We consider strongly convex distributed consensus optimization over connected networks. EFIX, the proposed method, is derived using quadratic penalty approach. In more detail, we use the standard reformulation { transforming the original problem into a constrained problem in a higher dimensional space { to define a sequence of suitable quadratic penalty subproblems with increasing penalty parameters. For quadratic objectives, the corresponding sequence consists of quadratic penalty subproblems. For the generic strongly convex case, the objective function is approximated with a quadratic model and hence the sequence of the resulting penalty subproblems is again quadratic. EFIX is then derived by solving each of the quadratic penalty subproblems via a fixed point (R)-linear solver, e.g., Jacobi Over-Relaxation method. The exact convergence is proved as well as the worst case complexity of order O(epsilon^-1) for the quadratic case. In the case of strongly convex generic functions, a standard result for penalty methods is obtained. Numerical results indicate that the method is highly competitive with state-of-the-art exact first order methods, requires smaller computational and communication effort, and is robust to the choice of algorithm parameters.
Since the initial proposal in the late 80s, spectral gradient methods continue to receive significant attention, especially due to their excellent numerical performance on various large scale applications. However, to date, they have not been sufficiently explored in the context of distributed optimization. In this paper, we consider unconstrained distributed optimization problems where $n$ nodes constitute an arbitrary connected network and collaboratively minimize the sum of their local convex cost functions. In this setting, building from existing exact distributed gradient methods, we prop…
This work develops a distributed optimization strategy with guaranteed exact convergence for a broad class of left-stochastic combination policies. The resulting exact diffusion strategy is shown in Part II to have a wider stability range and superior convergence performance than the EXTRA strategy. The exact diffusion solution is applicable to non-symmetric left-stochastic combination matrices, while many earlier developments on exact consensus implementations are limited to doubly-stochastic matrices; these latter matrices impose stringent constraints on the network topology. The derivation …
Part I of this work [2] developed the exact diffusion algorithm to remove the bias that is characteristic of distributed solutions for deterministic optimization problems. The algorithm was shown to be applicable to a larger set of combination policies than earlier approaches in the literature. In particular, the combination matrices are not required to be doubly stochastic, which impose stringent conditions on the graph topology and communications protocol. In this Part II, we examine the convergence and stability properties of exact diffusion in some detail and establish its linear convergen…
We study the problem of minimizing a sum of local objective convex functions over a network of processors/agents. This problem naturally calls for distributed optimization algorithms, in which the agents cooperatively solve the problem through local computations and communications with neighbors. While many of the existing distributed algorithms with constant stepsize can only converge to a neighborhood of optimal solution, some recent methods based on augmented Lagrangian and method of multipliers can achieve exact convergence with a fixed stepsize. However, these methods either suffer from s…
In this paper, a gradient-free distributed algorithm is introduced to solve a set constrained optimization problem under a directed communication network. Specifically, at each time-step, the agents locally compute a so-called pseudo-gradient to guide the updates of the decision variables, which can be applied in the fields where the gradient information is unknown, not available or non-existent. A surplus-based method is adopted to remove the doubly stochastic requirement on the weighting matrix, which enables the implementation of the algorithm in graphs having no associated doubly stochasti…
This paper develops column partition based distributed schemes for a class of large-scale convex sparse optimization problems, e.g., basis pursuit (BP), LASSO, basis pursuit denosing (BPDN), and their extensions, e.g., fused LASSO. We are particularly interested in the cases where the number of (scalar) decision variables is much larger than the number of (scalar) measurements, and each agent has limited memory or computing capacity such that it only knows a small number of columns of a measurement matrix. These problems in consideration are densely coupled and cannot be formulated as separabl…
Linear Convergence Rate Analysis of a Class of Exact First-Order Distributed Methods for Weight-Balanced Time-Varying Networks and Uncoordinated Step Sizes
This paper proposes a distributed algorithm for a network of agents to solve an optimization problem with separable objective function and locally coupled constraints. Our strategy is based on reformulating the original constrained problem as the unconstrained optimization of a smooth (continuously differentiable) exact penalty function. Computing the gradient of this penalty function in a distributed way is challenging even under the separability assumptions on the original optimization problem. Our technical approach shows that the distributed computation problem for the gradient can be form…
In this work, we propose an algorithm for solving exact sparse linear regression problems over a network in a distributed manner. Particularly, we consider the problem where data is stored among different computers or agents that seek to collaboratively find a common regressor with a specified sparsity k, i.e., the L0-norm is less than or equal to k. Contrary to existing literature that uses L1 regularization to approximate sparseness, we solve the problem with exact sparsity k. The main novelty in our proposal lies in showing a problem formulation with zero duality gap for which we adopt a du…
Recently, there has been significant progress in the development of distributed first order methods. (At least) two different types of methods, designed from very different perspectives, have been proposed that achieve both exact and linear convergence when a constant step size is used -- a favorable feature that was not achievable by most prior methods. In this paper, we unify, generalize, and improve convergence speed of these exact distributed first order methods. We first carry out a novel unifying analysis that sheds light on how the different existing methods compare. The analysis reveal…
Principal Component Analysis (PCA) is a fundamental data preprocessing tool in the world of machine learning. While PCA is often thought of as a dimensionality reduction method, the purpose of PCA is actually two-fold: dimension reduction and uncorrelated feature learning. Furthermore, the enormity of the dimensions and sample size in the modern day datasets have rendered the centralized PCA solutions unusable. In that vein, this paper reconsiders the problem of PCA when data samples are distributed across nodes in an arbitrarily connected network. While a few solutions for distributed PCA exi…
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…
MUSIC: Accelerated Convergence for Distributed Optimization With Inexact and Exact Methods
School of Computer Science and Technology, Hubei University of Science and Technology（湖北科技学院计算机科学与技术学院）；School of Electronic and Electrical Engineering, Wuhan Textile University（武汉纺织大学电子与电气工程学院）；The University of Manchester（曼彻斯特大学）
We consider minimizing a sum of non-smooth objective functions with set constraints in a distributed manner. As to this problem, we propose a distributed algorithm with an exponential convergence rate for the first time. By the exact penalty method, we reformulate the problem equivalently as a standard distributed one without consensus constraints. Then we design a distributed projected subgradient algorithm with the help of differential inclusions. Furthermore, we show that the algorithm converges to the optimal solution exponentially for strongly convex objective functions.
Optimal Design of Distributed Inexact Gradient Tracking: Exact Worst-case Convergence Rate and Explicit Parameters
State Key Laboratory of Industrial Control Technology and the College of Control Science and Engineering, Zhejiang University（浙江大学工业控制技术国家重点实验室与控制科学学院）
In this paper, we consider distributed Nash equilibrium seeking in monotone and hypomonotone games. We first assume that each player has knowledge of the opponents' decisions and propose a passivity-based modification of the standard gradient-play dynamics, that we call "Heavy Anchor". We prove that Heavy Anchor allows a relaxation of strict monotonicity of the pseudo-gradient, needed for gradient-play dynamics, and can ensure exact asymptotic convergence in merely monotone regimes. We extend these results to the setting where each player has only partial information of the opponents' decision…
We consider a standard distributed consensus optimization problem where a set of agents connected over an undirected network minimize the sum of their individual local strongly convex costs. Alternating Direction Method of Multipliers ADMM and Proximal Method of Multipliers PMM have been proved to be effective frameworks for design of exact distributed second order methods involving calculation of local cost Hessians. However, existing methods involve explicit calculation of local Hessian inverses at each iteration that may be very costly when the dimension of the optimization variable is larg…
This paper investigates distributed online convex optimization in the presence of an aggregative variable without any global/central coordinators over a multi-agent network, where each individual agent is only able to access partial information of time-varying global loss functions, thus requiring local information exchanges between neighboring agents. Motivated by many applications in reality, the considered local loss functions depend not only on their own decision variables, but also on an aggregative variable, such as the average of all decision variables. To handle this problem, an Online…
In this paper, we propose two exact distributed algorithms to solve mixed integer linear programming (MILP) problems with multiple agents where data privacy is important for the agents. A key challenge is that, because of the non-convex nature of MILPs, classical distributed and decentralized optimization approaches cannot be applied directly to find their optimal solutions. The proposed exact algorithms are based on adding primal cuts and restricting the Lagrangian relaxation of the original MILP problem. We show finite convergence of these algorithms for MILPs with only binary and continuous…
This paper solves the Sylvester equation in the form of AX+XB=C in a distributed way, and proposes three distributed continuous-time algorithms for three cases. We start with the basic algorithm for solving a least squares solution of the equation, and then give a simplified algorithm for the case when there is an exact solution to the equation, followed by an algorithm with regularization case. Based on local information and appropriate communication among neighbor agents, we solve the distributed computation problem of the Sylvester equation from the optimization viewpoint, and we prove the …
We consider distributed optimization methods for problems where forming the Hessian is computationally challenging and communication is a significant bottleneck. We leverage randomized sketches for reducing the problem dimensions as well as preserving privacy and improving straggler resilience in asynchronous distributed systems. We derive novel approximation guarantees for classical sketching methods and establish tight concentration results that serve as both upper and lower bounds on the error. We then extend our analysis to the accuracy of parameter averaging for distributed sketches. Furt…
Game theory has emerged as a fruitful paradigm for the design of networked multiagent systems. A fundamental component of this approach is the design of agents' utility functions so that their self-interested maximization results in a desirable collective behavior. In this work we focus on a well-studied class of distributed resource allocation problems where each agent is requested to select a subset of resources with the goal of optimizing a given system-level objective. Our core contribution is the development of a novel framework to tightly characterize the worst case performance of any re…
Certified Residual Quasi-Newton Methods for Distributed Variational Inequalities
Moscow Institute of Physics and Technology（莫斯科物理技术学院）；FusionBrain Lab（FusionBrain实验室）；Innopolis University（伊诺波利斯大学）；Mohamed bin Zayed University of Artificial Intelligence（穆罕默德·本·扎耶德人工智能大学）
We consider a distributed multi-agent optimization problem over a time-invariant undirected graph, where each agent possesses a local objective function and all agents collaboratively minimize the average of all objective functions through local computations and communications among neighbors. Recently, a class of distributed gradient methods has been proposed that achieves both exact and geometric convergence when a constant step size is used. The geometric convergence of these methods is ensured for conservatively selected step sizes, but how to choose an appropriate step size while running …
This paper develops a distributed primal-dual algorithm via event-triggered mechanism to solve a class of convex optimization problems subject to local set constraints, coupled equality and inequality constraints. Different from some existing distributed algorithms with the diminishing step-sizes, our algorithm uses the constant step-sizes, and is shown to achieve an exact convergence to an optimal solution with an $O(1/k)$ convergence rate for general convex objective functions, where $k>0$ is the iteration number. Moreover, by applying event-triggered communication mechanism, the proposed al…
In this work, we consider the distributed optimization problem in which each node has its own convex cost function and can communicate directly only with its neighbors, as determined by a directed communication topology (directed graph or digraph). First, we reformulate the optimization problem so that Alternating Direction Method of Multipliers (ADMM) can be utilized. Then, we propose an algorithm, herein called Distributed Alternating Direction Method of Multipliers using Finite-Time Exact Ratio Consensus (D-ADMM-FTERC), to solve the multi-node convex optimization problem, in which every nod…
Distributed linear algebraic equation over networks, where nodes hold a part of problem data and cooperatively solve the equation via node-to-node communications, is a basic distributed computation task receiving an increasing research attention. Communications over a network have a stochastic nature, with both temporal and spatial dependence due to link failures, packet dropouts or node recreation, etc. In this paper, we study the convergence and convergence rate of distributed linear equation protocols over a $\ast$-mixing random network, where the temporal and spatial dependencies between t…
This paper studies the distributed optimization problem when the objective functions might be nondifferentiable and subject to heterogeneous set constraints. Unlike existing subgradient methods, we focus on the case when the exact subgradients of the local objective functions can not be accessed by the agents. To solve this problem, we propose a projected primal-dual dynamics using only the objective function's approximate subgradients. We first prove that the formulated optimization problem can generally be solved with an error depending upon the accuracy of the available subgradients. Then, …
We consider distributed smooth nonconvex unconstrained optimization over networks, modeled as a connected graph. We examine the behavior of distributed gradient-based algorithms near strict saddle points. Specifically, we establish that (i) the renowned Distributed Gradient Descent (DGD) algorithm likely converges to a neighborhood of a Second-order Stationary (SoS) solution; and (ii) the more recent class of distributed algorithms based on gradient tracking--implementable also over digraphs--likely converges to exact SoS solutions, thus avoiding (strict) saddle-points. Furthermore, new conver…
In this paper, we study the distributed optimization problem using approximate first-order information. We suppose the agent can repeatedly call an inexact first-order oracle of each individual objective function and exchange information with its time-varying neighbors. We revisit the distributed subgradient method in this circumstance and show its suboptimality under square summable but not summable step sizes. We also present several conditions on the inexactness of the local oracles to ensure an exact convergence of the iterative sequences towards the global optimal solution. A numerical ex…
In this paper, we study secure distributed optimization against arbitrary gradient attack in multi-agent networks. In distributed optimization, there is no central server to coordinate local updates, and each agent can only communicate with its neighbors on a predefined network. We consider the scenario where out of $n$ networked agents, a fixed but unknown fraction $ρ$ of the agents are under arbitrary gradient attack in that their stochastic gradient oracles return arbitrary information to derail the optimization process, and the goal is to minimize the sum of local objective functions on un…
Driven by the need to solve increasingly complex optimization problems in signal processing and machine learning, there has been increasing interest in understanding the behavior of gradient-descent algorithms in non-convex environments. Most available works on distributed non-convex optimization problems focus on the deterministic setting where exact gradients are available at each agent. In this work and its Part II, we consider stochastic cost functions, where exact gradients are replaced by stochastic approximations and the resulting gradient noise persistently seeps into the dynamics of t…
This document describes our freely distributed Maple library {\sc spectra}, for Semidefinite Programming solved Exactly with Computational Tools of Real Algebra. It solves linear matrix inequalities with symbolic computation in exact arithmetic and it is targeted to small-size, possibly degenerate problems for which symbolic infeasibility or feasibility certificates are required.
KTH Royal Institute of Technology（瑞典皇家理工学院）；School of Electrical Engineering and Computer Science, and Digital Futures, KTH Royal Institute of Technology（电气工程和计算机科学与数字未来学院，瑞典皇家理工学院）
Distributed Optimization via Energy Conservation Laws in Dilated Coordinates
Data & Decision Sciences division of Tata Consultancy Services Research（塔塔咨询公司数据与决策科学部门）；Faculty of Systems & Control group, Indian Institute of Technology Bombay（印度理工学院孟买分校系统与控制学院）
University of California, Santa Barbara（加州大学圣巴巴拉分校）；University of Texas at Austin（得克萨斯大学奥斯汀分校）；University of Pennsylvania（宾夕法尼亚大学）
We consider the problem of decentralized consensus optimization, where the sum of $n$ smooth and strongly convex functions are minimized over $n$ distributed agents that form a connected network. In particular, we consider the case that the communicated local decision variables among nodes are quantized in order to alleviate the communication bottleneck in distributed optimization. We propose the Quantized Decentralized Gradient Descent (QDGD) algorithm, in which nodes update their local decision variables by combining the quantized information received from their neighbors with their local in…
