The push-sum algorithm is probably the most important distributed averaging approach over directed graphs, which has been applied to various problems including distributed optimization. This paper establishes the explicit absolute probability sequence for the push-sum algorithm, and based on which, constructs quadratic Lyapunov functions for push-sum based distributed optimization algorithms. As illustrative examples, the proposed novel analysis tool can improve the convergence rates of the subgradient-push and stochastic gradient-push, two important algorithms for distributed convex optimization over unbalanced directed graphs. Specifically, the paper proves that the subgradient-push algorithm converges at a rate of $O(1/\sqrt{t})$ for general convex functions and stochastic gradient-push algorithm converges at a rate of $O(1/t)$ for strongly convex functions, over time-varying unbalanced directed graphs. Both rates are respectively the same as the state-of-the-art rates of their single-agent counterparts and thus optimal, which closes the theoretical gap between the centralized and push-sum based (sub)gradient methods. The paper further proposes a heterogeneous push-sum based subgradient algorithm in which each agent can arbitrarily switch between subgradient-push and push-subgradient. The heterogeneous algorithm thus subsumes both subgradient-push and push-subgradient as special cases, and still converges to an optimal point at an optimal rate. The proposed tool can also be extended to analyze distributed weighted averaging.
The paper proposes a heterogeneous push-sum based subgradient algorithm for multi-agent distributed convex optimization in which each agent can arbitrarily switch between subgradient-push and push-subgradient at each time. It is shown that the heterogeneous algorithm converges to an optimal point at an optimal rate over time-varying directed graphs.
Decentralized optimization problems consist of multiple agents connected by a network. The agents have each local cost function, and the goal is to minimize the sum of the functions cooperatively. It requires the agents communicate with each other, and reducing the cost for communication is desired for a communication-limited environment. In this work, we propose a gradient-push algorithm involving event-triggered communication on directed network. Each agent sends its state information to its neighbors only when the difference between the latest sent state and the current state is larger than…
We propose an algorithm for average consensus over a directed graph which is both fully asynchronous and robust to unreliable communications. We show its convergence to the average, while allowing for slowly growing but potentially unbounded communication failures.
In this paper, we study the problem of achieving average consensus over a random time-varying sequence of directed graphs by extending the class of so-called push-sum algorithms to such random scenarios. Provided that an ergodicity notion, which we term the directed infinite flow property, holds and the auxiliary states of agents are uniformly bounded away from zero infinitely often, we prove the almost sure convergence of the evolutions of this class of algorithms to the average of initial states. Moreover, for a random sequence of graphs generated using a time-varying B-irreducible probabili…
In this paper, we study the distributed optimization problem for a system of agents embedded in time-varying directed communication networks. Each agent has its own cost function and agents cooperate to determine the global decision that minimizes the summation of all individual cost functions. We consider the so-called push-pull gradient-based algorithm (termed as AB/Push-Pull) which employs both row- and column-stochastic weights simultaneously to track the optimal decision and the gradient of the global cost while ensuring consensus and optimality. We show that the algorithm converges linea…
Facebook AI Research（Facebook人工智能研究院）；McGill University（麦吉尔大学）；University of Edinburgh（爱丁堡大学）
Distributed data-parallel algorithms aim to accelerate the training of deep neural networks by parallelizing the computation of large mini-batch gradient updates across multiple nodes. Approaches that synchronize nodes using exact distributed averaging (e.g., via AllReduce) are sensitive to stragglers and communication delays. The PushSum gossip algorithm is robust to these issues, but only performs approximate distributed averaging. This paper studies Stochastic Gradient Push (SGP), which combines PushSum with stochastic gradient updates. We prove that SGP converges to a stationary point of s…
This paper considers the problem of distributed optimization over time-varying graphs. For the case of undirected graphs, we introduce a distributed algorithm, referred to as DIGing, based on a combination of a distributed inexact gradient method and a gradient tracking technique. The DIGing algorithm uses doubly stochastic mixing matrices and employs fixed step-sizes and, yet, drives all the agents' iterates to a global and consensual minimizer. When the graphs are directed, in which case the implementation of doubly stochastic mixing matrices is unrealistic, we construct an algorithm that in…
The distributed convex optimization problem over the multi-agent system is considered in this paper, and it is assumed that each agent possesses its own cost function and communicates with its neighbours over a sequence of time-varying directed graphs. However, due to some reasons there exist communication delays while agents receive information from other agents, and we are going to seek the optimal value of the sum of agents' loss functions in this case. We desire to handle this problem with the push-sum distributed dual averaging (PS-DDA) algorithm. It is proved that this algorithm converge…
In this paper, we focus on solving a distributed convex optimization problem in a network, where each agent has its own convex cost function and the goal is to minimize the sum of the agents' cost functions while obeying the network connectivity structure. In order to minimize the sum of the cost functions, we consider a new distributed gradient-based method where each node maintains two estimates, namely, an estimate of the optimal decision variable and an estimate of the gradient for the average of the agents' objective functions. From the viewpoint of an agent, the information about the dec…
In this paper, we consider the problem of distributed consensus optimization over multi-agent networks with directed network topology. Assuming each agent has a local cost function that is smooth and strongly convex, the global objective is to minimize the average of all the local cost functions. To solve the problem, we introduce a robust gradient tracking method (R-Push-Pull) adapted from the recently proposed Push-Pull/AB algorithm. R-Push-Pull inherits the advantages of Push-Pull and enjoys linear convergence to the optimal solution with exact communication. Under noisy information exchang…
We consider the standard model of distributed optimization of a sum of functions $F(\bz) = \sum_{i=1}^n f_i(\bz)$, where node $i$ in a network holds the function $f_i(\bz)$. We allow for a harsh network model characterized by asynchronous updates, message delays, unpredictable message losses, and directed communication among nodes. In this setting, we analyze a modification of the Gradient-Push method for distributed optimization, assuming that \begin{enumerate*}[label=(\roman*)] \item node $i$ is capable of generating gradients of its function $f_i(\bz)$ corrupted by zero-mean bounded-support…
In this paper, we focus on solving a distributed convex optimization problem in a network, where each agent has its own convex cost function and the goal is to minimize the sum of the agents' cost functions while obeying the network connectivity structure. In order to minimize the sum of the cost functions, we consider new distributed gradient-based methods where each node maintains two estimates, namely, an estimate of the optimal decision variable and an estimate of the gradient for the average of the agents' objective functions. From the viewpoint of an agent, the information about the grad…
We propose a new belief update rule for Distributed Non-Bayesian learning in time-varying directed graphs, where a group of agents tries to collectively identify a hypothesis that best describes a sequence of observed data. We show that the proposed update rule, inspired by the Push-Sum algorithm, is consistent, moreover we provide an explicit characterization of its convergence rate. Our main result states that, after a transient time, all agents will concentrate their beliefs at a network independent rate. Network independent rates were not available for other consensus based distributed lea…
This paper considers a distributed convex optimization problem over a time-varying multi-agent network, where each agent has its own decision variables that should be set so as to minimize its individual objective subject to local constraints and global coupling equality constraints. Over directed graphs, a distributed algorithm is proposed that incorporates the push-sum protocol into dual subgradient methods. Under the convexity assumption, the optimality of primal and dual variables, and constraint violations is first established. Then the explicit convergence rates of the proposed algorithm…
In this paper, we propose a framework under which the decentralized optimization algorithms suggested in \cite{JKJJ,MA, NO,NO2} can be treated in a unified manner. More precisely, we show that the distributed subgradient descent algorithms \cite{JKJJ, NO}, the subgradient-push algorithm \cite{NO2}, and the distributed algorithm with row-stochastic matrix \cite{MA} can be derived by making suitable choices of consensus matrices, step-size and subgradient from the decentralized subgradient descent proposed in \cite{NO}. As a result of such unified understanding, we provide a convergence proof th…
Motivated by emerging applications in wireless sensor networks and large-scale data processing, we consider distributed optimization over directed networks where the agents communicate their information locally to their neighbors to cooperatively minimize a global cost function. We introduce a new unifying distributed constrained optimization model that is characterized as a bilevel optimization problem. This model captures a wide range of existing problems over directed networks including: (i) Distributed optimization with linear constraints; (ii) Distributed unconstrained nonstrongly convex …
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 presents a special type of distributed optimization problems, where the summation of agents' local cost functions (i.e., global cost function) is convex, but each individual can be non-convex. Unlike most distributed optimization algorithms by taking the advantages of gradient, the considered problem is allowed to be non-smooth, and the gradient information is unknown to the agents. To solve the problem, a Gaussian-smoothing technique is introduced and a gradient-free method is proposed. We prove that each agent's iterate approximately converges to the optimal solution both with pro…
This paper investigates the distributed online optimization problem over a multi-agent network subject to local set constraints and coupled inequality constraints, which has a lot of applications in many areas, such as wireless sensor networks, power systems and plug-in electric vehicles. In this problem, the cost function at each time step is the sum of local cost functions with each of them being gradually revealed to its corresponding agent, and meanwhile only local functions in coupled inequality constraints are accessible to each agent. To address this problem, a modified primal-dual algo…
In this paper, we consider a distributed learning problem in a subnetwork zero-sum game, where agents are competing in different subnetworks. These agents are connected through time-varying graphs where each agent has its own cost function and can receive information from its neighbors. We propose a distributed mirror descent algorithm for computing a Nash equilibrium and establish a sublinear regret bound on the sequence of iterates when the graphs are uniformly strongly connected and the cost functions are convex-concave. Moreover, we prove its convergence with suitably selected diminishing …
We consider the distributed optimization problem for the sum of convex functions where the underlying communications network connecting agents at each time is drawn at random from a collection of directed graphs. Building on an earlier work [15], where a modified version of the subgradient-push algorithm is shown to be almost surely convergent to an optimizer on sequences of random directed graphs, we find an upper bound of the order of $\sim O(\frac{1}{\sqrt{t}})$ on the convergence rate of our proposed algorithm, establishing the first convergence bound in such random settings.
This paper investigates the distributed Nash equilibrium seeking problem for two-network zero-sum games with set constraints, where the two networks have the opposite nonsmooth cost functions. The interaction of the agents in each network is characterized by an unbalanced directed graph. We are particularly interested in the case that the local cost function of each agent in the two networks is unknown in this paper. We first construct the stochastic subgradient-free two-variable oracles based on the measurements of the local cost functions. Instead of using subgradients of the local cost func…
We study distributed non-convex optimization on a time-varying multi-agent network. Each node has access to its own smooth local cost function, and the collective goal is to minimize the sum of these functions. We generalize the results obtained previously to the case of non-convex functions. Under some additional technical assumptions on the gradients we prove the convergence of the distributed push-sum algorithm to some critical point of the objective function. By utilizing perturbations on the update process, we show the almost sure convergence of the perturbed dynamics to a local minimum o…
In this paper, we study the problem of consensus-based distributed optimization where a network of agents, abstracted as a directed graph, aims to minimize the sum of all agents' cost functions collaboratively. In existing distributed optimization approaches (Push-Pull/AB) for directed graphs, all agents exchange their states with neighbors to achieve the optimal solution with a constant stepsize, which may lead to the disclosure of sensitive and private information. For privacy preservation, we propose a novel state-decomposition based gradient tracking approach (SD-Push-Pull) for distributed…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
We present a distributed solution to optimizing a convex function composed of several non-convex functions. Each non-convex function is privately stored with an agent while the agents communicate with neighbors to form a network. We show that coupled consensus and projected gradient descent algorithm proposed in [1] can optimize convex sum of non-convex functions under an additional assumption on gradient Lipschitzness. We further discuss the applications of this analysis in improving privacy in distributed optimization.
This paper studies the application of the blended dynamics approach towards distributed optimization problem where the global cost function is given by a sum of local cost functions. The benefits include (i) individual cost function need not be convex as long as the global cost function is strongly convex and (ii) the convergence rate of the distributed algorithm is arbitrarily close to the convergence rate of the centralized one. Two particular continuous-time algorithms are presented using the proportional-integral-type couplings. One has benefit of `initialization-free,' so that agents can …
Beijing Institute of Technology（北京理工大学）；Beijing Institute of Technology Chongqing Innovation Center（北京理工大学重庆创新中心）；Tongji University（同济大学）；Nanyang Technological University（南洋理工大学）
The non-smooth finite-sum minimization is a fundamental problem in machine learning. This paper develops a distributed stochastic proximal-gradient algorithm with random reshuffling to solve the finite-sum minimization over time-varying multi-agent networks. The objective function is a sum of differentiable convex functions and non-smooth regularization. Each agent in the network updates local variables with a constant step-size by local information and cooperates to seek an optimal solution. We prove that local variable estimates generated by the proposed algorithm achieve consensus and are a…
In this work, we study the problem of minimizing the sum of strongly convex functions split over a network of $n$ nodes. We propose the decentralized and asynchronous algorithm ADFS to tackle the case when local functions are themselves finite sums with $m$ components. ADFS converges linearly when local functions are smooth, and matches the rates of the best known finite sum algorithms when executed on a single machine. On several machines, ADFS enjoys a $O (\sqrt{n})$ or $O(n)$ speed-up depending on the leading complexity term as long as the diameter of the network is not too big with respect…
This paper considers nonconvex distributed constrained optimization over networks, modeled as directed (possibly time-varying) graphs. We introduce the first algorithmic framework for the minimization of the sum of a smooth nonconvex (nonseparable) function--the agent's sum-utility--plus a Difference-of-Convex (DC) function (with nonsmooth convex part). This general formulation arises in many applications, from statistical machine learning to engineering. The proposed distributed method combines successive convex approximation techniques with a judiciously designed perturbed push-sum consensus…
Aalto University（阿尔托大学）；Semnan University（塞姆南大学）；University of Melbourne（墨尔本大学）；Georgia State University（佐治亚州立大学）；University of Cyprus（塞浦路斯大学）
Motivated by recent development in networking and parallel data-processing, we consider a distributed and localized finite-sum (or fixed-sum) allocation technique to solve resource-constrained convex optimization problems over multi-agent networks (MANs). Such networks include (smart) agents representing an intelligent entity capable of communication, processing, and decision-making. In particular, we consider problems subject to practical nonlinear constraints on the dynamics of the agents in terms of their communications and actuation capabilities (referred to as the node dynamics), e.g., ne…
Tsinghua University（清华大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
This paper proposes a \emph{fully asynchronous} scheme for the policy evaluation problem of distributed reinforcement learning (DisRL) over directed peer-to-peer networks. Without waiting for any other node of the network, each node can locally update its value function at any time by using (possibly delayed) information from its neighbors. This is in sharp contrast to the gossip-based scheme where a pair of nodes concurrently update. Though the fully asynchronous setting involves a difficult multi-timescale decision problem, we design a novel stochastic average gradient (SAG) based distribute…
