We investigate fast and communication-efficient algorithms for the classic problem of minimizing a sum of strongly convex and smooth functions that are distributed among $n$ different nodes, which can communicate using a limited number of bits. Most previous communication-efficient approaches for this problem are limited to first-order optimization, and therefore have \emph{linear} dependence on the condition number in their communication complexity. We show that this dependence is not inherent: communication-efficient methods can in fact have sublinear dependence on the condition number. For this, we design and analyze the first communication-efficient distributed variants of preconditioned gradient descent for Generalized Linear Models, and for Newton's method. Our results rely on a new technique for quantizing both the preconditioner and the descent direction at each step of the algorithms, while controlling their convergence rate. We also validate our findings experimentally, showing fast convergence and reduced communication.
Many scientific and engineering challenges can be formulated as optimization problems which are constrained by partial differential equations (PDEs). These include inverse problems, control problems, and design problems. As a major challenge, the associated optimization procedures are inherently large-scale. To ensure computational tractability, the design of efficient and robust iterative methods becomes imperative. To meet this challenge, this paper introduces a fast and memory-efficient preconditioned iterative scheme for a class of distributed optimal control problems governed by convectio…
Communication efficiency is a major bottleneck in the applications of distributed networks. To address the problem, the problem of quantized distributed optimization has attracted a lot of attention. However, most of the existing quantized distributed optimization algorithms can only converge sublinearly. To achieve linear convergence, this paper proposes a novel quantized distributed gradient tracking algorithm (Q-DGT) to minimize a finite sum of local objective functions over directed networks. Moreover, we explicitly derive the update rule for the number of quantization levels, and prove th…
We come up with a class of distributed quantized averaging algorithms on asynchronous communication networks with fixed, switching and random topologies. The implementation of these algorithms is subject to the realistic constraint that the communication rate, the memory capacities of agents and the computation precision are finite. The focus of this paper is on the study of the convergence time of the proposed quantized averaging algorithms. By appealing to random walks on graphs, we derive polynomial bounds on the expected convergence time of the algorithms presented.
In this paper, we consider minimizing a sum of local convex objective functions in a distributed setting, where communication can be costly. We propose and analyze a class of nested distributed gradient methods with adaptive quantized communication (NEAR-DGD+Q). We show the effect of performing multiple quantized communication steps on the rate of convergence and on the size of the neighborhood of convergence, and prove R-Linear convergence to the exact solution with increasing number of consensus steps and adaptive quantization. We test the performance of the method, as well as some practical…
Consider a network whose nodes have some initial values, and it is desired to design an algorithm that builds on neighbor to neighbor interactions with the ultimate goal of convergence to the average of all initial node values or to some value close to that average. Such an algorithm is called generically "distributed averaging," and our goal in this paper is to study the performance of a subclass of deterministic distributed averaging algorithms where the information exchange between neighboring nodes (agents) is subject to uniform quantization. With such quantization, convergence to the prec…
Zhejiang University（浙江大学）；Rensselaer Polytechnic Institute（伦斯勒理工学院）；University of Minnesota, Twin Cities（明尼苏达大学双城分校）
The present paper develops a novel aggregated gradient approach for distributed machine learning that adaptively compresses the gradient communication. The key idea is to first quantize the computed gradients, and then skip less informative quantized gradient communications by reusing outdated gradients. Quantizing and skipping result in `lazy' worker-server communications, which justifies the term Lazily Aggregated Quantized gradient that is henceforth abbreviated as LAQ. Our LAQ can provably attain the same linear convergence rate as the gradient descent in the strongly convex case, while ef…
Multi-agent distributed optimization over a network minimizes a global objective formed by a sum of local convex functions using only local computation and communication. We develop and analyze a quantized distributed algorithm based on the alternating direction method of multipliers (ADMM) when inter-agent communications are subject to finite capacity and other practical constraints. While existing quantized ADMM approaches only work for quadratic local objectives, the proposed algorithm can deal with more general objective functions (possibly non-smooth) including the LASSO. Under certain co…
This paper considers a distributed optimization problem in a multi-agent system where a fraction of the agents act in an adversarial manner. Specifically, the malicious agents steer the network of agents away from the optimal solution by sending false information to their neighbors and consume significant bandwidth in the communication process. We propose a distributed gradient-based optimization algorithm in which the non-malicious agents exchange quantized information with one another. We prove convergence of the solution to a neighborhood of the optimal solution, and characterize the soluti…
Due to its efficiency and ease to implement, stochastic gradient descent (SGD) has been widely used in machine learning. In particular, SGD is one of the most popular optimization methods for distributed learning. Recently, quantized SGD (QSGD), which adopts quantization to reduce the communication cost in SGD-based distributed learning, has attracted much attention. Although several QSGD methods have been proposed, some of them are heuristic without theoretical guarantee, and others have high quantization variance which makes the convergence become slow. In this paper, we propose a new method…
The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；JD Explore Academy（京东探索研究院）；Tencent AI Lab（腾讯人工智能实验室）；Tencent（腾讯）
In this paper, we present a distributed variant of adaptive stochastic gradient method for training deep neural networks in the parameter-server model. To reduce the communication cost among the workers and server, we incorporate two types of quantization schemes, i.e., gradient quantization and weight quantization, into the proposed distributed Adam. Besides, to reduce the bias introduced by quantization operations, we propose an error-feedback technique to compensate for the quantized gradient. Theoretically, in the stochastic nonconvex setting, we show that the distributed adaptive gradient…
In this paper, we present a distributed algorithm which implements the $k$-means algorithm in a distributed fashion for multi-agent systems with directed communication links. The goal of $k$-means is to partition the network's agents in mutually exclusive sets (groups) such that agents in the same set have (and possibly share) similar information and are able to calculate a representative value for their group.During the operation of our distributed algorithm, each node (i) transmits quantized values in an event-driven fashion, and (ii) exhibits distributed stopping capabilities. Transmitting …
In this paper, we analyze the problem of optimally allocating resources in a distributed and privacy-preserving manner. We propose a novel distributed optimal resource allocation algorithm with privacy-preserving guarantees, which operates over a directed communication network. Our algorithm converges in finite time and allows each node to process and transmit quantized messages. Our algorithm utilizes a distributed quantized average consensus strategy combined with a privacy-preserving mechanism. We show that the algorithm converges in finite-time, and we prove that, under specific conditions…
Distributed Optimization with Nesterov Momentum and Finite-Time Quantized Consensus
The Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））；The Hong Kong University of Science and Technology（香港科技大学）；KTH Royal Institute of Technology（皇家理工学院）
This paper investigates Nash equilibrium (NE) seeking problems for noncooperative games over multi-players networks with finite bandwidth communication. A distributed quantized algorithm is presented, which consists of local gradient play, distributed decision estimating, and adaptive quantization. Exponential convergence of the algorithm is established, and a relationship between the convergence rate and the bandwidth is quantitatively analyzed. Finally, a simulation of an energy consumption game is presented to validate the proposed results.
In this paper, we consider the problem of privacy preservation in the average consensus problem when communication among nodes is quantized. More specifically, we consider a setting where some nodes in the network are curious but not malicious and they try to identify the initial states of other nodes based on the data they receive during their operation (without interfering in the computation in any other way), while some nodes in the network want to ensure that their initial states cannot be inferred exactly by the curious nodes. We propose two privacy-preserving event-triggered quantized av…
Communication is a crucial phase in the context of distributed training. Because parameter server (PS) frequently experiences network congestion, recent studies have found that training paradigms without a centralized server outperform the traditional server-based paradigms in terms of communication efficiency. However, with the increasing growth of model sizes, these server-free paradigms are also confronted with substantial communication overhead that seriously deteriorates the performance of distributed training. In this paper, we focus on communication efficiency of two serverless paradigm…
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…
We study efficient distributed algorithms for the fundamental problem of principal component analysis and leading eigenvector computation on the sphere, when the data are randomly distributed among a set of computational nodes. We propose a new quantized variant of Riemannian gradient descent to solve this problem, and prove that the algorithm converges with high probability under a set of necessary spherical-convexity properties. We give bounds on the number of bits transmitted by the algorithm under common initialization schemes, and investigate the dependency on the problem dimension in eac…
We consider the finite element discretization and the iterative solution of singularly perturbed elliptic reaction-diffusion equations in three-dimensional computational domains. These equations arise from the optimality conditions for elliptic distributed optimal control problems with energy regularization that were recently studied by M.~Neumüller and O.~Steinbach (2020). We provide quasi-optimal a priori finite element error estimates which depend both on the mesh size $h$ and on the regularization parameter $\varrho$. The choice $\varrho = h^2$ ensures optimal convergence which only depend…
We consider a convex unconstrained optimization problem that arises in a network of agents whose goal is to cooperatively optimize the sum of the individual agent objective functions through local computations and communications. For this problem, we use averaging algorithms to develop distributed subgradient methods that can operate over a time-varying topology. Our focus is on the convergence rate of these methods and the degradation in performance when only quantized information is available. Based on our recent results on the convergence time of distributed averaging algorithms, we derive …
Large-Scale Topology Optimisation of Time-dependent Thermal Conduction Using Space-Time Finite Elements and a Parallel Space-Time Multigrid Preconditioner
We analyze a class of distributed quantized consen- sus algorithms for arbitrary networks. In the initial setting, each node in the network has an integer value. Nodes exchange their current estimate of the mean value in the network, and then update their estimation by communicating with their neighbors in a limited capacity channel in an asynchronous clock setting. Eventually, all nodes reach consensus with quantized precision. We start the analysis with a special case of a distributed binary voting algorithm, then proceed to the expected convergence time for the general quantized consensus a…
Distributed quantized weight-balancing and average consensus over fixed digraphs are considered. A digraph with non-negative weights associated to its edges is weight-balanced if, for each node, the sum of the weights of its out-going edges is equal to that of its incoming edges. This paper proposes and analyzes the first distributed algorithm that solves the weight-balancing problem using only finite rate and simplex communications among nodes (compliant to the directed nature of the graph edges). Asymptotic convergence of the scheme is proved and a convergence rate analysis is provided. Buil…
We consider distributed iterative algorithms for the averaging problem over time-varying topologies. Our focus is on the convergence time of such algorithms when complete (unquantized) information is available, and on the degradation of performance when only quantized information is available. We study a large and natural class of averaging algorithms, which includes the vast majority of algorithms proposed to date, and provide tight polynomial bounds on their convergence time. We also describe an algorithm within this class whose convergence time is the best among currently available averagin…
In this paper we study the distributed average consensus problem in multi-agent systems with directed communication links that are subject to quantized information flow. Specifically, we present and analyze a distributed averaging algorithm which operates exclusively with quantized values (i.e., the information stored, processed and exchanged between neighboring agents is subject to deterministic uniform quantization) and relies on event-driven updates (e.g., to reduce energy consumption, communication bandwidth, network congestion, and/or processor usage). The main idea of the proposed algori…
This paper studies the network optimization problem about which a group of agents cooperates to minimize a global function under practical constraints of finite bandwidth communication. Particularly, we propose an adaptive encoding-decoding scheme to handle the constrained communication between agents. Based on this scheme, the continuous-time quantized distributed primal-dual (QDPD) algorithm is developed for network optimization problems. We prove that our algorithms can exactly track an optimal solution to the corresponding convex global cost function at a linear convergence rate. Furthermo…
We study distributed optimization problems over a network when the communication between the nodes is constrained, and so information that is exchanged between the nodes must be quantized. Recent advances using the distributed gradient algorithm with a quantization scheme at a fixed resolution have established convergence, but at rates significantly slower than when the communications are unquantized. In this paper, we introduce a novel quantization method, which we refer to as adaptive quantization, that allows us to match the convergence rates under perfect communications. Our approach adjus…
Decentralized Linearized Consensus ADMM with Efficient Quantized Communication
The Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））；The Hong Kong University of Science and Technology（香港科技大学）；KTH Royal Institute of Technology（瑞典皇家理工学院）
Yale University（耶鲁大学）；Massachusetts Institute of Technology（麻省理工学院）；University of Pennsylvania（宾夕法尼亚大学）
How can we efficiently mitigate the overhead of gradient communications in distributed optimization? This problem is at the heart of training scalable machine learning models and has been mainly studied in the unconstrained setting. In this paper, we propose Quantized-Frank-Wolfe (QFW), the first projection-free and communication-efficient algorithm for solving constrained optimization problems at scale. We consider both convex and non-convex objective functions, expressed as a finite-sum or more generally a stochastic optimization problem, and provide strong theoretical guarantees on the conv…
Arizona State University（亚利桑那州立大学）；Purdue University（普渡大学）；Bloomberg（彭博社）
This paper studies distributed algorithms for (strongly convex) composite optimization problems over mesh networks, subject to quantized communications. Instead of focusing on a specific algorithmic design, a black-box model is proposed, casting linearly convergent distributed algorithms in the form of fixed-point iterates. The algorithmic model is equipped with a novel random or deterministic Biased Compression (BC) rule on the quantizer design, and a new Adaptive encoding Nonuniform Quantizer (ANQ) coupled with a communication-efficient encoding scheme, which implements the BC-rule using a f…
