Unbalanced Digraphs Research Articles

Distributed Online Optimization in Time-Varying Unbalanced Networks Without Explicit Subgradients

This paper studies a distributed online constrained optimization problem over time-varying unbalanced digraphs without explicit subgradients. In sharp contrast to the existing algorithms, we design a novel consensus-based distributed online algorithm with a local randomized zeroth-order oracle and then rescale the oracle by constructing row-stochastic matrices, which aims to address the unbalancedness of time-varying digraphs. Under mild conditions, the average dynamic regret over a time horizon is shown to asymptotically converge at a sublinear rate provided that the accumulated variation grows sublinearly with a specific order. Moreover, the counterpart of the proposed algorithm when subgradients are available is also provided, along with its dynamic regret bound, which reflects that the convergence of our algorithm is essentially not affected by the zeroth-order oracle. Simulations on distributed targets tracking problem and dynamic sparse signal recovery problem in sensor networks are employed to demonstrate the effectiveness of the proposed algorithm.

A Privacy-Masking Learning Algorithm for Online Distributed Optimization over Time-Varying Unbalanced Digraphs

This paper investigates a constrained distributed optimization problem enabled by differential privacy where the underlying network is time-changing with unbalanced digraphs. To solve such a problem, we first propose a differentially private online distributed algorithm by injecting adaptively adjustable Laplace noises. The proposed algorithm can not only protect the privacy of participants without compromising a trusted third party, but also be implemented on more general time-varying unbalanced digraphs. Under mild conditions, we then show that the proposed algorithm can achieve a sublinear expected bound of regret for general local convex objective functions. The result shows that there is a trade-off between the optimization accuracy and privacy level. Finally, numerical simulations are conducted to validate the efficiency of the proposed algorithm.

Online distributed dual averaging algorithm for multi-agent bandit optimization over time-varying general directed networks

The paper deals with online distributed convex optimization over a multi-agent network, where a swarm of agents attempts to minimize coordinately a sequence of time-changing global loss functions. At per time slot, the global loss function is decomposed into a sum of several local loss functions, each of which is available sequentially and held privately by one agent over the network, and individual agent does not possess prior knowledge of the future global loss function. We are interested in a bandit setting, where only the values of local loss functions at queried points are disclosed to individual agent. Meanwhile, we consider a general multi-agent network where the agents’ communication is represented as a sequence of time-changing unbalanced digraphs and the corresponding weight matrices are only row stochastic. We investigate two bandit scenarios including one- and two-point bandit feedback, and then design two corresponding online distributed bandit algorithms for such a class of problems by exploiting the classical dual averaging method. We show that both algorithms can achieve the sub-linear expected dynamic regret provided that the accumulative variation of the benchmark sequence grows sub-linearly with time. In particular, the bounds of the expected static regret obtained in this paper can reduce the relevant results when restricted to the centralized bandit online convex optimization. In contrast to existing algorithms with gradient feedback, numerical tests verify the competitive performances of the proposed algorithms.

Distributed Online Bandit Learning in Dynamic Environments Over Unbalanced Digraphs

This work is concerned with distributed online <i>bandit learning</i> over a multi-agent network, where a group of agents aim to seek the minimizer of a time-changing global loss function cooperatively. At each epoch, the global loss function can be represented as the sum of local loss functions known privately by individual agent over the network. Furthermore, local functions are sequentially accessible to agents, and all the agents have no knowledge of future loss functions. Thus, agents of the network must interchange messages to pursue an online estimation of the global loss function. In this paper, we are interested in a bandit setup, where only values of local loss functions at sampling points are disclosed to agents. Meanwhile, we consider a more general network with unbalanced digraphs that the corresponding weight matrix is allowed to be only row stochastic. By extending the celebrated mirror descent algorithm, we first design a distributed bandit online leaning method for the online distributed convex problem. We then establish the sub-linear expected dynamic regret attained by the algorithm for convex and strongly convex loss functions, respectively, when the accumulative deviation of the minimizer sequence increases sub-linearly. Moreover, the expected dynamic regret bound is analysed for strongly convex loss functions. In addition, the expected static regret bound with the order of <inline-formula><tex-math notation="LaTeX">$O(\sqrt{T})$</tex-math></inline-formula> is obtained in the bandit setting while the corresponding static regret bound with the order of <inline-formula><tex-math notation="LaTeX">$O(\ln {T})$</tex-math></inline-formula> is also provided for the strongly convex case. Finally, numerical examples are provided to illustrate the efficiency of the method and to verify the theoretical findings.

Distributed Adaptive Optimization With Weight-Balancing

This article addresses the continuous-time distributed optimization of a strictly convex summation-separable cost function with possibly nonconvex local functions over strongly connected digraphs. Distributed optimization methods in the literature require convexity of local functions, or balanced weights, or vanishing step sizes, or algebraic information (eigenvalues or eigenvectors) of the Laplacian matrix. The solution proposed here covers both weight-balanced and unbalanced digraphs in a unified way, without any of the aforementioned requirements.

Distributed heterogeneous energy storage systems synchronization control with dynamic weights and time-varying unbalanced digraphs

This paper provides a distributed control strategy for battery energy storage systems (BESS) based on multi-agent system. The proposed control laws can guarantee that the state of charge (SoC) is synchronized, and BESS active power can be shared proportionally. Moreover, the heterogeneous characteristics of BESS are considered in modeling BESS and designing control laws, while they are not carefully considered in the existed literature. Besides, the proposed control laws are robust to the changing communication topologies and link broken since they can be applied over the time-varying directed unbalanced communication networks. Simulation studies, which are carried out on matlab/simulink, validate the effectiveness of the control strategy and laws.

Privacy-Preserving Distributed Online Optimization Over Unbalanced Digraphs via Subgradient Rescaling

In this article, we investigate a distributed online constrained optimization problem with differential privacy where the network is modeled by an unbalanced digraph with a row-stochastic adjacency matrix. To address such a problem, a distributed differentially private algorithm without introducing a trusted third-party is proposed to preserve the privacy of the participating nodes. Under mild conditions, we show that the proposed algorithm attains an O(log T) expected regret bound for strongly convex local cost functions, where T is the time horizon. Moreover, we remove the need for knowing the time horizon T in advance by adopting doubling trick scheme, and derive an O(√T) expected regret bound for general convex local cost functions. Our results coincide with the best theoretical regrets that can be achieved in the state-of-the-art algorithms. Finally, simulation results are conducted to validate the efficiency of our proposed algorithm.

Distributed Optimization Under Unbalanced Digraphs With Node Errors: Robustness of Surplus-Based Dual Averaging Algorithm

In this article, the robustness of distributed constrained optimization algorithms for weight-unbalanced directed multiagent networks is studied. Specifically, it is assumed that each agent is subject to additive random node errors , which are caused by the imperfect communication in networks. Under this framework, it is shown that a typical algorithm, called surplus-based dual averaging (SDA), can successfully achieve the convergence to the optimal value of the considered optimization problem, which exhibits the robustness to the node errors. Technically, in the proof of this result, an important augmentation of the node errors in the fusion and surplus variables is first introduced, and then by means of the random process theory and the attenuation effect of diminishing step size on node errors, the robustness of SDA is proved. In addition, inspired by the SDA, a robust algorithm is developed to solve the distributed online optimization problem with node errors. Finally, simulations are given to verify the theoretical results.

Cluster Consensus of Multiagent Systems With Weighted Antagonistic Interactions.

This article addresses the problem of cluster consensus for multiagent systems (MASs) associated with weighted antagonistic interactions. Compared with some existing results, a general communication topology among agents is introduced in this article, where there is no need to confine directed cycles to the root of a spanning tree. By taking into consideration the structures of directed cycles and multiple paths, the judgment of structural balance is simplified significantly. A novel cluster consensus protocol is also proposed for structurally unbalanced digraphs. Moreover, two necessary and sufficient conditions are derived, by which cluster consensus can be achieved in an MAS if and only if its communication topology contains a directed spanning tree. Then, by employing an algebra theorem, a sufficient criterion for the unstable system under a directed cycle is obtained, whether the number of agents on this cycle is odd or even. Some illustrative examples are given to demonstrate the effectiveness of theoretical results.

Primal-Dual Subgradient Algorithm for Distributed Constraint Optimization Over Unbalanced Digraphs

This paper investigates the distributed convex optimization problem with coupled inequality constraints over unbalanced digraphs depicted as row-stochastic matrices, where each agent in the network only has access to its local information, while the local constraint functions of all the agents are coupled in inequality constraints. To solve this kind of problems, a novel distributed iterative algorithm is proposed via consensus scheme and the projected primal-dual subgradient method. Different from the previous related results, our algorithm can not only deal with coupling inequality constraints but also conquer the unbalanced topology caused by directed graphs. Moreover, it is proved that under certain assumptions, the optimal solution of the problem can be asymptotically obtained by performing the designed algorithm. Finally, the numerical examples are presented to further illustrate the efficacy of the proposed approach.

Distributed Convex Optimization with Inequality Constraints over Time-Varying Unbalanced Digraphs

This paper considers a distributed convex optimization problem with inequality constraints over time-varying unbalanced digraphs, where the cost function is a sum of local objective functions, and each node of the graph only knows its local objective and inequality constraints. Although there is a vast body of literature on distributed optimization, most of them require the graph to be balanced, which is quite restrictive and not necessary. To solve it, this work proposes a novel idea of using the epigraph form of the constrained optimization, which can be easily used to study time-varying unbalanced digraphs. Under local communications, a simple iterative algorithm is then designed for each node. We prove that if the graph is uniformly jointly strongly connected, each node asymptotically converges to some common optimal solution.

Tight bound on parameter of surplus-based averaging algorithm over balanced digraphs

ABSTRACT We study a continuous-time surplus-based algorithm for multi-agent average consensus, and derive a tight upper bound on the key parameter included in this algorithm that ensures convergence over strongly connected and balanced digraphs. We specialise the upper bound result to undirected (connected) graphs and unweighted cyclic digraphs; in particular, for undirected graphs the algorithm converges for arbitrary positive values of the parameter, and for cyclic digraphs the upper bound on the parameter depends only on the number of agents and may be easily calculated. Moreover, it is suggested through extensive simulation that, for the same number of agents, the upper bound for cyclic digraphs be smaller than that for other strongly connected and possibly unbalanced digraphs; this implies that as long as the parameter satisfies the upper bound for cyclic digraphs, this parameter can work for other digraphs.

Finite-Time Consensus of Opinion Dynamics and its Applications to Distributed Optimization Over Digraph.

In this paper, some efficient criteria for finite-time consensus of a class of nonsmooth opinion dynamics over a digraph are established. The lower and upper bounds on the finite settling time are obtained based respectively on the maximal and minimal cut capacity of the digraph. By using tools of the nonsmooth theory and algebraic graph theory, the Carathéodory and Filippov solutions of nonsmooth opinion dynamics are analyzed and compared in detail. In the sense of Filippov solutions, the dynamic consensus is demonstrated without a leader and the finite-time bipartite consensus is also investigated in a signed digraph correspondingly. To achieve a predetermined consensus, a leader agent is introduced to the considered agent networks. As an application, the nonsmooth compartmental dynamics in the presence of a leader is embedded in the proposed continuous-time protocol to solve the distributed optimization problems over an unbalanced digraph. The convergence to the optimal solution by using the proposed distributed algorithm is guaranteed with appropriately selected parameters. To verify the effectiveness of the proposed protocols, three numerical examples are performed.

Cluster Consensus in Networks of Agents With Weighted Cooperative–Competitive Interactions

This brief concerns a general cooperative–competitive network with weighted interactions, based on which the cluster consensus problem is addressed. A directed network with the weighted cooperative–competitive interactions is classified into three categories: 1) interactively balanced; 2) sub-balanced; and 3) unbalanced digraphs. In interactively balanced or sub-balanced networks, cluster consensus can be achieved, while in interactively unbalanced networks, cluster consensus can be reached in need of additional control inputs exerted on specific pinned agents. Simulations are finally presented to verify the theoretical results.

Coordinated Control for Flywheel Energy Storage Matrix Systems for Wind Farm Based on Charging/Discharging Ratio Consensus Algorithms

This paper proposes a distributed algorithm for coordination of flywheel energy storage matrix system (FESMS) cooperated with wind farm. A simple and distributed ratio consensus algorithm is proposed to solve FESMS dispatch problem. The algorithm is based on average consensus for both undirected and unbalanced directed graphs. Average consensus is guaranteed in unbalanced digraphs by updating the weight matrix with both its row sums and column sums being 1. Simulation examples illustrate the effectiveness of the proposed control method.

Phase synchronization control of complex networks of Lagrangian systems on adaptive digraphs

This paper presents a formation control and synchronization method that utilizes adaptive network topologies for a class of complex dynamical networks comprised of a large number of highly-nonlinear Euler–Lagrange (EL) systems. A time-varying and switching network topology, constructed by the adaptive graph Laplacian matrix, relaxes the standard requirement of consensus stability, even permitting exponential synchronization on an unbalanced digraph or a weakly connected digraph that can sporadically lose connectivity. The time-varying graph Laplacian matrix is adapted by an adaptive control scheme based on relative positions and errors of synchronization and tracking. The adaptive graph Laplacian is integrated with a phase synchronization controller that synchronizes the relative motions of EL systems moving in elliptical orbits, thereby yielding a smaller synchronization error than an uncoupled tracking control law in the presence of bounded disturbances and modeling errors. An example of reconfiguring hundreds of spacecraft in Low Earth Orbit shows the effectiveness of the proposed phase synchronization controller for a large number of complex EL systems moving in periodic elliptical orbits.

Constructing continuum many countable, primitive, unbalanced digraphs

We construct continuum many non-isomorphic countable digraphs which are highly arc transitive, have finite out-valency and infinite in-valency, and whose automorphism groups are primitive.