Distributed Subgradient Methods Research Articles

Distributed Subgradient Method With Random Quantization and Flexible Weights: Convergence Analysis.

The distributed subgradient (DSG) method is a widely used algorithm for coping with large-scale distributed optimization problems in machine-learning applications. Most existing works on DSG focus on ideal communication between cooperative agents, where the shared information between agents is exact and perfect. This assumption, however, can lead to potential privacy concerns and is not feasible when wireless transmission links are of poor quality. To meet this challenge, a common approach is to quantize the data locally before transmission, which avoids exposure of raw data and significantly reduces the size of the data. Compared with perfect data, quantization poses fundamental challenges to maintaining data accuracy, which further impacts the convergence of the algorithms. To overcome this problem, we propose a DSG method with random quantization and flexible weights and provide comprehensive results on the convergence of the algorithm for (strongly/weakly) convex objective functions. We also derive the upper bounds on the convergence rates in terms of the quantization error, the distortion, the step sizes, and the number of network agents. Our analysis extends the existing results, for which special cases of step sizes and convex objective functions are considered, to general conclusions on weakly convex cases. Numerical simulations are conducted in convex and weakly convex settings to support our theoretical results.

Subgradient averaging for multi-agent optimisation with different constraint sets

We consider a multi-agent setting with agents exchanging information over a possibly time-varying network, aiming at minimising a separable objective function subject to constraints. To achieve this objective we propose a novel subgradient averaging algorithm that allows for non-differentiable objective functions and different constraint sets per agent. Allowing different constraints per agent simultaneously with a time-varying communication network constitutes a distinctive feature of our approach, extending existing results on distributed subgradient methods. To highlight the necessity of dealing with a different constraint set within a distributed optimisation context, we analyse a problem instance where an existing algorithm does not exhibit a convergent behaviour if adapted to account for different constraint sets. For our proposed iterative scheme we show asymptotic convergence of the iterates to a minimum of the underlying optimisation problem for step sizes of the form ηk+1, η>0. We also analyse this scheme under a step size choice of ηk+1, η>0, and establish a convergence rate of O(lnkk) in objective value. To demonstrate the efficacy of the proposed method, we investigate a robust regression problem and an ℓ2 regression problem with regularisation.

Distributed Resource Allocation Over Dynamic Networks With Uncertainty

Motivated by broad applications in various fields of engineering, we study a network resource allocation problem where the goal is to optimally allocate a fixed quantity of the resources over a network of nodes. We consider large scale networks with complex interconnection structures, thus any solution must be implemented in parallel and based only on local data resulting in a need for distributed algorithms. In this article, we study a distributed Lagrangian method for such problems. By utilizing the so-called distributed subgradient methods to solve the dual problem, our approach eliminates the need for central coordination in updating the dual variables, which is often required in classic Lagrangian methods. Our focus is to understand the performance of this distributed algorithm when the number of resources is unknown and may be time-varying. In particular, we obtain an upper bound on the convergence rate of the algorithm to the optimal value, in expectation, as a function of network topology. The effectiveness of the proposed method is demonstrated by its application to the economic dispatch problem in power systems, with simulations completed on the benchmark IEEE-14 and IEEE-118 bus test systems.

Fast Convergence Rates of Distributed Subgradient Methods With Adaptive Quantization

We study distributed optimization problems over a network when the communication between the nodes is constrained, and therefore, 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 article, 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 adjusts the quantization scheme used by each node as the algorithm progresses: as we approach the solution, we become more certain about where the state variables are localized and adapt the quantizer codebook accordingly. We bound the convergence rates of the proposed method as a function of the communication bandwidth, the underlying network topology, and structural properties of the constituent objective functions. In particular, we show that if the objective functions are convex or strongly convex, then using adaptive quantization does not affect the rate of convergence of the distributed subgradient methods when the communications are quantized, except for a constant that depends on the resolution of the quantizer. To the best of our knowledge, the rates achieved in this article are better than any existing work in the literature for distributed gradient methods under finite communication bandwidths. We also provide numerical simulations that compare convergence properties of the distributed gradient methods with and without quantization for solving distributed regression problems for both quadratic and absolute loss functions.

Distributed Subgradient Method for Constrained Convex Optimization with Quantized and Event-Triggered Communication

Distributed Subgradient Method for Constrained Convex Optimization with Quantized and Event-Triggered Communication

Distributed optimization over directed graphs with row stochasticity and constraint regularity

This paper deals with an optimization problem over a network of agents, where the cost function is the sum of the individual (possibly nonsmooth) objectives of the agents and the constraint set is the intersection of local constraints. Most existing methods employing subgradient and consensus steps for solving this problem require the weight matrix associated with the network to be column stochastic or even doubly stochastic, conditions that can be hard to arrange in directed networks. Moreover, known convergence analyses for distributed subgradient methods vary depending on whether the problem is unconstrained or constrained, and whether the local constraint sets are identical or nonidentical and compact. The main goals of this paper are: (i) removing the common column stochasticity requirement; (ii) relaxing the compactness assumption, and (iii) providing a unified convergence analysis. Specifically, assuming the communication graph to be fixed and strongly connected and the weight matrix to (only) be row stochastic, a distributed projected subgradient algorithm and a variation of this algorithm are presented to solve the problem for cost functions that are convex and Lipschitz continuous. The key component of the algorithms is to adjust the subgradient of each agent by an estimate of its corresponding entry in the normalized left Perron eigenvector of the weight matrix. These estimates are obtained locally from an augmented consensus iteration using the same row stochastic weight matrix and requiring very limited global information about the network. Moreover, based on a regularity assumption on the local constraint sets, a unified analysis is given that can be applied to both unconstrained and constrained problems and without assuming compactness of the constraint sets or an interior point in their intersection. Further, we also establish an upper bound on the absolute objective error evaluated at each agent’s available local estimate under a nonincreasing step size sequence. This bound allows us to analyze the convergence rate of both algorithms.

Distributed Subgradient Method With Edge-Based Event-Triggered Communication

This paper proposes a distributed subgradient method for constrained optimization with event-triggered communications. In the proposed method, each agent has an estimate of an optimal solution as a state and iteratively updates it by a consensus-based subgradient algorithm with a projection to a common constraint set. The local communications are carried out by the edge-based triggering mechanism when the difference between the current state and the last triggered state exceeds a threshold. We show that the states of all agents asymptotically converge to one of the optimal solutions under a diminishing and summability condition on a stepsize and a threshold for a trigger condition. We also investigate the convergence rate with respect to the time-averaged state of each agent. The simulation results show that the proposed event-triggered algorithm can reduce the number of communications compared to the time-triggered algorithms.

Distributed iterative thresholding for ℓ0/ℓ1-regularized linear inverse problems

The $\ell _{0}/\ell _{1}$ -regularized least-squares approach is used to deal with linear inverse problems under sparsity constraints, which arise in mathematical and engineering fields. In particular, multiagent models have recently emerged in this context to describe diverse kinds of networked systems, ranging from medical databases to wireless sensor networks. In this paper, we study methods for solving $\ell _{0}/\ell _{1}$ -regularized least-squares problems in such multiagent systems. We propose a novel class of distributed protocols based on iterative thresholding and input driven consensus techniques, which are well-suited to work in-network when the communication to a central processing unit is not allowed. Estimation is performed by the agents themselves, which typically consist of devices with limited computational capabilities. This motivates us to develop low-complexity and low-memory algorithms that are feasible in real applications. Our main result is a rigorous proof of the convergence of these methods in regular networks. We introduce a suitable distributed, regularized, least-squares functional, and we prove that our algorithms reach their minima using results from dynamical systems theory. Furthermore, we propose numerical comparisons with the alternating direction method of multipliers and the distributed subgradient methods, in terms of performance, complexity, and memory usage. We conclude that our techniques are preferable for their good memory-accuracy tradeoff.

Network Synchronization with Convexity

In this paper, we establish a few new synchronization conditions for complex networks with nonlinear and nonidentical self-dynamics with switching directed communication graphs. In light of the recent works on distributed subgradient methods, we impose integral convexity for the nonlinear node self-dynamics in the sense that the self-dynamics of a given node is the gradient of some concave function corresponding to that node. The node couplings are assumed to be linear but with switching directed communication graphs. Several sufficient and/or necessary conditions are established for exact or approximate synchronization over the considered complex networks. These results show when and how nonlinear node self-dynamics may cooperate with the linear diffusive coupling, which eventually leads to network synchronization conditions under relaxed connectivity requirements.

Performance Evaluation of the Consensus-Based Distributed Subgradient Method Under Random Communication Topologies

We investigate collaborative optimization of an objective function expressed as a sum of local convex functions, when the agents make decisions in a distributed manner using local information, while the communication topology used to exchange messages and information is modeled by a graph-valued random process, assumed independent and identically distributed. Specifically, we study the performance of the consensus-based multi-agent distributed subgradient method and show how it depends on the probability distribution of the random graph. For the case of a constant stepsize, we first give an upper bound on the difference between the objective function, evaluated at the agents' estimates of the optimal decision vector, and the optimal value. Second, for a particular class of convex functions, we give an upper bound on the distances between the agents' estimates of the optimal decision vector and the minimizer. In addition, we provide the rate of convergence to zero of the time varying component of the aforementioned upper bound. The addressed metrics are evaluated via their expected values. As an application, we show how the distributed optimization algorithm can be used to perform collaborative system identification and provide numerical experiments under the randomized and broadcast gossip protocols.

Distributed Subgradient Methods for Convex Optimization Over Random Networks

We consider the problem of cooperatively minimizing the sum of convex functions, where the functions represent local objective functions of the agents. We assume that each agent has information about his local function, and communicate with the other agents over a time-varying network topology. For this problem, we propose a distributed subgradient method that uses averaging algorithms for locally sharing information among the agents. In contrast to previous works on multi-agent optimization that make worst-case assumptions about the connectivity of the agents (such as bounded communication intervals between nodes), we assume that links fail according to a given stochastic process. Under the assumption that the link failures are independent and identically distributed over time (possibly correlated across links), we provide almost sure convergence results for our subgradient algorithm.

Distributed Subgradient Methods for Multi-Agent Optimization

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> We study a distributed computation model for optimizing a sum of convex objective functions corresponding to multiple agents. For solving this (not necessarily smooth) optimization problem, we consider a subgradient method that is distributed among the agents. The method involves every agent minimizing his/her own objective function while exchanging information locally with other agents in the network over a time-varying topology. We provide convergence results and convergence rate estimates for the subgradient method. Our convergence rate results explicitly characterize the tradeoff between a desired accuracy of the generated approximate optimal solutions and the number of iterations needed to achieve the accuracy. </para>

Randomized gossip algorithms

Motivated by applications to sensor, peer-to-peer, and ad hoc networks, we study distributed algorithms, also known as algorithms, for exchanging information and for computing in an arbitrarily connected network of nodes. The topology of such networks changes continuously as new nodes join and old nodes leave the network. Algorithms for such networks need to be robust against changes in topology. Additionally, nodes in sensor networks operate under limited computational, communication, and energy resources. These constraints have motivated the design of gossip algorithms: schemes which distribute the computational burden and in which a node communicates with a randomly chosen neighbor. We analyze the averaging problem under the constraint for an arbitrary network graph, and find that the averaging time of a algorithm depends on the second largest eigenvalue of a doubly stochastic matrix characterizing the algorithm. Designing the fastest algorithm corresponds to minimizing this eigenvalue, which is a semidefinite program (SDP). In general, SDPs cannot be solved in a distributed fashion; however, exploiting problem structure, we propose a distributed subgradient method that solves the optimization problem over the network. The relation of averaging time to the second largest eigenvalue naturally relates it to the mixing time of a random walk with transition probabilities derived from the algorithm. We use this connection to study the performance and scaling of algorithms on two popular networks: Wireless Sensor Networks, which are modeled as Geometric Random Graphs, and the Internet graph under the so-called Preferential Connectivity (PC) model.