We present convergence and error estimates of the time-discrete consensus-based optimization(CBO) algorithms proposed in [arXiv:1909.09249] for general nonconvex functions. In authors' recent work [arxiv: 1910.08239], rigorous error analysis of the first-order consensus-based optimization algorithm proposed in [arXiv:1909.09249] was studied at the particle level without resorting to the kinetic equation via a mean-field limit. However, the error analysis for the corresponding time-discrete algorithm was not done mainly due to lack of discrete analogue of Itô's stochastic calculus. In this paper, we provide a simple and elementary convergence and error analysis for a general time-discrete consensus-based optimization algorithm, which includes the three discrete algorithms in [arXiv:1909.09249]. Our analysis provides numerical stability and convergence conditions for the three algorithms, as well as error estimates to the global minimum.
We present stochastic consensus and convergence of the discrete consensus-based optimization (CBO) algorithm with random batch interactions and heterogeneous external noises. Despite the wide applications and successful performance in many practical simulations, the convergence of the discrete CBO algorithm was not rigorously investigated in such a generality. In this work, we introduce a generalized discrete CBO algorithm with a weighted representative point and random batch interactions, and show that the proposed discrete CBO algorithm exhibits stochastic consensus and convergence toward th…
A new theorem on conditions for convergence to consensus of a multiagent time-dependent time-discrete dynamical system is presented. The theorem is build up on the notion of averaging maps. We compare this theorem to results by Moreau (IEEE Transactions on Automatic Control, vol. 50, no. 2, 2005) about set-valued Lyapunov theory and convergence under switching communication topologies. We give examples that point out differences of approaches including examples where Moreau's theorem is not applicable but ours is. Further on, we give examples that demonstrate that the theory of convergence to …
Global optimization of a non-convex objective function often appears in large-scale machine-learning and artificial intelligence applications. Recently, consensus-based optimization (in short CBO) methods have been introduced as one of the gradient-free optimization methods. In this paper, we provide a convergence analysis for the first-order CBO method in \cite{C-J-L-Z}. Prior to the current work, the convergence study was carried out for CBO methods on corresponding mean-field limit, a Fokker-Planck equation, which does not imply the convergence of the CBO method {\it per se}. Based on the c…
We consider the celebrated Cucker-Smale model in finite dimension, modelling interacting collective dynamics and their possible evolution to consensus. The objective of this paper is to study the effect of time delays in the general model. By a Lyapunov functional approach, we provide convergence results to consensus for symmetric as well as nonsymmetric communication weights under some structural conditions.
We study the state consensus problem for linear shift-invariant discrete-time homogeneous multi-agent systems (MASs) over time-varying graphs. A novel approach based on the small gain theorem is proposed to design the consensus control protocols for both neutrally stable and neutrally unstable MASs, assuming the uniformly connected graphs. It is shown that the state consensus can be achieved for neutrally stable MASs under a weak uniform observability condition; for neutrally unstable MASs, the state consensus entails a strong uniform observability condition. Two numerical examples are worked …
Consensus conditions and convergence speeds are crucial for distributed consensus algorithms of networked systems. Based on a basic first-order average-consensus protocol with time-varying topologies and additive noises, this paper first investigates its critical consensus condition on network topology by stochastic approximation frameworks. A new joint-connectivity condition called extensible joint-connectivity that contains a parameter $δ$ (termed the extensible exponent) is proposed. With this and a balanced topology condition, we show that a critical value of $δ$ for consensus is $1/2$. Op…
This article investigates discrete-time matrix-weighted consensus of multi-agent networks over undirected and connected graphs. We first present consensus protocols for the agents in common networks of symmetric matrix weights with possibly different update rates and switching network topologies. A special type of matrix-weighted consensus with non-symmetric matrix-weights that can render several consensus control scenarios such as ones with scaled/rotated updates and affine motion constraints is also considered. We employ Lyapunov stability theory for discrete-time systems and occasionally ut…
This paper studies the multi-agent average consensus problem under the requirement of differential privacy of the agents' initial states against an adversary that has access to all the messages. We first establish that a differentially private consensus algorithm cannot guarantee convergence of the agents' states to the exact average in distribution, which in turn implies the same impossibility for other stronger notions of convergence. This result motivates our design of a novel differentially private Laplacian consensus algorithm in which agents linearly perturb their state-transition and me…
This paper utilizes the agent's memory in accelerated consensus for second-order multi-agent systems (MASs). In the case of one-tap memory, explicit formulas for the optimal consensus convergence rate and control parameters are derived by applying the Jury stability criterion. It is proved that the optimal consensus convergence rate with one-tap memory is faster than that without memory. In the case of M-tap memory, an iterative algorithm is given to derive the control parameters to accelerate the convergence rate. Moreover, the accelerated consensus with one-tap memory is extended to the form…
This paper is concerned with the leader-following consensus problem for a class of Lipschitz nonlinear multi-agent systems with uncertain dynamics, where each agent only transmits its noisy output, at discrete instants and independently from its neighbors. The proposed consensus protocol is based on a continuous-discrete time observer, which provides a continuous time estimation of the state of the neighbors from their discrete-time output measurements, together with a continuous control law. The stability of the multi-agent system is analyzed with a Lyapunov approach and the exponential pract…
Results for estimating the convergence rate of non-stationary distributed consensus algorithms are provided, on the basis of qualitative (mainly topological) as well as basic quantitative information (lower-bounds on the matrix entries). The results appear to be tight in a number of instances and are illustrated through simple as well as more sophisticated examples. The main idea is to follow propagation of information along certain spanning trees which arise in the communication graph.
In this paper, the fast consensus problem of high-order multi-agent systems under undirected topologies is considered. The direct link between the consensus convergence rate and the control gains is established. An accelerated consensus algorithm based on gradient descent is proposed to optimize the convergence rate. By applying the Routh-Hurwitz stability criterion, the lower bound on the convergence rate is derived, and explicit control gains are derived as the necessary condition to achieve the optimal convergence rate. Moreover, a protocol with time-varying control gains is designed to ach…
A theorem on (partial) convergence to consensus of multiagent systems is presented. It is proven with tools studying the convergence properties of products of row stochastic matrices with positive diagonals which are infinite to the left. Thus, it can be seen as a switching linear system in discrete time. It is further shown that the result is strictly more general than results of Moreau (IEEE Transactions on Automatic Control, vol. 50, no. 2, 2005), although Moreau's results are formulated for generally nonlinear updating maps. This is shown by demonstrating the existence of an appropriate sw…
We propose a weight design method to increase the convergence rate of distributed consensus. Prior work has focused on symmetric weight design due to computational tractability. We show that with proper choice of asymmetric weights, the convergence rate can be improved significantly over even the symmetric optimal design. In particular, we prove that the convergence rate in a lattice graph can be made independent of the size of the graph with asymmetric weights. We then use a Sturm-Liouville operator to approximate the graph Laplacian of more general graphs. A general weight design method is p…
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 converge…
Convergence analysis of consensus algorithms is revisited in the light of the Hilbert distance. Tsitsiklis Lyapunov function is shown to be the Hilbert distance to consensus in log coordinates. Birkhoff theorem, which proves contraction of the Hilbert metric for any positive homogeneous monotone map, provides an early yet general convergence result for consensus algorithms. Because Birkhoff theorem holds in arbitrary cones, we extend consensus algorithms to the cone of positive definite matrices. The proposed generalization finds applications in the convergence analysis of quantum stochastic m…
This paper investigates the weighted-averaging dynamic for unconstrained and constrained consensus problems. Through the use of a suitably defined adjoint dynamic, quadratic Lyapunov comparison functions are constructed to analyze the behavior of weighted-averaging dynamic. As a result, new convergence rate results are obtained that capture the graph structure in a novel way. In particular, the exponential convergence rate is established for unconstrained consensus with the exponent of the order of $1-O(1/(m\log_2m))$. Also, the exponential convergence rate is established for constrained conse…
In this paper, we consider minimizing a sum of local convex objective functions in a distributed setting, where the cost of communication and/or computation can be expensive. We extend and generalize the analysis for a class of nested gradient-based distributed algorithms (NEAR-DGD; Berahas, Bollapragada, Keskar and Wei, 2018) to account for multiple gradient steps at every iteration. We show the effect of performing multiple gradient 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 a fixed number of …
The objective of the paper is to establish a computable upper bound for the almost sure convergence rate for a class of ratio consensus algorithms defined via column-stochastic matrices. Our result extends the works of Iutzeler et al. (2013) on similar bounds that have been obtained in a more restrictive setup with limited conclusions. The present paper complements the results of Gerencsér and Gerencsér (2021), identifying the exact almost sure convergence rate of a wide class of ratio consensus algorithms in terms of a spectral gap, which is, however, not computable in general. The upper boun…
The present paper is devoted to estimating the speed of convergence towards consensus for a general class of discrete-time multi-agent systems. In the systems considered here, both the topology of the interconnection graph and the weight of the arcs are allowed to vary as a function of time. Under the hypothesis that some spanning tree structure is preserved along time, and that some nonzero minimal weight of the information transfer along this tree is guaranteed, an estimate of the contraction rate is given. The latter is expressed explicitly as the spectral radius of some matrix depending up…
We consider the problems of reaching average consensus and solving consensus-based optimization over unreliable communication networks wherein packets may be dropped accidentally during transmission. Existing work either assumes that the link failures affect the communication on both directions or that the message senders {\em know exactly}, in each iteration, how many of their outgoing links are functioning properly. In this paper, we consider directed links, and we {\em do not} require each node know its current outgoing degree. First, we propose and characterize the convergence rate of reac…
In this letter, we study the problem of accelerating reaching average consensus over connected graphs in a discrete-time communication setting. Literature has shown that consensus algorithms can be accelerated by increasing the graph connectivity or optimizing the weights agents place on the information received from their neighbors. In this letter instead of altering the communication graph, we investigate two methods that use buffered states to accelerate reaching average consensus over a given graph. In the first method, we study how convergence rate of the well-known first-order Laplacian …
Institut de Matemàtica Multidisciplinària, Universitat Politècnica de València（多学科数学研究所，瓦伦西亚理工大学）；Departament de Sistemes Informàtics i Computació, Universitat Politècnica de València（信息系统与计算系，瓦伦西亚理工大学）
This paper addresses the robust consensus problem under switching topologies. Contrary to existing methods, the proposed approach provides decentralized protocols that achieve consensus for networked multi-agent systems in a predefined time. Namely, the protocol design provides a tuning parameter that allows setting the convergence time of the agents to a consensus state. An appropriate Lyapunov analysis exposes the capability of the current proposal to achieve predefined-time consensus over switching topologies despite the presence of bounded perturbations. Finally, the paper presents a compa…
Gossip protocols are popular methods for average consensus problems in distributed computing. We prove new convergence guarantees for a variety of such protocols, including path, clique, and synchronous pairwise gossip. These arise by exploiting the connection between these protocols and the block randomized Kaczmarz method for solving linear systems. Moreover, we extend existing convergence results for block randomized Kaczmarz to allow for a more general choice of blocks, rank-deficient systems, and provide a tighter convergence rate guarantee. We furthermore apply this analysis to inconsist…
Consensus and synchronization in discrete-time networks of multi-agents with stochastically switching topologies and time delays
Fudan University（复旦大学）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学促进部数理研究所）；Santa Fe Institute for the Sciences of Complexity（圣塔菲复杂性科学研究所）
Distributed optimization algorithms have been studied extensively in the literature; however, underlying most algorithms is a linear consensus scheme, i.e. averaging variables from neighbors via doubly stochastic matrices. We consider nonlinear consensus schemes with a set of time-varying and agent-dependent monotonic Lipschitz nonlinear transformations, where we note the operation is a subprojection onto the consensus plane, which has been identified via stochastic approximation theory. For the proposed nonlinear consensus schemes, we establish convergence when combined with the NEXT algorith…
We study the popular distributed consensus method over networks composed of a number of densely connected clusters with a sparse connection between them. In these cluster networks, the method often constitutes two-time-scale dynamics, where the internal nodes within each cluster reach consensus quickly relative to the aggregate nodes across clusters. Our main contribution is to provide the rate of the distributed consensus method, which characterize explicitly the impacts of the internal and external graphs on the performance of this method. Our main result shows that this rate converges expon…
This paper presents new classes of consensus protocols with fixed-time convergence, which enable the definition of an upper bound for consensus state as a parameter of the consensus protocol, ensuring its independence from the initial condition of the nodes. We demonstrate that our methodology subsumes current classes of fixed-time consensus protocols that are based on homogeneous in the bi-limit vector fields. Moreover, the proposed framework enables for the development of independent consensus protocols that are not needed to be homogeneous in the bi-limit. This proposal offers extra degrees…
This paper presents a finite time convergence analysis for a decentralized stochastic approximation (SA) scheme. The scheme generalizes several algorithms for decentralized machine learning and multi-agent reinforcement learning. Our proof technique involves separating the iterates into their respective consensual parts and consensus error. The consensus error is bounded in terms of the stationarity of the consensual part, while the updates of the consensual part can be analyzed as a perturbed SA scheme. Under the Markovian noise and time varying communication graph assumptions, the decentrali…
We study the problem of achieving average consensus between a group of agents over a network with erasure links. In the context of consensus problems, the unreliability of communication links between nodes has been traditionally modeled by allowing the underlying graph to vary with time. In other words, depending on the realization of the link erasures, the underlying graph at each time instant is assumed to be a subgraph of the original graph. Implicit in this model is the assumption that the erasures are symmetric: if at time t the packet from node i to node j is dropped, the same is true fo…
We consider a linear consensus system with n agents that can switch between r different connectivity patterns. A natural question is which switching law yields the best (or worst) possible speed of convergence to consensus? We formulate this question in a rigorous manner by relaxing the switched system into a bilinear consensus control system, with the control playing the role of the switching law. A best (or worst) possible switching law then corresponds to an optimal control. We derive a necessary condition for optimality, stated in the form of a maximum principle (MP). Our approach, combine…
Multi-agent coordination algorithms with randomized interactions have seen use in a variety of settings in the multi-agent systems literature. In some cases, these algorithms can be random by design, as in a gossip-like algorithm, and in other cases they are random due to external factors, as in the case of intermittent communications. Targeting both of these scenarios, we present novel convergence rate estimates for consensus problems solved over random graphs. Established results provide asymptotic convergence in this setting, and we provide estimates of the rate of convergence in two forms.…
We present a results about convergence of products of row-stochastic matrices which are infinite to the left and all have positive diagonals. This is regarded as in inhomogeneous consensus process where confidence weights may change in every time step but where each agent has a little bit of self confidence. The positive diagonal leads to a fixed zero pattern in certain subproducts of the infinite product. We discuss the use of the joint spectral radius on the set of the evolving subproducts and conditions on the subprodutcs to ensure convergence of parts of the infinite product to fixed rank-…
We apply the Min-Sum message-passing protocol to solve the consensus problem in distributed optimization. We show that while the ordinary Min-Sum algorithm does not converge, a modified version of it known as Splitting yields convergence to the problem solution. We prove that a proper choice of the tuning parameters allows Min-Sum Splitting to yield subdiffusive accelerated convergence rates, matching the rates obtained by shift-register methods. The acceleration scheme embodied by Min-Sum Splitting for the consensus problem bears similarities with lifted Markov chains techniques and with mult…
We revisit an algorithm for distributed consensus optimization proposed in 2010 by J. Wang and N. Elia. By means of a Lyapunov-based analysis, we prove input-to-state stability of the algorithm relative to a closed invariant set composed of optimal equilibria and with respect to perturbations affecting the algorithm's dynamics. In the absence of perturbations, this result implies linear convergence of the local estimates and Lyapunov stability of the optimal steady state. Moreover, we unveil fundamental connections with the well-known Gradient Tracking and with distributed integral control. Ov…
Previous researches have shown that adding local memory can accelerate the consensus. It is natural to ask questions like what is the fastest rate achievable by the $M$-tap memory acceleration, and what are the corresponding control parameters. This paper introduces a set of effective and previously unused techniques to analyze the convergence rate of accelerated consensus with $M$-tap memory of local nodes and to design the control protocols. These effective techniques, including the Kharitonov stability theorem, the Routh stability criterion and the robust stability margin, have led to the f…
Optimal transport has been used extensively in resource matching to promote the efficiency of resources usages by matching sources to targets. However, it requires a significant amount of computations and storage spaces for large-scale problems. In this paper, we take a consensus-based approach to decentralize discrete optimal transport problems and develop fully distributed algorithms with alternating direction method of multipliers. We show that our algorithms guarantee certain levels of efficiency and privacy besides the distributed nature. We further derive primal and dual algorithms by ex…
In this paper, a distributed velocity-constrained consensus problem is studied for discrete-time multi-agent systems, where each agent's velocity is constrained to lie in a nonconvex set. A distributed constrained control algorithm is proposed to enable all agents to converge to a common point using only local information. {The gains of the algorithm for all agents need not to be the same or predesigned and can be adjusted by each agent itself based on its own and neighbors' information.} It is shown that the algorithm is robust to arbitrarily bounded communication delays and arbitrarily switc…
From Consensus-Based Optimization to Evolution Strategies: Proof of Global Convergence
Technical University of Munich（慕尼黑工业大学）；Munich Center for Machine Learning (MCML)（慕尼黑机器学习中心）；Munich Data Science Institute (MDSI)（慕尼黑数据科学研究所）；Hunan University（湖南大学）；University of Florence（佛罗伦萨大学）
In this paper, an optimal output consensus problem is studied for discrete-time linear multiagent systems subject to external disturbances. Each agent is assigned with a local cost function which is known only to itself. Distributed protocols are to be designed to guarantee an output consensus for these high-order agents and meanwhile minimize the aggregate cost as the sum of these local costs. To overcome the difficulties brought by high-order dynamics and external disturbances, we develop an embedded design and constructively present a distributed rule to solve this problem. The proposed con…
This paper establishes novel results regarding the global convergence properties of a large class of consensus protocols for multi-agent systems that evolve in continuous time on the n-dimensional unit sphere or n-sphere. For any connected, undirected graph and all n in N\{1}, each protocol in said class is shown to yield almost global consensus. The feedback laws are negative gradients of Lyapunov functions and one instance generates the canonical intrinsic gradient descent protocol. This convergence result sheds new light on the general problem of consensus on Riemannian manifolds; the n-sph…
