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.
Consensus algorithms have been proven to be useful in the design of distributed optimization methods @cite_18 @cite_28 @cite_6 @cite_31 @cite_23 @cite_2 . Consensus methods have also been extensively studied for both discrete-time and continuous-time models in the past decade, some references related to the current paper include @cite_26 @cite_40 @cite_35 @cite_19 @cite_8 @cite_24 @cite_13 @cite_7 @cite_22 @cite_11 @cite_36 @cite_12 .
@cite_35 @cite_18 @cite_26 @cite_22 @cite_7 @cite_8 @cite_28 @cite_36 @cite_6 @cite_24 @cite_19 @cite_40 @cite_23 @cite_2 @cite_31 @cite_13 @cite_12 @cite_11
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This paper introduces several new algorithms for consensus over the special orthogonal group. By relying on a convex relaxation of the space of rotation matrices, consensus over rotation elements is reduced to solving a convex problem with a unique global solution. The consensus protocol is then implemented as a distributed optimization using (i) dual decomposition, and (ii) both semi and fully distributed variants of the alternating direction method of multipliers technique -- all with strong convergence guarantees. The convex relaxation is shown to be exact at all iterations of the dual decomposition based method, and exact once consensus is reached in the case of the alternating direction method of multipliers. Further, analytic and or efficient solutions are provided for each iteration of these distributed computation schemes, allowing consensus to be reached without any online optimization. Examples in satellite attitude alignment with up to 100 agents, an estimation problem from computer vision, and a rotation averaging problem on @math validate the approach. We investigate the approximate consensus problem in highly dynamic networks in which topology may change continually and unpredictably. We prove that in both synchronous and partially synchronous networks, approximate consensus is solvable if and only if the communication graph in each round has a rooted spanning tree. Interestingly, the class of averaging algorithms, which have the benefit of being memoryless and requiring no process identifiers, entirely captures the solvability issue of approximate consensus in that the problem is solvable if and only if it can be solved using any averaging algorithm. We develop a proof strategy which for each positive result consists in a reduction to the nonsplit networks. It dramatically improves the best known upper bound on the decision times of averaging algorithms and yields a quadratic time non-averaging algorithm for approximate consensus in non-anonymous networks. We also prove that a general upper bound on the decision times of averaging algorithms have to be exponential, shedding light on the price of anonymity. Finally we apply our results to networked systems with a fixed topology and benign fault models to show that with n processes, up to @math of link faults per round can be tolerated for approximate consensus, increasing by a factor 2 the bound of Santoro and Widmayer for exact consensus. This paper presents analysis and design results for distributed consensus algorithms in multi-agent networks. We consider arbitrary consensus functions of the initial state of the network agents. Under mild smoothness assumptions, we obtain necessary and sufficient conditions characterizing any algorithm that asymptotically achieves consensus. This characterization is the building block to obtain various design results. We first identify a class of smooth functions for which one can synthesize in a systematic way distributed algorithms that achieve consensus. We apply this result to the family of weighted power mean functions, and characterize the exponential convergence properties of the resulting algorithms. We conclude with two distributed algorithms that achieve, respectively, max and min consensus in finite time. Consensus is a fundamental feature of distributed systems, and it is the prerequisite for several complex tasks, such as flocking of mobile robots, localization in wireless-sensor networks, or decentralized control of smart grids. Average consensus, in particular, is quite challenging, because it is typically obtained asymptotically, while few finite-time algorithms are available. In this paper, we provide a methodology to achieve distributed average consensus in finite time, while maintaining low computational and memory requirements, and small completion times. The provided solution, namely, finite-time average-consensus by iterated max-consensus (FAIM) is based on several runs of the max-consensus algorithm, and has low memory requirements for each node. Compared to existing Flooding approaches, the proposed algorithm requires less memory, at the cost of a slight increase in the number of steps required for termination. The FAIM algorithm assumes that the nodes are aware of an upper bound on the network diameter. To relax this assumption, we complement this paper with a novel distributed algorithm that, in the case of undirected graphs, provides an upper bound on the network diameter which, in the worst case, is twice the actual diameter. A comparison of the proposed finite-time algorithm against the state of the art concludes this paper. When solving consensus optimization problems over a graph, there is often an explicit characterization of the convergence rate of Gradient Descent (GD) using the spectrum of the graph Laplacian. The same type of problems under the Alternating Direction Method of Multipliers (ADMM) are, however, poorly understood. For instance, simple but important non-strongly-convex consensus problems have not yet being analyzed, especially concerning the dependency of the convergence rate on the graph topology. Recently, for a non-strongly-convex consensus problem, a connection between distributed ADMM and lifted Markov chains was proposed, followed by a conjecture that ADMM is faster than GD by a square root factor in its convergence time, in close analogy to the mixing speedup achieved by lifting several Markov chains. Nevertheless, a proof of such a claim is is still lacking. Here we provide a full characterization of the convergence of distributed over-relaxed ADMM for the same type of consensus problem in terms of the topology of the underlying graph. Our results provide explicit formulas for optimal parameter selection in terms of the second largest eigenvalue of the transition matrix of the graph's random walk. Another consequence of our results is a proof of the aforementioned conjecture, which interestingly, we show it is valid for any graph, even the ones whose random walks cannot be accelerated via Markov chain lifting. In the past few years, the problem of distributed consensus has received a lot of attention, particularly in the framework of ad hoc sensor networks. Most methods proposed in the literature address the consensus averaging problem by distributed linear iterative algorithms, with asymptotic convergence of the consensus solution. The convergence rate of such distributed algorithms typically depends on the network topology and the weights given to the edges between neighboring sensors, as described by the network matrix. In this paper, we propose to accelerate the convergence rate for given network matrices by the use of polynomial filtering algorithms. The main idea of the proposed methodology is to apply a polynomial filter on the network matrix that will shape its spectrum in order to increase the convergence rate. Such an algorithm is equivalent to periodic updates in each of the sensors by aggregating a few of its previous estimates. We formulate the computation of the coefficients of the optimal polynomial as a semidefinite program that can be efficiently and globally solved for both static and dynamic network topologies. We finally provide simulation results that demonstrate the effectiveness of the proposed solutions in accelerating the convergence of distributed consensus averaging problems. Recently there has been a significant amount of research on developing consensus based algorithms for distributed optimization motivated by applications that vary from large scale machine learning to wireless sensor networks. This work describes and proves convergence of a new algorithm called Push-Sum Distributed Dual Averaging which combines a recent optimization algorithm [1] with a push-sum consensus protocol [2]. As we discuss, the use of push-sum has significant advantages. Restricting to doubly stochastic consensus protocols is not required and convergence to the true average consensus is guaranteed without knowing the stationary distribution of the update matrix in advance. Furthermore, the communication semantics of just summing the incoming information make this algorithm truly asynchronous and allow a clean analysis when varying intercommunication intervals and communication delays are modelled. We include experiments in simulation and on a small cluster to complement the theoretical analysis. We consider the following distributed consensus problem: Each node in a complete communication network of size n initially holds an opinion, which is chosen arbitrarily from a finite set Σ. The system must converge toward a consensus state in which all, or almost all nodes, hold the same opinion. Moreover, this opinion should be valid, i.e., it should be one among those initially present in the system. This condition should be met even in the presence of a malicious adversary who can modify the opinions of a bounded subset of nodes, adaptively chosen in every round. We consider the 3-majority dynamics: At every round, every node pulls the opinion from three random neighbors and sets his new opinion to the majority one (ties are broken arbitrarily). Let k be the number of valid opinions. We show that, if k ≤ nα, where α is a suitable positive constant, the 3-majority dynamics converges in time polynomial in k and log n with high probability even in the presence of an adversary who can affect up to o([EQUATION]) nodes at each round. Previously, the convergence of the 3-majority protocol was known for |Σ| = 2 only, with an argument that is robust to adversarial errors. On the other hand, no anonymous, uniform-gossip protocol that is robust to adversarial errors was known for |Σ| > 2. 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. We study several variants of coordinated consensus in dynamic networks. We assume a synchronous model, where the communication graph for each round is chosen by a worst-case adversary. The network topology is always connected, but can change completely from one round to the next. The model captures mobile and wireless networks, where communication can be unpredictable. In this setting we study the fundamental problems of eventual, simultaneous, and Δ-coordinated consensus, as well as their relationship to other distributed problems, such as determining the size of the network. We show that in the absence of a good initial upper bound on the size of the network, eventual consensus is as hard as computing deterministic functions of the input, e.g., the minimum or maximum of inputs to the nodes. We also give an algorithm for computing such functions that is optimal in every execution. Next, we show that simultaneous consensus can never be achieved in less than n - 1 rounds in any execution, where n is the size of the network; consequently, simultaneous consensus is as hard as computing an upper bound on the number of nodes in the network. For Δ-coordinated consensus, we show that if the ratio between nodes with input 0 and input 1 is bounded away from 1, it is possible to decide in time n-Θ(√ nΔ), where Δ bounds the time from the first decision until all nodes decide. If the dynamic graph has diameter D, the time to decide is min O(nD Δ),n-Ω(nΔ D) , even if D is not known in advance. Finally, we show that (a) there is a dynamic graph such that for every input, no node can decide before time n-O(Δ0.28n0.72); and (b) for any diameter D = O(Δ), there is an execution with diameter D where no node can decide before time Ω(nD Δ). To our knowledge, our work constitutes the first study of Δ-coordinated consensus in general graphs. We present distributed algorithms that can be used by multiple agents to align their estimates with a particular value over a network with time-varying connectivity. Our framework is general in that this value can represent a consensus value among multiple agents or an optimal solution of an optimization problem, where the global objective function is a combination of local agent objective functions. Our main focus is on constrained problems where the estimates of each agent are restricted to lie in different convex sets. To highlight the effects of constraints, we first consider a constrained consensus problem and present a distributed "projected consensus algorithm" in which agents combine their local averaging operation with projection on their individual constraint sets. This algorithm can be viewed as a version of an alternating projection method with weights that are varying over time and across agents. We establish convergence and convergence rate results for the projected consensus algorithm. We next study a constrained optimization problem for optimizing the sum of local objective functions of the agents subject to the intersection of their local constraint sets. We present a distributed "projected subgradient algorithm" which involves each agent performing a local averaging operation, taking a subgradient step to minimize its own objective function, and projecting on its constraint set. We show that, with an appropriately selected stepsize rule, the agent estimates generated by this algorithm converge to the same optimal solution for the cases when the weights are constant and equal, and when the weights are time-varying but all agents have the same constraint set. Methods for distributed optimization are necessary to solve large-scale problems such as those becoming more common in machine learning. The communication cost associated with transmitting large messages can become a serious performance bottleneck. We propose a consensus-based distributed algorithm to minimize a convex separable objective. Each node holds one component of the objective function, and the nodes alternate between a computation phase, where local gradient steps are performed based on the local objective, and a communication phase, where consensus steps are performed to bring the local states into agreement. The nodes use local decision rules to adaptively determine when communication is not necessary. This results in significantly lower communication costs and allows a user to tradeoff the amount of communication with the accuracy of the final output. Experiments on a cluster using simulated and real datasets illustrate the tradeoff. We investigate the vulnerabilities of consensus-based distributed optimization protocols to nodes that deviate from the prescribed update rule (e.g., due to failures or adversarial attacks). After characterizing certain fundamental limitations on the performance of any distributed optimization algorithm in the presence of adversaries, we propose a robust consensus-based distributed optimization algorithm that is guaranteed to converge to the convex hull of the set of minimizers of the non-adversarial nodes' functions. We also study the distance-to-optimality properties of our proposed robust algorithm in terms of F-local sets of the graph. We show that finding the largest size of such sets is NP-hard. This paper provides a theoretical framework for analysis of consensus algorithms for multi-agent networked systems with an emphasis on the role of directed information flow, robustness to changes in network topology due to link node failures, time-delays, and performance guarantees. An overview of basic concepts of information consensus in networks and methods of convergence and performance analysis for the algorithms are provided. Our analysis framework is based on tools from matrix theory, algebraic graph theory, and control theory. We discuss the connections between consensus problems in networked dynamic systems and diverse applications including synchronization of coupled oscillators, flocking, formation control, fast consensus in small-world networks, Markov processes and gossip-based algorithms, load balancing in networks, rendezvous in space, distributed sensor fusion in sensor networks, and belief propagation. We establish direct connections between spectral and structural properties of complex networks and the speed of information diffusion of consensus algorithms. A brief introduction is provided on networked systems with nonlocal information flow that are considerably faster than distributed systems with lattice-type nearest neighbor interactions. Simulation results are presented that demonstrate the role of small-world effects on the speed of consensus algorithms and cooperative control of multivehicle formations The paper studies average consensus with random topologies (intermittent links) and noisy channels. Consensus with noise in the network links leads to the bias-variance dilemma-running consensus for long reduces the bias of the final average estimate but increases its variance. We present two different compromises to this tradeoff: the A-ND algorithm modifies conventional consensus by forcing the weights to satisfy a persistence condition (slowly decaying to zero;) and the A-NC algorithm where the weights are constant but consensus is run for a fixed number of iterations [^(iota)], then it is restarted and rerun for a total of [^(p)] runs, and at the end averages the final states of the [^(p)] runs (Monte Carlo averaging). We use controlled Markov processes and stochastic approximation arguments to prove almost sure convergence of A-ND to a finite consensus limit and compute explicitly the mean square error (mse) (variance) of the consensus limit. We show that A-ND represents the best of both worlds-zero bias and low variance-at the cost of a slow convergence rate; rescaling the weights balances the variance versus the rate of bias reduction (convergence rate). In contrast, A-NC, because of its constant weights, converges fast but presents a different bias-variance tradeoff. For the same number of iterations [^(iota)][^(p)] , shorter runs (smaller [^(iota)] ) lead to high bias but smaller variance (larger number [^(p)] of runs to average over.) For a static nonrandom network with Gaussian noise, we compute the optimal gain for A-NC to reach in the shortest number of iterations [^(iota)][^(p)] , with high probability (1-delta), (epsiv, delta)-consensus (epsiv residual bias). Our results hold under fairly general assumptions on the random link failures and communication noise. We study the scalability of consensus-based distributed optimization algorithms by considering two questions: How many processors should we use for a given problem, and how often should they communicate when communication is not free? Central to our analysis is a problem-specific value r which quantifies the communication computation tradeoff. We show that organizing the communication among nodes as a k-regular expander graph [1] yields speedups, while when all pairs of nodes communicate (as in a complete graph), there is an optimal number of processors that depends on r. Surprisingly, a speedup can be obtained, in terms of the time to reach a fixed level of accuracy, by communicating less and less frequently as the computation progresses. Experiments on a real cluster solving metric learning and non-smooth convex minimization tasks demonstrate strong agreement between theory and practice. In decentralized consensus optimization, a connected network of agents collaboratively minimize the sum of their local objective functions over a common decision variable, where their information exchange is restricted between the neighbors. To this end, one can first obtain a problem reformulation and then apply the alternating direction method of multipliers (ADMM). The method applies iterative computation at the individual agents and information exchange between the neighbors. This approach has been observed to converge quickly and deemed powerful. This paper establishes its linear convergence rate for the decentralized consensus optimization problem with strongly convex local objective functions. The theoretical convergence rate is explicitly given in terms of the network topology, the properties of local objective functions, and the algorithm parameter. This result is not only a performance guarantee but also a guideline toward accelerating the ADMM convergence. We consider distributed plurality consensus in a complete graph of size @math with @math initial opinions. We design an efficient and simple protocol in the asynchronous communication model that ensures that all nodes eventually agree on the initially most frequent opinion. In this model, each node is equipped with a random Poisson clock with parameter @math . Whenever a node's clock ticks, it samples some neighbors, uniformly at random and with replacement, and adjusts its opinion according to the sample. A prominent example is the so-called two-choices algorithm in the synchronous model, where in each round, every node chooses two neighbors uniformly at random, and if the two sampled opinions coincide, then that opinion is adopted. This protocol is very efficient and well-studied when @math . If @math for some small @math , we show that it converges to the initial plurality opinion within @math rounds, w.h.p., as long as the initial difference between the largest and second largest opinion is @math . On the other side, we show that there are cases in which @math rounds are needed, w.h.p. One can beat this lower bound in the synchronous model by combining the two-choices protocol with randomized broadcasting. Our main contribution is a non-trivial adaptation of this approach to the asynchronous model. If the support of the most frequent opinion is at least @math times that of the second-most frequent one and @math , then our protocol achieves the best possible run time of @math , w.h.p. We relax full synchronicity by allowing @math nodes to be poorly synchronized, and the well synchronized nodes are only required to be within a certain time difference from one another. We enforce this synchronicity by introducing a novel gadget into the protocol.