Distributed inertial online game algorithm for tracking generalized Nash equilibria.

This paper is concerned with the distributed generalized Nash equilibrium (GNE) tracking problem of noncooperative games in dynamic environments, where the cost function and/or the coupled constraint function are time-varying and revealed to each agent after it makes a decision. We first consider the case without coupled constraints and propose a distributed inertial online game (D-IOG) algorithm based on the mirror descent method. The proposed algorithm is capable of tracking Nash equilibrium (NE) through a time-varying communication graph and has the potential of achieving a low average regret. With an appropriate non-increasing stepsize sequence and an inertial parameter, the regrets can grow sublinearly if the deviation of the NE sequence grows sublinearly. Second, the time-varying coupled constraints are further investigated, and a modified D-IOG algorithm for tracking GNE is proposed based on the primal-dual and mirror descent methods. Then, the upper bounds of regrets and constraint violation are derived. Moreover, inertia and two information transmission modes are discussed. Finally, two simulation examples are provided to illustrate the effectiveness of the D-IOG algorithms.

Communication-efficient distributed cubic Newton with compressed lazy Hessian.

Recently, second-order distributed optimization algorithms have been becoming a research hot in distributed learning, due to their faster convergence rate than the first-order algorithms. However, second-order algorithms always suffer from serious communication bottleneck. To conquer such challenge, we propose communication-efficient second-order distributed optimization algorithms in the parameter-server framework, by incorporating cubic Newton methods with compressed lazy Hessian. Specifically, our algorithms require each worker communicate compressed Hessians with the server only at some particular iterations, which can save both communication bits and communication rounds. For non-convex problems, we theoretically prove that our algorithms can reduce the communication cost comparing to the state-of-the-art second-order algorithms, while maintaining the same iteration complexity order O(ϵ-3/2) as the centralized cubic Newton methods. By further using gradient regularization technique, our algorithms can achieve global convergence for convex problems. Moreover, for strongly convex problems, our algorithms can achieve local superlinear convergence rate without any requirement on initial conditions. Finally, numerical experiments are conducted to show the high efficiency of the proposed algorithms.

A Distributed k-Winners-Take-All Model With Binary Consensus Protocols.

This article concentrates on solving the k -winners-take-all (k WTA) problem with large-scale inputs in a distributed setting. We propose a multiagent system with a relatively simple structure, in which each agent is equipped with a 1-D system and interacts with others via binary consensus protocols. That is, only the signs of the relative state information between neighbors are required. By virtue of differential inclusion theory, we prove that the system converges from arbitrary initial states. In addition, we derive the convergence rate as O(1/t) . Furthermore, in comparison to the existing models, we introduce a novel comparison filter to eliminate the resolution ratio requirement on the input signal, that is, the difference between the k th and (k+1) th largest inputs must be larger than a positive threshold. As a result, the proposed distributed k WTA model is capable of solving the k WTA problem, even when more than two elements of the input signal share the same value. Finally, we validate the effectiveness of the theoretical results through two simulation examples.

Decentralized ADMM with compressed and event-triggered communication.

This paper considers the decentralized optimization problem, where agents in a network cooperate to minimize the sum of their local objective functions by communication and local computation. We propose a decentralized second-order communication-efficient algorithm called communication-censored and communication-compressed quadratically approximated alternating direction method of multipliers (ADMM), termed as CC-DQM, by combining event-triggered communication with compressed communication. In CC-DQM, agents are allowed to transmit the compressed message only when the current primal variables have changed greatly compared to its last estimate. Moreover, to relieve the computation cost, the update of Hessian is also scheduled by the trigger condition. Theoretical analysis shows that the proposed algorithm can still maintain an exact linear convergence, despite the existence of compression error and intermittent communication, if the local objective functions are strongly convex and smooth. Finally, numerical experiments demonstrate its satisfactory communication efficiency.

A Second-Order Projected Primal-Dual Dynamical System for Distributed Optimization and Learning.

This article focuses on developing distributed optimization strategies for a class of machine learning problems over a directed network of computing agents. In these problems, the global objective function is an addition function, which is composed of local objective functions. Such local objective functions are convex and only endowed by the corresponding computing agent. A second-order Nesterov accelerated dynamical system with time-varying damping coefficient is developed to address such problems. To effectively deal with the constraints in the problems, the projected primal-dual method is carried out in the Nesterov accelerated system. By means of the cocoercive maximal monotone operator, it is shown that the trajectories of the Nesterov accelerated dynamical system can reach consensus at the optimal solution, provided that the damping coefficient and gains meet technical conditions. In the end, the validation of the theoretical results is demonstrated by the email classification problem and the logistic regression problem in machine learning.

A new protocol for finite-time consensus of detail-balanced multi-agent networks.

In this paper, a finite-time consensus protocol for multi-agent networks is discussed from a new perspective. The order ß of the nonlinear function in the protocol is shown to be a crucial parameter in analyzing the finite-time consensus property of multi-agent networks with a detail-balanced communication topology. When ß>0, the corresponding protocol can guarantee the consensus of the multi-agent networks. In particular, if ß∈(0,1), the consensus can be realized within finite time. A leader-follow model is also investigated in this paper. Finally, several concrete protocols are proposed based on our theoretical analysis, and numerical examples are given to make a comparison among different protocols from the aspect of convergence speed.

Distributed k-winners-take-all via multiple neural networks with inertia.

This paper is dedicated to solving the k-winners-take-all problem with large-scale input signals in a distributed manner. According to the decomposition of global input signals, a novel dynamical system consisting of multiple coordinated neural networks is proposed for finding the k largest inputs. In the system, each neural network is designed to tackle its available partial inputs only for a local objective ki (ki≤k). Simultaneously, a consensus-based approach is adopted to coordinate multiple neural networks for achieving the global objective k. In addition, an inertial term is introduced in each neural network for regulating its transient behavior, which has the potential of accelerating the convergence. By developing a cocoercive operator, we theoretically prove that the multiple neural networks with inertial terms converge asymptotically/exponentially to the k-winners-take-all solution exactly from arbitrary initial states for whatever decomposition of inputs and objective. Furthermore, some extensions to distributed constrained k-winners-take-all are also investigated. Finally, simulation results are presented to substantiate the effectiveness of the proposed system as well as its superior performance over existing distributed networks.

Privacy-Preserving Distributed ADMM With Event-Triggered Communication.

This article addresses distributed optimization problems, in which a group of agents cooperatively minimize the sum of their private objective functions via information exchanging. Building on alternating direction method of multipliers (ADMM), we propose a privacy-preserving and communication-efficient decentralized quadratically approximated ADMM algorithm, termed PC-DQM, for solving such type of problems under the scenario of limited communication. In PC-DQM, an event-triggered mechanism is designed to schedule the communication instants for reducing communication cost. Simultaneously, for privacy preservation, a Hessian matrix with perturbed noise is introduced to quadratically approximate the objective function, which results in a closed form of primal vector update and then avoids solving a subproblem at each iteration with possible high computation cost. In addition, the triggered scheme is also utilized to schedule the update of Hessian, which can also reduce computation cost. We theoretically show that PC-DQM can protect privacy but without losing accuracy. In addition, we rigorously prove that PC-DQM converges linearly to the exact optimal solution for strongly convex and smooth objective functions. Finally, numerical simulation is presented to illustrate the effectiveness and efficiency of our algorithm.

A Collaborative Neurodynamic Approach to Multiple-Objective Distributed Optimization.

This paper is concerned with multiple-objective distributed optimization. Based on objective weighting and decision space decomposition, a collaborative neurodynamic approach to multiobjective distributed optimization is presented. In the approach, a system of collaborative neural networks is developed to search for Pareto optimal solutions, where each neural network is associated with one objective function and given constraints. Sufficient conditions are derived for ascertaining the convergence to a Pareto optimal solution of the collaborative neurodynamic system. In addition, it is proved that each connected subsystem can generate a Pareto optimal solution when the communication topology is disconnected. Then, a switching-topology-based method is proposed to compute multiple Pareto optimal solutions for discretized approximation of Pareto front. Finally, simulation results are discussed to substantiate the performance of the collaborative neurodynamic approach. A portfolio selection application is also given.

Global exponential synchronization of multiple coupled inertial memristive neural networks with time-varying delay via nonlinear coupling.

In this paper, global exponential synchronization of multiple coupled inertial memristive neural networks with time-varying delay is investigated. First, by choosing suitable variable substitution, the inertial memristive neural networks are transformed into first-order differential equations. Next, a novel coupling scheme with linear diffusive term and discontinuous sign function term depending on the first order derivative of state variables is introduced. Based on this coupling scheme, several sufficient conditions for global exponential synchronization of multiple inertial memristive neural networks are derived by using Lyapunov stability theory and some inequality techniques. Finally, several numerical examples are presented to substantiate the effectiveness of the theoretical results.

Global exponential synchronization of inertial memristive neural networks with time-varying delay via nonlinear controller.

The paper is concerned with the synchronization problem of inertial memristive neural networks with time-varying delay. First, by choosing a proper variable substitution, inertial memristive neural networks described by second-order differential equations can be transformed into first-order differential equations. Then, a novel controller with a linear diffusive term and discontinuous sign term is designed. By using the controller, the sufficient conditions for assuring the global exponential synchronization of the derive and response neural networks are derived based on Lyapunov stability theory and some inequality techniques. Finally, several numerical simulations are provided to substantiate the effectiveness of the theoretical results.

A Collective Neurodynamic Approach to Distributed Constrained Optimization.

This paper presents a collective neurodynamic approach with multiple interconnected recurrent neural networks (RNNs) for distributed constrained optimization. The objective function of the distributed optimization problems to be solved is a sum of local convex objective functions, which may be nonsmooth. Subject to its local constraints, each local objective function is minimized individually by using an RNN, with consensus among others. In contrast to existing continuous-time distributed optimization methods, the proposed collective neurodynamic approach is capable of solving more general distributed optimization problems. Simulation results on three numerical examples are discussed to substantiate the effectiveness and characteristics of the proposed approach. In addition, an application to the optimal placement problem is delineated to demonstrate the viability of the approach.

Global Synchronization of Multiple Recurrent Neural Networks With Time Delays via Impulsive Interactions.

In this paper, new results on the global synchronization of multiple recurrent neural networks (NNs) with time delays via impulsive interactions are presented. Impulsive interaction means that a number of NNs communicate with each other at impulse instants only, while they are independent at the remaining time. The communication topology among NNs is not required to be always connected and can switch ON and OFF at different impulse instants. By using the concept of sequential connectivity and the properties of stochastic matrices, a set of sufficient conditions depending on time delays is derived to ascertain global synchronization of multiple continuous-time recurrent NNs. In addition, a counterpart on the global synchronization of multiple discrete-time NNs is also discussed. Finally, two examples are presented to illustrate the results.

Global synchronization of memristive neural networks subject to random disturbances via distributed pinning control.

This paper presents theoretical results on global exponential synchronization of multiple memristive neural networks in the presence of external noise by means of two types of distributed pinning control. The multiple memristive neural networks are coupled in a general structure via a nonlinear function, which consists of a linear diffusive term and a discontinuous sign term. A pinning impulsive control law is introduced in the coupled system to synchronize all neural networks. Sufficient conditions are derived for ascertaining global exponential synchronization in mean square. In addition, a pinning adaptive control law is developed to achieve global exponential synchronization in mean square. Both pinning control laws utilize only partial state information received from the neighborhood of the controlled neural network. Simulation results are presented to substantiate the theoretical results.

Global exponential synchronization of multiple memristive neural networks with time delay via nonlinear coupling.

This paper presents theoretical results on the global exponential synchronization of multiple memristive neural networks with time delays. A novel coupling scheme is introduced, in a general topological structure described by a directed or undirected graph, with a linear diffusive term and discontinuous sign term. Several criteria are derived based on the Lyapunov stability theory to ascertain the global exponential stability of synchronization manifold in the coupling scheme. Simulation results for several examples are given to substantiate the effectiveness of the theoretical results.