Time-varying Directed Graph Research Articles

Distributed online constrained convex optimization with event-triggered communication

This paper focuses on the distributed online convex optimization problem with time-varying inequality constraints over a network of agents, where each agent collaborates with its neighboring agents to minimize the cumulative network-wide loss over time. To reduce communication overhead between the agents, we propose a distributed event-triggered online primal–dual algorithm over a time-varying directed graph. With several classes of appropriately chose decreasing parameter sequences and non-increasing event-triggered threshold sequences, we establish dynamic network regret and network cumulative constraint violation bounds. Finally, a numerical simulation example is provided to verify the theoretical results.

Differentially private distributed online optimization via push-sum one-point bandit dual averaging

This paper focuses on the distributed online optimization problem in multi-agent systems considering privacy preservation. Each agent exchanges local information with neighboring agents on the strongly connected time-varying directed graphs. Since the process of information transmission is prone to information leakage, a distributed push-sum dual averaging algorithm based on the differential privacy mechanism is proposed to protect the privacy of the data. In addition, to handle situations where the gradient information of the node cost function is unknown, the one-point gradient estimation is designed to calculate the true gradient information and guide the update of the decision variables. With the appropriate choice of the stepsizes and the exploration parameters, the algorithm can effectively protect the privacy of agents while achieving sublinear regret with the convergence rate O(T34). Furthermore, this paper also explores the effect of one-point estimation parameters on the regret in the online setting and investigates the relation between the convergence effect of individual regret and differential privacy levels. Finally, several federated learning experiments were conducted to verify the efficacy of the algorithm.

A game-theory-based scheme to facilitate consensus latency minimization in sharding blockchain

Unbalanced consensus latency in reaching a consensus on transactions occurs due to differences in the ability of different committees to process transactions. As the consensus latency accumulates, the transaction throughput of the sharding blockchain significantly decreases. This article proposes a consensus latency minimization scheme to improve the transaction throughput. According to a time-varying directed graph and a non-cooperative game model solved by a distributed algorithm, we show that the generalized Nash equilibrium for all committees are to balance the differences in consensus latency among various committees and to minimize the consensus latency throughout the whole period. The security of the proposed scheme is analyzed by considering the peaceful equilibrium, and the committee actions asymptotically converge to a generalized Nash equilibrium. Through the comparisons with other baseline algorithms via simulation experiments, we demonstrate that our proposed scheme could result in the optimal consensus latency and throughput. Therefore, the proposed consensus latency minimization scheme effectively avoids the existing problem of the transaction throughput in sharding blockchain and provides a viable and security instrument for the sharding blockchain operator to improve the transaction throughput performance throughout the whole period.

Online Distributed Optimization With Nonconvex Objective Functions via Dynamic Regrets

In this paper, the problem of online distributed optimization subject to a convex set is studied by employing a network of agents, where the objective functions allocated to agents are nonconvex. Each agent only has access to its own objective function information at the previous time, and can only communicate with its immediate neighbors via a time-varying directed graph. To tackle this problem, first, a new online distributed algorithm with gradient information is proposed based on consensus algorithms and projection-free strategies. Of particular interest is that dynamic regrets, whose offline benchmarks are to pursue the stationary points at each time, are employed to measure the performance of the algorithm. In the worst case, the difficulty in achieving sublinear bounds of dynamic regrets is characterized by the deviation in the objective function sequence, as well as the deviation in the gradient sequence. Under mild assumptions on the graph and the objective functions, we prove that if the deviation in the objective function sequence is sublinear with the square root of the time horizon, and if the deviation in the gradient sequence is sublinear with the time horizon, then dynamic regrets grow sublinearly. Second, considering the case where the gradient information of the objective functions is not available, we propose a zeroth-order online distributed projection-free algorithm, by which agents make decisions only depending on the random zeroth-order oracle. It turns out that under the same conditions as in the first case, if the smoothing parameters in the random zeroth-order oracles scale inversely with the time horizon, then the expectations of dynamic regrets increase sublinearly. Finally, simulations are presented to demonstrate the effectiveness of our theoretical results.

Finite-time error bounds for distributed linear stochastic approximation

This paper considers a novel multi-agent linear stochastic approximation algorithm driven by Markovian noise and general consensus-type interaction, in which each agent evolves according to its local stochastic approximation process which depends on the information from its neighbors. The interconnection structure among the agents is described by a time-varying directed graph. While the convergence of consensus-based stochastic approximation algorithms when the interconnection among the agents is described by doubly stochastic matrices (at least in expectation) has been studied, less is known about the case when the interconnection matrix is simply stochastic. For any uniformly strongly connected graph sequences whose associated interaction matrices are stochastic, the paper derives finite-time bounds on the mean-square error, defined as the deviation of the output of the algorithm from the unique equilibrium point of the associated ordinary differential equation. For the case of interconnection matrices being stochastic, the equilibrium point can be any unspecified convex combination of the local equilibria of all the agents in the absence of communication. Both the cases with constant and time-varying step-sizes are considered. In the case when the convex combination is required to be a straight average and interaction between any pair of neighboring agents may be uni-directional, so that doubly stochastic matrices cannot be implemented in a distributed manner, the paper proposes a push-sum-type distributed stochastic approximation algorithm and provides its finite-time bound for the time-varying step-size case by leveraging the analysis for the consensus-type algorithm with stochastic matrices and developing novel properties of the push-sum algorithm. Distributed temporal difference learning is discussed as an illustrative application.

A hybrid observer for estimating the state of a distributed linear system

A hybrid observer is described for estimating the state of an m>0 channel, n-dimensional, continuous-time, linear system of the form ẋ=Ax,yi=Cix,i∈{1,2,…,m}. The system’s state x is simultaneously estimated by m agents assuming each agent i senses yi and receives appropriately defined data from each of its current neighbors. Neighbor relations are characterized by a time-varying directed graph N(t) whose vertices correspond to agents and whose arcs depict neighbor relations. Agent i updates its estimate xi of x at “event times” ti1,ti2,ti3,… using a local continuous-time linear observer and a local parameter estimator which iterates q times during each event time interval [ti(s−1),tis),s≥1 to obtain an estimate of x(tis). Subject to the assumptions that none of the Ci’s are zero, the neighbor graph N(t) is strongly connected for all time, and the system whose state is to be estimated is jointly observable, it is shown that for any number λ>0, it is possible to choose q and the local observer gains so that each estimate xi converges to x at least as fast as e−λt does. This result holds whether or not agents communicate synchronously, although in the asynchronous case it is necessary to assume that N(t) changes in a suitably defined sense. Exponential convergence is also assured if the event time sequences of the m agents are slightly different than each other, although in this case only if the system being observed is exponentially stable; this limitation however, is primarily a robustness issue shared by all state estimators, centralized or not, which are operating in “open loop” in the face of small modeling errors. The result also holds facing abrupt changes in the number of vertices and arcs in the inter-agent communication graph upon which the algorithm depends.

Distributed quantized mirror descent for strongly convex optimization over time-varying directed graph

Distributed quantized mirror descent for strongly convex optimization over time-varying directed graph

Dual Averaging Push for Distributed Convex Optimization Over Time-Varying Directed Graph

Inspired by the subgradient push method developed recently by Nedic et al. we present a distributed dual averaging push algorithm for constrained nonsmooth convex optimization over time-varying directed graph. Our algorithm combines the dual averaging method with the push-sum technique and achieves an $O(1/ \sqrt{k})$ convergence rate. Compared with the subgradient push algorithm, our algorithm, first, addresses the constrained problems, and, second, has a faster convergence rate, and, third, simplifies the convergence analysis. We also generalize the proposed algorithm so that input variables of subgradient oracles have guaranteed convergence.

A Distributed Algorithm for Computing a Common Fixed Point of a Finite Family of Paracontractions

A distributed algorithm is described for finding a common fixed point of a family of m>1 nonlinear maps M_i : R^n -> R^n assuming that each map is a paracontraction and that at least one such common fixed point exists. The common fixed point is simultaneously computed by m agents assuming each agent i knows only M_i, the current estimates of the fixed point generated by its neighbors, and nothing more. Each agent recursively updates its estimate of a fixed point by utilizing the current estimates generated by each of its neighbors. Neighbor relations are characterized by a time-varying directed graph N(t). It is shown under suitably general conditions on N(t), that the algorithm causes all agents estimates to converge to the same common fixed point of the m nonlinear maps.

Event-Triggering Sampling Based Leader-Following Consensus in Second-Order Multi-Agent Systems

In this note, the problem of second-order leader-following consensus by a novel distributed event-triggered sampling scheme in which agents exchange information via a limited communication medium is studied. Event-based distributed sampling rules are designed, where each agent decides when to measure its own state value and requests its neighbor agents broadcast their state values across the network when a locally-computed measurement error exceeds a state-dependent threshold. For the case of fixed topology, a necessary and sufficient condition is established. For the case of switching topology, a sufficient condition is obtained under the assumption that the time-varying directed graph is uniformly jointly connected. It is shown that the inter-event intervals are lower bounded by a strictly positive constant, which excludes the Zeno-behavior before the consensus is achieved. Numerical simulation examples are provided to demonstrate the correctness of theoretical results.