Asynchronous Optimization Over Heterogeneous Networks via Consensus ADMM

This paper considers the distributed optimization of a sum of locally observable, non-convex functions. The optimization is performed over a multi-agent networked system, and each local function depends only on a subset of the variables. An asynchronous and distributed alternating directions method of multipliers (ADMM) method that allows the nodes to defer or skip the computation and transmission of updates is proposed in the paper. The proposed algorithm utilizes different approximations in the update step, resulting in proximal and majorized ADMM variants. Both variants are shown to converge to a local minimum, under certain regularity conditions. The proposed asynchronous algorithms are also applied to the problem of cooperative localization in wireless ad hoc networks, where it is shown to outperform the other state-of-the-art localization algorithms.Comment: Submitted to Transactions on signal and information processing over Network

Impact of Communication Delay on Asynchronous Distributed Optimal Power Flow Using ADMM

Distributed optimization has attracted lots of attention in the operation of power systems in recent years, where a large area is decomposed into smaller control regions each solving a local optimization problem with periodic information exchange with neighboring regions. However, most distributed optimization methods are iterative and require synchronization of all regions at each iteration, which is hard to achieve without a centralized coordinator and might lead to under-utilization of computation resources due to the heterogeneity of the regions. To address such limitations of synchronous schemes, this paper investigates the applicability of asynchronous distributed optimization methods to power system optimization. Particularly, we focus on solving the AC Optimal Power Flow problem and propose an algorithmic framework based on the Alternating Direction Method of Multipliers (ADMM) method that allows the regions to perform local updates with information received from a subset of but not all neighbors. Through experimental studies, we demonstrate that the convergence performance of the proposed asynchronous scheme is dependent on the communication delay of passing messages among the regions. Under mild communication delays, the proposed scheme can achieve comparable or even faster convergence compared with its synchronous counterpart, which can be used as a good alternative to centralized or synchronous distributed optimization approaches.Comment: SmartGridComm 201

Asynchronous ADMM for Distributed Non-Convex Optimization in Power Systems

Large scale, non-convex optimization problems arising in many complex networks such as the power system call for efficient and scalable distributed optimization algorithms. Existing distributed methods are usually iterative and require synchronization of all workers at each iteration, which is hard to scale and could result in the under-utilization of computation resources due to the heterogeneity of the subproblems. To address those limitations of synchronous schemes, this paper proposes an asynchronous distributed optimization method based on the Alternating Direction Method of Multipliers (ADMM) for non-convex optimization. The proposed method only requires local communications and allows each worker to perform local updates with information from a subset of but not all neighbors. We provide sufficient conditions on the problem formulation, the choice of algorithm parameter and network delay, and show that under those mild conditions, the proposed asynchronous ADMM method asymptotically converges to the KKT point of the non-convex problem. We validate the effectiveness of asynchronous ADMM by applying it to the Optimal Power Flow problem in multiple power systems and show that the convergence of the proposed asynchronous scheme could be faster than its synchronous counterpart in large-scale applications

Toward Model Parallelism for Deep Neural Network based on Gradient-free ADMM Framework

Alternating Direction Method of Multipliers (ADMM) has recently been proposed as a potential alternative optimizer to the Stochastic Gradient Descent(SGD) for deep learning problems. This is because ADMM can solve gradient vanishing and poor conditioning problems. Moreover, it has shown good scalability in many large-scale deep learning applications. However, there still lacks a parallel ADMM computational framework for deep neural networks because of layer dependency among variables. In this paper, we propose a novel parallel deep learning ADMM framework (pdADMM) to achieve layer parallelism: parameters in each layer of neural networks can be updated independently in parallel. The convergence of the proposed pdADMM to a critical point is theoretically proven under mild conditions. The convergence rate of the pdADMM is proven to be $o(1/k)$ where $k$ is the number of iterations. Extensive experiments on six benchmark datasets demonstrated that our proposed pdADMM can lead to more than 10 times speedup for training large-scale deep neural networks, and outperformed most of the comparison methods. Our code is available at: https://github.com/xianggebenben/pdADMM.Comment: ICDM202

Distributed Optimization with Coupling Constraints via Dual Proximal Gradient Method with Applications to Asynchronous Networks

In this paper, we consider solving a distributed optimization problem (DOP) with coupling constraints in a multi-agent network based on proximal gradient method. In this problem, each agent aims to minimize an individual cost function composed of both smooth and non-smooth parts. To this end, we derive the dual problem by the concept of Fenchel conjugate, which results in two kinds of dual problems: consensus based constrained and augmented unconstrained problems. In the first scenario, we propose a fully distributed dual proximal gradient (D-DPG) algorithm, where the agents can make updates only with the dual information of their neighbours and local step-sizes. Moreover, if the non-smooth parts of the objective functions are with certain simple structures, the agents only need to update dual variables with some simple operations, which can reduce the overall computational complexity. In the second scenario, an augmented dual proximal gradient (A-DPG) algorithm is proposed, which allows for the asymmetric interpretations of the global constraints for the agents and can be more efficient than D-DGP algorithm in some special-structured DOPs. Based on A-DPG algorithm, an asynchronous dual proximal gradient (Asyn-DPG) algorithm is proposed for the asynchronous networks where each agent updates its strategy with heterogenous step-size and possible outdated dual information of others. In all the discussed scenarios, analytical (ergodic) convergence rates are derived. The effectiveness of the proposed algorithms is verified by solving a social welfare optimization problem in the electricity market.Comment: 20 pages, 11 figure

Distributed Inexact Successive Convex Approximation ADMM: Analysis-Part I

In this two-part work, we propose an algorithmic framework for solving non-convex problems whose objective function is the sum of a number of smooth component functions plus a convex (possibly non-smooth) or/and smooth (possibly non-convex) regularization function. The proposed algorithm incorporates ideas from several existing approaches such as alternate direction method of multipliers (ADMM), successive convex approximation (SCA), distributed and asynchronous algorithms, and inexact gradient methods. Different from a number of existing approaches, however, the proposed framework is flexible enough to incorporate a class of non-convex objective functions, allow distributed operation with and without a fusion center, and include variance reduced methods as special cases. Remarkably, the proposed algorithms are robust to uncertainties arising from random, deterministic, and adversarial sources. The part I of the paper develops two variants of the algorithm under very mild assumptions and establishes first-order convergence rate guarantees. The proof developed here allows for generic errors and delays, paving the way for different variance-reduced, asynchronous, and stochastic implementations, outlined and evaluated in part II

Asynchronous Decentralized Successive Convex Approximation

We study decentralized asynchronous multiagent optimization over networks, modeled as static (possibly directed) graphs. The optimization problem consists of minimizing a (possibly nonconvex) smooth function--the sum of the agents' local costs--plus a convex (possibly nonsmooth) regularizer, subject to convex constraints. Agents can perform their local computations as well as communicate with their immediate neighbors at any time, without any form of coordination or centralized scheduling; furthermore, when solving their local subproblems, they can use outdated information from their neighbors. We propose the first distributed asynchronous algorithm, termed ASY-DSCA, that converges at an R-linear rate to the optimal solution of convex problems whose objective function satisfies a general error bound condition; this condition is weaker than the more frequently used strong convexity, and it is satisfied by several empirical risk functions that are not strongly convex; examples include LASSO and logistic regression problems. When the objective function is nonconvex, ASY-DSCA converges to a stationary solution of the problem at a sublinear rate

Asynchronous Gradient-Push

We consider a multi-agent framework for distributed optimization where each agent has access to a local smooth strongly convex function, and the collective goal is to achieve consensus on the parameters that minimize the sum of the agents' local functions. We propose an algorithm wherein each agent operates asynchronously and independently of the other agents. When the local functions are strongly-convex with Lipschitz-continuous gradients, we show that the iterates at each agent converge to a neighborhood of the global minimum, where the neighborhood size depends on the degree of asynchrony in the multi-agent network. When the agents work at the same rate, convergence to the global minimizer is achieved. Numerical experiments demonstrate that Asynchronous Gradient-Push can minimize the global objective faster than state-of-the-art synchronous first-order methods, is more robust to failing or stalling agents, and scales better with the network size.Comment: 33 pages, 9 figures, accepted to IEEE Transactions on Automatic Contro

Composite Optimization with Coupling Constraints via Penalized Proximal Gradient Method in Partially Asynchronous Networks

In this paper, we consider a composite optimization problem with linear coupling constraints in a multi-agent network. In this problem, all the agents jointly optimize a global composite cost function which is the linear sum of individual cost functions composed of both smooth and non-smooth components. To solve this problem, we propose an asynchronous penalized proximal gradient (Asyn-PPG) algorithm, a variant of classical proximal gradient method, by considering the asynchronous update instants of the agents and communication delays in the network. Specifically, we consider a slot-based asynchronous network (SAN), where the whole time domain is split into sequential time slots and each agent is permitted to make multiple updates during a slot by accessing the historical state information of others. Moreover, we consider a set of global linear constraints and impose some violation penalties on the updating algorithms. By the Asyn-PPG algorithm, we will show that a periodically convergence with rate O(1/K) (K is the index of time slots) can be guaranteed if the coefficient of the penalties for all agents is synchronized at the end of the time slots and the step-size of the Asyn-PPG algorithm is properly determined. The feasibility of the proposed algorithm is verified by solving a consensus based distributed LASSO problem and a social welfare optimization problem in the electricity market respectively.Comment: 16 pages, 8 figure

Distributed Optimization for Smart Cyber-Physical Networks

The presence of embedded electronics and communication capabilities as well as sensing and control in smart devices has given rise to the novel concept of cyber-physical networks, in which agents aim at cooperatively solving complex tasks by local computation and communication. Numerous estimation, learning, decision and control tasks in smart networks involve the solution of large-scale, structured optimization problems in which network agents have only a partial knowledge of the whole problem. Distributed optimization aims at designing local computation and communication rules for the network processors allowing them to cooperatively solve the global optimization problem without relying on any central unit. The purpose of this survey is to provide an introduction to distributed optimization methodologies. Principal approaches, namely (primal) consensus-based, duality-based and constraint exchange methods, are formalized. An analysis of the basic schemes is supplied, and state-of-the-art extensions are reviewed