approach to facilitate understandingPresents unique insights applicable to multiple computing fields, including multicore microprocessors, wireless networks, distributed systems, and Internet protocols Synthesizes and distills material into a simple, unified presentation with examples, illustrations, and exercises We study distributed optimization algorithms for minimizing the average of convex functions. The applications include empirical risk minimization problems in statistical machine learning where the datasets are large and have to be stored on different machines. We design a distributed stochastic variance reduced gradient algorithm that, under certain conditions on the condition number, simultaneously achieves the optimal parallel runtime, amount of communication and rounds of communication among all distributed first-order methods up to constant factors. Our method and its accelerated extension also outperform existing distributed algorithms in terms of the rounds of communication as long as the condition number is not too large compared to the size of data in each machine. We also prove a lower bound for the number of rounds of communication for a broad class of distributed first-order methods including the proposed algorithms in this paper. We show that our accelerated distributed stochastic variance reduced gradient algorithm achieves this lower bound so that it uses the fewest rounds of communication among all distributed first-order algorithms. This third edition of a classic textbook can be used to teach at the senior undergraduate and graduate levels. The material concentrates on fundamental theories as well as techniques and algorithms. The advent of the Internet and the World Wide Web, and, more recently, the emergence of cloud computing and streaming data applications, has forced a renewal of interest in distributed and parallel data management, while, at the same time, requiring a rethinking of some of the traditional techniques. This book covers the breadth and depth of this re-emerging field. The coverage consists of two parts. The first part discusses the fundamental principles of distributed data management and includes distribution design, data integration, distributed query processing and optimization, distributed transaction management, and replication. The second part focuses on more advanced topics and includes discussion of parallel database systems, distributed object management, peer-to-peer data management, web data management, data stream systems, and cloud computing. New in this Edition: New chapters, covering database replication, database integration, multidatabase query processing, peer-to-peer data management, and web data management. Coverage of emerging topics such as data streams and cloud computing Extensive revisions and updates based on years of class testing and feedback Ancillary teaching materials are available. In this paper, we investigate a distributed learning scheme for a broad class of stochastic optimization problems and games that arise in signal processing and wireless communications. The proposed algorithm relies on the method of matrix exponential learning (MXL) and only requires locally computable gradient observations that are possibly imperfect. To analyze it, we introduce the notion of a stable Nash equilibrium and we show that the algorithm is globally convergent to such equilibria—or locally convergent when an equilibrium is only locally stable. To complement our convergence analysis, we also derive explicit bounds for the algorithm's convergence speed and we test it in realistic multicarrier multiple-antenna wireless scenarios where several users seek to maximize their energy efficiency. Our results show that learning allows users to attain a net increase between @math and @math in energy efficiency, even under very high uncertainty. We consider distributed convex optimization problems originating from sample average approximation of stochastic optimization, or empirical risk minimization in machine learning. We assume that each machine in the distributed computing system has access to a local empirical loss function, constructed with i.i.d. data sampled from a common distribution. We propose a communication-efficient distributed algorithm to minimize the overall empirical loss, which is the average of the local empirical losses. The algorithm is based on an inexact damped Newton method, where the inexact Newton steps are computed by a distributed preconditioned conjugate gradient method. We analyze its iteration complexity and communication efficiency for minimizing self-concordant empirical loss functions, and discuss the results for ridge regression, logistic regression and binary classification with a smoothed hinge loss. In a standard setting for supervised learning where the condition number of the problem grows with square root of the sample size, the required number of communication rounds of the algorithm does not increase with the sample size, and only grows slowly with the number of machines. This paper proposes parallel and distributed algorithms for solving very large-scale sparse optimization problems on computer clusters and clouds. Modern datasets usually have a large number of features or training samples, and they are usually stored in a distributed manner. Motivated by the need of solving sparse optimization problems with large datasets, we propose two approaches including (i) distributed implementations of prox-linear algorithms and (ii) GRock, a parallel greedy block coordinate descent method. Different separability properties of the objective terms in the problem enable different data distributed schemes along with their corresponding algorithm implementations. We also establish the convergence of GRock and explain why it often performs exceptionally well for sparse optimization. Numerical results on a computer cluster and Amazon EC2 demonstrate the efficiency and elasticity of our algorithms. Distributed optimization algorithms are highly attractive for solving big data problems. In particular, many machine learning problems can be formulated as the global consensus optimization problem, which can then be solved in a distributed manner by the alternating direction method of multipliers (ADMM) algorithm. However, this suffers from the straggler problem as its updates have to be synchronized. In this paper, we propose an asynchronous ADMM algorithm by using two conditions to control the asynchrony: partial barrier and bounded delay. The proposed algorithm has a simple structure and good convergence guarantees (its convergence rate can be reduced to that of its synchronous counterpart). Experiments on different distributed ADMM applications show that asynchrony reduces the time on network waiting, and achieves faster convergence than its synchronous counterpart in terms of the wall clock time. Historically, centrally computed algorithms have been the primary means of power system optimization and control. With increasing penetrations of distributed energy resources requiring optimization and control of power systems with many controllable devices, distributed algorithms have been the subject of significant research interest. This paper surveys the literature of distributed algorithms with applications to optimization and control of power systems. In particular, this paper reviews distributed algorithms for offline solution of optimal power flow (OPF) problems as well as online algorithms for real-time solution of OPF, optimal frequency control, optimal voltage control, and optimal wide-area control problems. Consider a set of @math agents seeking to solve distributively the minimization problem @math where the convex functions @math are local to the agents. The popular Alternating Direction Method of Multipliers has the potential to handle distributed optimization problems of this kind. We provide a general reformulation of the problem and obtain a class of distributed algorithms which encompass various network architectures. The rate of convergence of our method is considered. It is assumed that the infimum of the problem is reached at a point @math , the functions @math are twice differentiable at this point and @math in the positive definite ordering of symmetric matrices. With these assumptions, it is shown that the convergence to the consensus @math is linear and the exact rate is provided. Application examples where this rate can be optimized with respect to the ADMM free parameter @math are also given. 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. 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. The distributed @math -coloring problem is one of the most fundamental and well-studied problems in distributed algorithms. Starting with the work of Cole and Vishkin in 1986, a long line of gradually improving algorithms has been published. The state-of-the-art running time, prior to our work, is @math , due to Kuhn and Wattenhofer [Proceedings of the @math th Annual ACM Symposium on Principles of Distributed Computing, Denver, CO, 2006, pp. 7--15]. Linial [Proceedings of the @math th Annual IEEE Symposium on Foundation of Computer Science, Los Angeles, CA, 1987, pp. 331--335] proved a lower bound of @math for the problem, and Szegedy and Vishwanathan [Proceedings of the 25th Annual ACM Symposium on Theory of Computing, San Diego, CA, 1993, pp. 201--207] provided a heuristic argument that shows that algorithms from a wide family of locally iterative algorithms are unlikely to achieve a running time smaller than @math . We present a de...