A Block-wise, Asynchronous and Distributed ADMM Algorithm for General Form Consensus Optimization

Many machine learning models, including those with non-smooth regularizers, can be formulated as consensus optimization problems, which can be solved by the alternating direction method of multipliers (ADMM). Many recent efforts have been made to develop asynchronous distributed ADMM to handle large amounts of training data. However, all existing asynchronous distributed ADMM methods are based on full model updates and require locking all global model parameters to handle concurrency, which essentially serializes the updates from different workers. In this paper, we present a novel block-wise, asynchronous and distributed ADMM algorithm, which allows different blocks of model parameters to be updated in parallel. The lock-free block-wise algorithm may greatly speedup sparse optimization problems, a common scenario in reality, in which most model updates only modify a subset of all decision variables. We theoretically prove the convergence of our proposed algorithm to stationary points for non-convex general form consensus problems with possibly non-smooth regularizers. We implement the proposed ADMM algorithm on the Parameter Server framework and demonstrate its convergence and near-linear speedup performance as the number of workers increases

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

Asynchronous Distributed ADMM for Large-Scale Optimization- Part I: Algorithm and Convergence Analysis

Aiming at solving large-scale learning problems, this paper studies distributed optimization methods based on the alternating direction method of multipliers (ADMM). By formulating the learning problem as a consensus problem, the ADMM can be used to solve the consensus problem in a fully parallel fashion over a computer network with a star topology. However, traditional synchronized computation does not scale well with the problem size, as the speed of the algorithm is limited by the slowest workers. This is particularly true in a heterogeneous network where the computing nodes experience different computation and communication delays. In this paper, we propose an asynchronous distributed ADMM (AD-AMM) which can effectively improve the time efficiency of distributed optimization. Our main interest lies in analyzing the convergence conditions of the AD-ADMM, under the popular partially asynchronous model, which is defined based on a maximum tolerable delay of the network. Specifically, by considering general and possibly non-convex cost functions, we show that the AD-ADMM is guaranteed to converge to the set of Karush-Kuhn-Tucker (KKT) points as long as the algorithm parameters are chosen appropriately according to the network delay. We further illustrate that the asynchrony of the ADMM has to be handled with care, as slightly modifying the implementation of the AD-ADMM can jeopardize the algorithm convergence, even under a standard convex setting.Comment: 37 page

A Partition-Based Implementation of the Relaxed ADMM for Distributed Convex Optimization over Lossy Networks

In this paper we propose a distributed implementation of the relaxed Alternating Direction Method of Multipliers algorithm (R-ADMM) for optimization of a separable convex cost function, whose terms are stored by a set of interacting agents, one for each agent. Specifically the local cost stored by each node is in general a function of both the state of the node and the states of its neighbors, a framework that we refer to as `partition-based' optimization. This framework presents a great flexibility and can be adapted to a large number of different applications. We show that the partition-based R-ADMM algorithm we introduce is linked to the relaxed Peaceman-Rachford Splitting (R-PRS) operator which, historically, has been introduced in the literature to find the zeros of sum of functions. Interestingly, making use of non expansive operator theory, the proposed algorithm is shown to be provably robust against random packet losses that might occur in the communication between neighboring nodes. Finally, the effectiveness of the proposed algorithm is confirmed by a set of compelling numerical simulations run over random geometric graphs subject to i.i.d. random packet losses.Comment: Full version of the paper to be presented at Conference on Decision and Control (CDC) 201

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

ARock: an Algorithmic Framework for Asynchronous Parallel Coordinate Updates

Finding a fixed point to a nonexpansive operator, i.e., $x^*=Tx^*$, abstracts many problems in numerical linear algebra, optimization, and other areas of scientific computing. To solve fixed-point problems, we propose ARock, an algorithmic framework in which multiple agents (machines, processors, or cores) update $x$ in an asynchronous parallel fashion. Asynchrony is crucial to parallel computing since it reduces synchronization wait, relaxes communication bottleneck, and thus speeds up computing significantly. At each step of ARock, an agent updates a randomly selected coordinate $x_i$ based on possibly out-of-date information on $x$. The agents share $x$ through either global memory or communication. If writing $x_i$ is atomic, the agents can read and write $x$ without memory locks. Theoretically, we show that if the nonexpansive operator $T$ has a fixed point, then with probability one, ARock generates a sequence that converges to a fixed points of $T$. Our conditions on $T$ and step sizes are weaker than comparable work. Linear convergence is also obtained. We propose special cases of ARock for linear systems, convex optimization, machine learning, as well as distributed and decentralized consensus problems. Numerical experiments of solving sparse logistic regression problems are presented.Comment: updated the linear convergence proof

An Asynchronous Distributed Proximal Gradient Method for Composite Convex Optimization

We propose a distributed first-order augmented Lagrangian (DFAL) algorithm to minimize the sum of composite convex functions, where each term in the sum is a private cost function belonging to a node, and only nodes connected by an edge can directly communicate with each other. This optimization model abstracts a number of applications in distributed sensing and machine learning. We show that any limit point of DFAL iterates is optimal; and for any $\epsilon>0$, an $\epsilon$-optimal and $\epsilon$-feasible solution can be computed within $\mathcal{O}(\log(\epsilon^{-1}))$ DFAL iterations, which require $\mathcal{O}(\frac{\psi_{\max}^{1.5}}{d_{\min}} \epsilon^{-1})$ proximal gradient computations and communications per node in total, where $\psi_{\max}$ denotes the largest eigenvalue of the graph Laplacian, and $d_{\min}$ is the minimum degree of the graph. We also propose an asynchronous version of DFAL by incorporating randomized block coordinate descent methods; and demonstrate the efficiency of DFAL on large scale sparse-group LASSO problems.Comment: The manuscript will appear in the Proceedings of the 32nd International Conference on Machine Learning, Lille, France, 2015. JMLR: W&CP volume 37. Copyright 2015 by the author(s

Cloud-Assisted Remote Sensor Network Virtualization for Distributed Consensus Estimation

We develop cloud-assisted remote sensing techniques for enabling distributed consensus estimation of unknown parameters in a given geographic area. We first propose a distributed sensor network virtualization algorithm that searches for, selects, and coordinates Internet-accessible sensors to perform a sensing task in a specific region. The algorithm converges in linearithmic time for large-scale networks, and requires exchanging a number of messages that is at most linear in the number of sensors. Second, we design an uncoordinated, distributed algorithm that relies on the selected sensors to estimate a set of parameters without requiring synchronization among the sensors. Our simulation results show that the proposed algorithm, when compared to conventional ADMM (Alternating Direction Method of Multipliers), reduces communication overhead significantly without compromising the estimation error. In addition, the convergence time, though increases slightly, is still linear as in the case of conventional ADMM.Comment: 11 pages, double column, pre-submissio

Newton-Raphson Consensus under asynchronous and lossy communications for peer-to-peer networks

In this work we study the problem of unconstrained convex-optimization in a fully distributed multi-agent setting which includes asynchronous computation and lossy communication. In particular, we extend a recently proposed algorithm named Newton-Raphson Consensus by integrating it with a broadcast-based average consensus algorithm which is robust to packet losses. We show via the separation of time scales principle that under mild conditions (i.e., persistency of the agents activation and bounded consecutive communication failures) the proposed algorithm is proved to be locally exponentially stable with respect to the optimal global solution. Finally, we complement the theoretical analysis with numerical simulations that are based on real datasets

Distributed Partitioned Big-Data Optimization via Asynchronous Dual Decomposition

In this paper we consider a novel partitioned framework for distributed optimization in peer-to-peer networks. In several important applications the agents of a network have to solve an optimization problem with two key features: (i) the dimension of the decision variable depends on the network size, and (ii) cost function and constraints have a sparsity structure related to the communication graph. For this class of problems a straightforward application of existing consensus methods would show two inefficiencies: poor scalability and redundancy of shared information. We propose an asynchronous distributed algorithm, based on dual decomposition and coordinate methods, to solve partitioned optimization problems. We show that, by exploiting the problem structure, the solution can be partitioned among the nodes, so that each node just stores a local copy of a portion of the decision variable (rather than a copy of the entire decision vector) and solves a small-scale local problem