We study distributed algorithms for solving global optimization problems in which the objective function is the sum of local objective functions of agents and the constraint set is given by the intersection of local constraint sets of agents. We assume that each agent knows only his own local objective function and constraint set, and exchanges information with the other agents over a randomly varying network topology to update his information state. We assume a state-dependent communication model over this topology: communication is Markovian with respect to the states of the agents and the probability with which the links are available depends on the states of the agents. In this paper, we study a projected multi-agent subgradient algorithm under state-dependent communication. The algorithm involves each agent performing a local averaging to combine his estimate with the other agents' estimates, taking a subgradient step along his local objective function, and projecting the estimates on his local constraint set. The state-dependence of the communication introduces significant challenges and couples the study of information exchange with the analysis of subgradient steps and projection errors. We first show that the multi-agent subgradient algorithm when used with a constant stepsize may result in the agent estimates to diverge with probability one. Under some assumptions on the stepsize sequence, we provide convergence rate bounds on a "disagreement metric" between the agent estimates. Our bounds are time-nonhomogeneous in the sense that they depend on the initial starting time. Despite this, we show that agent estimates reach an almost sure consensus and converge to the same optimal solution of the global optimization problem with probability one under different assumptions on the local constraint sets and the stepsize sequence.
This paper studies resilient multi-agent distributed estimation of an unknown vector parameter when a subset of the agents is adversarial. We present and analyze a Flag Raising Distributed Estimator ($\mathcal{FRDE}$) that allows the agents under attack to perform accurate parameter estimation and detect the adversarial agents. The $\mathcal{FRDE}$ algorithm is a consensus+innovations estimator in which agents combine estimates of neighboring agents (consensus) with local sensing information (innovations). We establish that, under $\mathcal{FRDE}$, either the uncompromised agents' estimates ar…
This work builds on our recent work on a distributed optimization algorithm for graphs with directed unreliable communications. We show its linear convergence when we take either the proximal of each function or an affine minorant for when the function is smooth.
Distributed multi-agent optimization finds many applications in distributed learning, control, estimation, etc. Most existing algorithms assume knowledge of first-order information of the objective and have been analyzed for convex problems. However, there are situations where the objective is nonconvex, and one can only evaluate the function values at finitely many points. In this paper we consider derivative-free distributed algorithms for nonconvex multi-agent optimization, based on recent progress in zero-order optimization. We develop two algorithms for different settings, provide detaile…
We show that the averaged consensus algorithm on directed graphs with unreliable communications by Bof-Carli-Schenato has a dual optimization interpretation, which could be extended to the case of distributed optimization. We report on our numerical simulations for the distributed optimization algorithm for smooth and nonsmooth functions.
This work studies the problem of distributed optimization in heterogeneous linear multi-agent systems. Instead of relying on a perfect communication network as in many existing distributed optimization approaches, we considered two important issues related to communication networks. First, assumed that communication delays exist when each agent receives information from its neighbors. Further, since communications networks are generally unreliable, we assumed each agent interacted with the other agents through random digraphs. Finally, to prove convergence to optimal solutions, delay-dependent…
In this paper, we consider minimizing a sum of local convex objective functions in a distributed setting, where communication can be costly. We propose and analyze a class of nested distributed gradient methods with adaptive quantized communication (NEAR-DGD+Q). We show the effect of performing multiple quantized communication steps on the rate of convergence and on the size of the neighborhood of convergence, and prove R-Linear convergence to the exact solution with increasing number of consensus steps and adaptive quantization. We test the performance of the method, as well as some practical…
Motivated by applications in machine learning and statistics, we study distributed optimization problems over a network of processors, where the goal is to optimize a global objective composed of a sum of local functions. In these problems, due to the large scale of the data sets, the data and computation must be distributed over processors resulting in the need for distributed algorithms. In this paper, we consider a popular distributed gradient-based consensus algorithm, which only requires local computation and communication. An important problem in this area is to analyze the convergence r…
This paper is concerned with a leader-follower problem for a multi-agent system with a switching interconnection topology. Distributed observers are designed for the second-order follower-agents, under the common assumption that the velocity of the active leader cannot be measured in real time. Some dynamic neighbor-based rules, consisting of distributed controllers and observers for the autonomous agents, are developed to keep updating the information of the leader. With the help of an explicitly constructed common Lyapunov function (CLF), it is proved that each agent can follow the active le…
This paper proposes a new framework for distributed optimization, called distributed aggregative optimization, which allows local objective functions to be dependent not only on their own decision variables, but also on the average of summable functions of decision variables of all other agents. To handle this problem, a distributed algorithm, called distributed gradient tracking (DGT), is proposed and analyzed, where the global objective function is strongly convex, and the communication graph is balanced and strongly connected. It is shown that the algorithm can converge to the optimal varia…
We consider the problem of distributed Kalman filtering for sensor networks in the case there is a limit in data transmission and there is model uncertainty. More precisely, we propose a distributed filtering strategy with event-triggered communication in which the state estimators are computed according to the least favorable model. The latter belongs to a ball (in Kullback-Leibler topology) about the nominal model. We also present a preliminary numerical example in order to test the performance of the proposed strategy.
This paper considers distributed nonconvex optimization with the cost functions being distributed over agents. Noting that information compression is a key tool to reduce the heavy communication load for distributed algorithms as agents iteratively communicate with neighbors, we propose three distributed primal--dual algorithms with compressed communication. The first two algorithms are applicable to a general class of compressors with bounded relative compression error and the third algorithm is suitable for two general classes of compressors with bounded absolute compression error. We show t…
We consider the distributed leader-follower consensus problem with event-triggered communications. The system under consideration is a non-linear input-affine multi agent system. The agents are assumed to have identical dynamics structure with uncertain parameters and satisfying an incremental stabilisability condition. A distributed control law is proposed which achieves consensus based on two novel Communication Triggering Conditions (CTCs): the first one to achieve an asymptotic consensus but without any guarantees on Zeno behaviour and the second one to exclude Zeno behaviour but with prac…
This paper addresses distributed average tracking of physical second-order agents with nonlinear dynamics, where the interaction among the agents is described by an undirected graph. In both agents' and reference inputs' dynamics, there is a nonlinear term that satisfying the Lipschitz-type condition. To achieve the distributed average tracking problem in the presence of nonlinear term, a non-smooth filter and a control input are designed for each agent. The idea is that each filter outputs converge to the average of the reference inputs and the reference velocities asymptotically and in paral…
In this work, we design distributed control laws for spatial self-organization of multi-agent swarms in 1D and 2D spatial domains. The objective is to achieve a desired density distribution over a simply-connected spatial domain. Since individual agents in a swarm are not themselves of interest and we are concerned only with the macroscopic objective, we view the network of agents in the swarm as a discrete approximation of a continuous medium and design control laws to shape the density distribution of the continuous medium. The key feature of this work is that the agents in the swarm do not …
We study the policy evaluation problem in multi-agent reinforcement learning. In this problem, a group of agents works cooperatively to evaluate the value function for the global discounted accumulative reward problem, which is composed of local rewards observed by the agents. Over a series of time steps, the agents act, get rewarded, update their local estimate of the value function, then communicate with their neighbors. The local update at each agent can be interpreted as a distributed consensus-based variant of the popular temporal difference learning algorithm TD(0). While distributed rei…
In this work, we address the distributed optimization problem with event-triggered communication by the notion of input feedforward passivity (IFP). First, we analyze the distributed continuous-time algorithm over uniformly jointly strongly connected balanced digraphs in an IFP-based framework. Then, we propose a distributed event-triggered communication mechanism for this algorithm. Next, we discretize the continuous-time algorithm by the forward Euler method with a constant stepsize irrelevant to network size, and show that the discretization can be seen as a stepsize-dependent passivity deg…
This work develops effective distributed strategies for the solution of constrained multi-agent stochastic optimization problems with coupled parameters across the agents. In this formulation, each agent is influenced by only a subset of the entries of a global parameter vector or model, and is subject to convex constraints that are only known locally. Problems of this type arise in several applications, most notably in disease propagation models, minimum-cost flow problems, distributed control formulations, and distributed power system monitoring. This work focuses on stochastic settings, whe…
We consider the problem of distributed Kalman filtering for sensor networks in the case there are constraints in data transmission and there is model uncertainty. More precisely, we propose two distributed filtering strategies with event-triggered communication where the state estimators are computed according to the least favorable model. The latter belongs to a ball about the nominal model. We also show that both the methods are stable in the sense that the mean-square of the state estimation error is bounded in all the nodes.
In this paper, we first present an adaptive distributed observer for a discrete-time leader system. This adaptive distributed observer will provide, to each follower, not only the estimation of the leader's signal, but also the estimation of the leader's system matrix. Then, based on the estimation of the matrix S, we devise a discrete adaptive algorithm to calculate the solution to the regulator equations associated with each follower, and obtain an estimated feedforward control gain. Finally, we solve the cooperative output regulation problem for discrete-time linear multi-agent systems by b…
This paper presents the formulation and analysis of a fully distributed dynamic event-triggered communication based robust dynamic average consensus algorithm. Dynamic average consensus problem involves a networked set of agents estimating the time-varying average of dynamic reference signals locally available to individual agents. We propose an asymptotically stable solution to the dynamic average consensus problem that is robust to network disruptions. Since this robust algorithm requires continuous communication among agents, we introduce a novel dynamic event-triggered communication scheme…
Heterogeneous networks comprise agents with varying capabilities in terms of computation, storage, and communication. In such settings, it is crucial to factor in the operating characteristics in allowing agents to choose appropriate updating schemes, so as to better distribute computational tasks and utilize the network more efficiently. We consider the multi-agent optimization problem of cooperatively minimizing the sum of local strongly convex objectives. We propose an asynchronous distributed primal-dual protocol, which allows for the primal update steps to be agent-dependent (an agent can…
This paper addresses distributed average tracking for a group of heterogeneous physical agents consisting of single-integrator, double-integrator and Euler-Lagrange dynamics. Here, the goal is that each agent uses local information and local interaction to calculate the average of individual time-varying reference inputs, one per agent. Two nonsmooth algorithms are proposed to achieve the distributed average tracking goal. In our first proposed algorithm, each agent tracks the average of the reference inputs, where each agent is required to have access to only its own position and the relative…
Recent developments on large-scale distributed machine learning applications, e.g., deep neural networks, benefit enormously from the advances in distributed non-convex optimization techniques, e.g., distributed Stochastic Gradient Descent (SGD). A series of recent works study the linear speedup property of distributed SGD variants with reduced communication. The linear speedup property enable us to scale out the computing capability by adding more computing nodes into our system. The reduced communication complexity is desirable since communication overhead is often the performance bottleneck…
In this paper we study the effect of the way that the data is partitioned in distributed optimization. The original DiSCO algorithm [Communication-Efficient Distributed Optimization of Self-Concordant Empirical Loss, Yuchen Zhang and Lin Xiao, 2015] partitions the input data based on samples. We describe how the original algorithm has to be modified to allow partitioning on features and show its efficiency both in theory and also in practice.
This paper investigates Nash equilibrium (NE) seeking problems for noncooperative games over multi-players networks with finite bandwidth communication. A distributed quantized algorithm is presented, which consists of local gradient play, distributed decision estimating, and adaptive quantization. Exponential convergence of the algorithm is established, and a relationship between the convergence rate and the bandwidth is quantitatively analyzed. Finally, a simulation of an energy consumption game is presented to validate the proposed results.
Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）；Weizmann Institute of Science（魏茨曼科学研究所）
We resolve the min-max complexity of distributed stochastic convex optimization (up to a log factor) in the intermittent communication setting, where $M$ machines work in parallel over the course of $R$ rounds of communication to optimize the objective, and during each round of communication, each machine may sequentially compute $K$ stochastic gradient estimates. We present a novel lower bound with a matching upper bound that establishes an optimal algorithm.
We investigate fast and communication-efficient algorithms for the classic problem of minimizing a sum of strongly convex and smooth functions that are distributed among $n$ different nodes, which can communicate using a limited number of bits. Most previous communication-efficient approaches for this problem are limited to first-order optimization, and therefore have \emph{linear} dependence on the condition number in their communication complexity. We show that this dependence is not inherent: communication-efficient methods can in fact have sublinear dependence on the condition number. For …
In recent years, variance-reducing stochastic methods have shown great practical performance, exhibiting linear convergence rate when other stochastic methods offered a sub-linear rate. However, as datasets grow ever bigger and clusters become widespread, the need for fast distribution methods is pressing. We propose here a distribution scheme for SAGA which maintains a linear convergence rate, even when communication between nodes is limited.
UC Berkeley（加州大学伯克利分校）；The University of Iowa（爱荷华大学）；Princeton University（普林斯顿大学）
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…
In this paper, we consider an unconstrained distributed optimization problem over a network of agents, in which some agents are adversarial. We solve the problem via gradient-based distributed optimization algorithm and characterize the effect of the adversarial agents on the convergence of the algorithm to the optimal solution. The attack model considered is such that agents locally perturb their iterates before broadcasting it to neighbors; and we analyze the case in which the adversarial agents cooperate in perturbing their estimates and the case where each adversarial agent acts independen…
University of Science and Technology of China（中国科学技术大学）；Rensselaer Polytechnic Institute（伦斯勒理工学院）；Sun Yat-Sen University（中山大学）
This paper develops a communication-efficient algorithm to solve the stochastic optimization problem defined over a distributed network, aiming at reducing the burdensome communication in applications such as distributed machine learning.Different from the existing works based on quantization and sparsification, we introduce a communication-censoring technique to reduce the transmissions of variables, which leads to our communication-Censored distributed Stochastic Gradient Descent (CSGD) algorithm. Specifically, in CSGD, the latest mini-batch stochastic gradient at a worker will be transmitte…
This paper focuses on the problem of distributed consensus control of multi-agent systems while considering two main practical concerns (i) stochastic noise in the agent dynamics and (ii) predefined performance constraints over evolutions of multi-agent systems. In particular, we consider that each agent is driven by a stochastic differential equation with state-dependent noise which makes the considered problem more challenging compared to non-stochastic agents. The work provides sufficient conditions under which the proposed timevarying distributed control laws ensure consensus in expectatio…
We study cost-effective communication strategies that can be used to improve the performance of distributed learning systems in resource-constrained environments. For distributed learning in sequential decision making, we propose a new cost-effective partial communication protocol. We illustrate that with this protocol the group obtains the same order of performance that it obtains with full communication. Moreover, we prove that under the proposed partial communication protocol the communication cost is $O(\log T)$, where $T$ is the time horizon of the decision-making process. This improves s…
This work is concerned with the problem of distributed resource allocation in continuous-time setting but with discrete-time communication over infinitely jointly connected and balanced digraphs. We provide a passivity-based perspective for the continuous-time algorithm, based on which an intermittent communication scheme is developed. Particularly, a periodic communication scheme is first derived through analyzing the passivity degradation over output sampling of the distributed dynamics at each node. Then, an asynchronous distributed event-triggered scheme is further developed. The sampled-b…
We consider distributed convex optimization problems originated 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 c…
Average consensus protocols emerge with a central role in distributed systems and decision-making such as distributed information fusion, distributed optimization, distributed estimation, and control. A key advantage of these protocols is that agents exchange and reveal their state information only to their neighbors. Yet, it can raise privacy concerns in situations where the agents' states contain sensitive information. In this paper, we propose a novel (noiseless) privacy preserving distributed algorithms for multi-agent systems to reach an average consensus. The main idea of the algorithms …
This paper discusses distributed optimization over a directed graph. We begin with some well known algorithms which achieve consensus among agents including FROST [1], which possesses the quickest convergence to the optimum. It is a well known fact FROST has a linear convergence. However FROST works only over fixed topology of underlying network. Moreover the updates proposed therein require perfectly synchronized communication among nodes. Hence communication delays among nodes, which are inevitable in a realistic scenario, preclude the possibility of implementing FROST in real time. In this …
This paper is concerned with the consensus problem for multi-agent systems subject to communication delays between the neighboring agents. We consider a scenario where each agent is characterized by a general high-order linear system and the communication delays between the agents are non-uniform and time-varying. We design a distributed control protocol for the agents and provide an equivalent stability problem to be solved that guarantees the state consensus in the group of agents. Moreover, a delay-dependent stability criterion is provided by combining the Lyapunov-Krasovskii method with th…
This paper addresses the coverage control problem of multi-agent systems in the uncertain environment. With the aid of Voronoi partition, a distributed coverage control formulation of multi-agent system is proposed to complete the workload in uncertain environments. Driven by the gradient of thermal field, each agent is able to move around for clearing the workload on its own subregion. Theoretical analysis is conducted to ensure the completion of workload in finite time. Finally, numerical simulations are carried out to demonstrate the effectiveness and advantages of the proposed coverage con…
We study the fundamental limits to communication-efficient distributed methods for convex learning and optimization, under different assumptions on the information available to individual machines, and the types of functions considered. We identify cases where existing algorithms are already worst-case optimal, as well as cases where room for further improvement is still possible. Among other things, our results indicate that without similarity between the local objective functions (due to statistical data similarity or otherwise) many communication rounds may be required, even if the machines…
This paper considers a distributed optimization problem in a multi-agent system where a fraction of the agents act in an adversarial manner. Specifically, the malicious agents steer the network of agents away from the optimal solution by sending false information to their neighbors and consume significant bandwidth in the communication process. We propose a distributed gradient-based optimization algorithm in which the non-malicious agents exchange quantized information with one another. We prove convergence of the solution to a neighborhood of the optimal solution, and characterize the soluti…
Due to its efficiency and ease to implement, stochastic gradient descent (SGD) has been widely used in machine learning. In particular, SGD is one of the most popular optimization methods for distributed learning. Recently, quantized SGD (QSGD), which adopts quantization to reduce the communication cost in SGD-based distributed learning, has attracted much attention. Although several QSGD methods have been proposed, some of them are heuristic without theoretical guarantee, and others have high quantization variance which makes the convergence become slow. In this paper, we propose a new method…
We address the problem of model-free distributed stabilization of heterogeneous multi-agent systems using reinforcement learning (RL). Two algorithms are developed. The first algorithm solves a centralized linear quadratic regulator (LQR) problem without knowing any initial stabilizing gain in advance. The second algorithm builds upon the results of the first algorithm, and extends it to distributed stabilization of multi-agent systems with predefined interaction graphs. Rigorous proofs are provided to show that the proposed algorithms achieve guaranteed convergence if specific conditions hold…
H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology（乔治亚理工学院H. Milton Stewart工业与系统工程学院）；Department of Electrical Engineering, Technion（以色列理工学院电气工程系）
Distributed optimization has gained a surge of interest in recent years. In this paper we propose a distributed projection free algorithm named Distributed Conditional Gradient Sliding(DCGS). Compared to the state-of-the-art distributed Frank-Wolfe algorithm, our algorithm attains the same communication complexity under much more realistic assumptions. In contrast to the consensus based algorithm, DCGS is based on the primal-dual algorithm, yielding a modular analysis that can be exploited to improve linear oracle complexity whenever centralized Frank-Wolfe can be improved. We demonstrate this…
This paper studies the distributed multi-agent resilient optimization problem under the f-total Byzantine attacks. Compared with the previous work on Byzantineresilient multi-agent exact optimization problems, we do not require the communication topology to be fully connected. Under the redundancy of cost functions, we propose the distributed comparative gradient elimination resilient optimization algorithm based on the traditional assumptions on strongly convex global costs and Lipschitz continuous gradients. Under this algorithm, we successfully prove that if the number of inneighbors of eac…
In this work we study a distributed optimal output consensus problem for heterogeneous linear multi-agent systems where the agents aim to reach consensus with the purpose of minimizing the sum of private convex costs. Based on output feedback, a fully distributed control law is proposed by using the proportional-integral (PI) control technique. For strongly convex cost functions with Lipschitz gradients, the designed controller can achieve convergence exponentially in an undirected and connected network. Furthermore, to remove the requirement of continuous communications, the proposed control …
Pennsylvania State University（宾夕法尼亚州立大学）；University of Virginia（弗吉尼亚大学）
Due to the explosion in the size of the training datasets, distributed learning has received growing interest in recent years. One of the major bottlenecks is the large communication cost between the central server and the local workers. While error feedback compression has been proven to be successful in reducing communication costs with stochastic gradient descent (SGD), there are much fewer attempts in building communication-efficient adaptive gradient methods with provable guarantees, which are widely used in training large-scale machine learning models. In this paper, we propose a new com…
Zhejiang University（浙江大学）；Rensselaer Polytechnic Institute（伦斯勒理工学院）；University of Minnesota, Twin Cities（明尼苏达大学双城分校）
The present paper develops a novel aggregated gradient approach for distributed machine learning that adaptively compresses the gradient communication. The key idea is to first quantize the computed gradients, and then skip less informative quantized gradient communications by reusing outdated gradients. Quantizing and skipping result in `lazy' worker-server communications, which justifies the term Lazily Aggregated Quantized gradient that is henceforth abbreviated as LAQ. Our LAQ can provably attain the same linear convergence rate as the gradient descent in the strongly convex case, while ef…
