Achieving Geometric Convergence for Distributed Optimization over Time-Varying Graphs

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 incorporates the push-sum protocol into the DIGing structure, thus obtaining Push-DIGing algorithm. The Push-DIGing uses column stochastic matrices and fixed step-sizes, but it still converges to a global and consensual minimizer. Under the strong convexity assumption, we prove that the algorithms converge at R-linear (geometric) rates as long as the step-sizes do not exceed some upper bounds. We establish explicit estimates for the convergence rates. When the graph is undirected it shows that DIGing scales polynomially in the number of agents. We also provide some numerical experiments to demonstrate the efficacy of the proposed algorithms and to validate our theoretical findings

Improved Convergence Rates for Distributed Resource Allocation

In this paper, we develop a class of decentralized algorithms for solving a convex resource allocation problem in a network of $n$ agents, where the agent objectives are decoupled while the resource constraints are coupled. The agents communicate over a connected undirected graph, and they want to collaboratively determine a solution to the overall network problem, while each agent only communicates with its neighbors. We first study the connection between the decentralized resource allocation problem and the decentralized consensus optimization problem. Then, using a class of algorithms for solving consensus optimization problems, we propose a novel class of decentralized schemes for solving resource allocation problems in a distributed manner. Specifically, we first propose an algorithm for solving the resource allocation problem with an $o(1/k)$ convergence rate guarantee when the agents' objective functions are generally convex (could be nondifferentiable) and per agent local convex constraints are allowed; We then propose a gradient-based algorithm for solving the resource allocation problem when per agent local constraints are absent and show that such scheme can achieve geometric rate when the objective functions are strongly convex and have Lipschitz continuous gradients. We have also provided scalability/network dependency analysis. Based on these two algorithms, we have further proposed a gradient projection-based algorithm which can handle smooth objective and simple constraints more efficiently. Numerical experiments demonstrates the viability and performance of all the proposed algorithms

Accelerated Consensus via Min-Sum Splitting

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 multi-step first order methods in convex optimization

A Robust Gradient Tracking Method for Distributed Optimization over Directed Networks

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 exchange, R-Push-Pull is more robust than the existing gradient tracking based algorithms; the solutions obtained by each agent reach a neighborhood of the optimum in expectation exponentially fast under a constant stepsize policy. We provide a numerical example that demonstrate the effectiveness of R-Push-Pull

Distributed Nesterov gradient methods over arbitrary graphs

In this letter, we introduce a distributed Nesterov method, termed as $\mathcal{ABN}$, that does not require doubly-stochastic weight matrices. Instead, the implementation is based on a simultaneous application of both row- and column-stochastic weights that makes this method applicable to arbitrary (strongly-connected) graphs. Since constructing column-stochastic weights needs additional information (the number of outgoing neighbors at each agent), not available in certain communication protocols, we derive a variation, termed as FROZEN, that only requires row-stochastic weights but at the expense of additional iterations for eigenvector learning. We numerically study these algorithms for various objective functions and network parameters and show that the proposed distributed Nesterov methods achieve acceleration compared to the current state-of-the-art methods for distributed optimization

FROST -- Fast row-stochastic optimization with uncoordinated step-sizes

In this paper, we discuss distributed optimization over directed graphs, where doubly-stochastic weights cannot be constructed. Most of the existing algorithms overcome this issue by applying push-sum consensus, which utilizes column-stochastic weights. The formulation of column-stochastic weights requires each agent to know (at least) its out-degree, which may be impractical in e.g., broadcast-based communication protocols. In contrast, we describe FROST (Fast Row-stochastic-Optimization with uncoordinated STep-sizes), an optimization algorithm applicable to directed graphs that does not require the knowledge of out-degrees; the implementation of which is straightforward as each agent locally assigns weights to the incoming information and locally chooses a suitable step-size. We show that FROST converges linearly to the optimal solution for smooth and strongly-convex functions given that the largest step-size is positive and sufficiently small.Comment: Submitted for journal publication, currently under revie

Distributed stochastic optimization with gradient tracking over strongly-connected networks

In this paper, we study distributed stochastic optimization to minimize a sum of smooth and strongly-convex local cost functions over a network of agents, communicating over a strongly-connected graph. Assuming that each agent has access to a stochastic first-order oracle ($\mathcal{SFO}$), we propose a novel distributed method, called $\mathcal{S}$-$\mathcal{AB}$, where each agent uses an auxiliary variable to asymptotically track the gradient of the global cost in expectation. The $\mathcal{S}$-$\mathcal{AB}$ algorithm employs row- and column-stochastic weights simultaneously to ensure both consensus and optimality. Since doubly-stochastic weights are not used, $\mathcal{S}$-$\mathcal{AB}$ is applicable to arbitrary strongly-connected graphs. We show that under a sufficiently small constant step-size, $\mathcal{S}$-$\mathcal{AB}$ converges linearly (in expected mean-square sense) to a neighborhood of the global minimizer. We present numerical simulations based on real-world data sets to illustrate the theoretical results

Distributed Subgradient Projection Algorithm over Directed Graphs: Alternate Proof

We propose Directed-Distributed Projected Subgradient (D-DPS) to solve a constrained optimization problem over a multi-agent network, where the goal of agents is to collectively minimize the sum of locally known convex functions. Each agent in the network owns only its local objective function, constrained to a commonly known convex set. We focus on the circumstance when communications between agents are described by a \emph{directed} network. The D-DPS combines surplus consensus to overcome the asymmetry caused by the directed communication network. The analysis shows the convergence rate to be $O(\frac{\ln k}{\sqrt{k}})$.Comment: Disclaimer: This manuscript provides an alternate approach to prove the results in \textit{C. Xi and U. A. Khan, Distributed Subgradient Projection Algorithm over Directed Graphs, in IEEE Transactions on Automatic Control}. The changes, colored in blue, result into a tighter result in Theorem~1". arXiv admin note: text overlap with arXiv:1602.0065

Optimal Algorithms for Distributed Optimization

In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb) is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers

Improving Convergence Rate of Distributed Consensus Through Asymmetric Weights

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 proposed based on this continuum approximation. Numerical computations show that the resulting convergence rate with asymmetric weight design is improved considerably over that with symmetric optimal weights and Metropolis-Hastings weights.Comment: 2012 American Control Conferenc