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 improved upper bounds on the convergence rate of the unquantized subgradient method. We then propose a distributed subgradient method under the additional constraint that agents can only store and communicate quantized information, and we provide bounds on its convergence rate that highlight the dependence on the number of quantization levels.
In this paper, we consider a distributed constrained optimization problem with delayed subgradient information over the time-varying communication network, where each agent can only communicate with its neighbors and the communication channel has a limited data rate. We propose an adaptive quantization method to address this problem. A mirror descent algorithm with delayed subgradient information is established based on the theory of Bregman divergence. With non-Euclidean Bregman projection-based scheme, the proposed method essentially generalizes many previous classical Euclidean projection-b…
The paper proposes a heterogeneous push-sum based subgradient algorithm for multi-agent distributed convex optimization in which each agent can arbitrarily switch between subgradient-push and push-subgradient at each time. It is shown that the heterogeneous algorithm converges to an optimal point at an optimal rate over time-varying directed graphs.
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…
We propose a distributed algorithm, termed the 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 directed network. The D-DPS augments an additional variable for each agent, to overcome the asymmetry caused by the directed communication n…
This paper investigates the distributed Nash equilibrium seeking problem for two-network zero-sum games with set constraints, where the two networks have the opposite nonsmooth cost functions. The interaction of the agents in each network is characterized by an unbalanced directed graph. We are particularly interested in the case that the local cost function of each agent in the two networks is unknown in this paper. We first construct the stochastic subgradient-free two-variable oracles based on the measurements of the local cost functions. Instead of using subgradients of the local cost func…
This paper studies a distributed online constrained optimization problem over time-varying unbalanced digraphs without explicit subgradients. In sharp contrast to the existing algorithms, we design a novel consensus-based distributed online algorithm with a local randomized zeroth-order oracle and then rescale the oracle by constructing row-stochastic matrices, which aims to address the unbalancedness of time-varying digraphs. Under mild conditions, the average dynamic regret over a time horizon is shown to asymptotically converge at a sublinear rate provided that the accumulated variation gro…
We consider a multi agent optimization problem where a set of agents collectively solves a global optimization problem with the objective function given by the sum of locally known convex functions. We focus on the case when information exchange among agents takes place over a directed network and propose a distributed subgradient algorithm in which each agent performs local processing based on information obtained from his incoming neighbors. Our algorithm uses weight balancing to overcome the asymmetries caused by the directed communication network, i.e., agents scale their outgoing informat…
We present distributed subgradient methods for min-max problems with agreement constraints on a subset of the arguments of both the convex and concave parts. Applications include constrained minimization problems where each constraint is a sum of convex functions in the local variables of the agents. In the latter case, the proposed algorithm reduces to primal-dual updates using local subgradients and Laplacian averaging on local copies of the multipliers associated to the global constraints. For the case of general convex-concave saddle-point problems, our analysis establishes the convergence…
In this paper, we propose a framework under which the decentralized optimization algorithms suggested in \cite{JKJJ,MA, NO,NO2} can be treated in a unified manner. More precisely, we show that the distributed subgradient descent algorithms \cite{JKJJ, NO}, the subgradient-push algorithm \cite{NO2}, and the distributed algorithm with row-stochastic matrix \cite{MA} can be derived by making suitable choices of consensus matrices, step-size and subgradient from the decentralized subgradient descent proposed in \cite{NO}. As a result of such unified understanding, we provide a convergence proof th…
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 …
Privacy preservation is becoming an increasingly important issue in data mining and machine learning. In this paper, we consider the privacy preserving features of distributed subgradient optimization algorithms. We first show that a well-known distributed subgradient synchronous optimization algorithm, in which all agents make their optimization updates simultaneously at all times, is not privacy preserving in the sense that the malicious agent can learn other agents' subgradients asymptotically. Then we propose a distributed subgradient projection asynchronous optimization algorithm without …
We consider a distributed multi-agent network system where the goal is to minimize a sum of convex objective functions of the agents subject to a common convex constraint set. Each agent maintains an iterate sequence and communicates the iterates to its neighbors. Then, each agent combines weighted averages of the received iterates with its own iterate, and adjusts the iterate by using subgradient information (known with stochastic errors) of its own function and by projecting onto the constraint set. The goal of this paper is to explore the effects of stochastic subgradient errors on the conv…
In this paper we consider convergence rate problems for stochastic strongly-convex optimization in the non-Euclidean sense with a constraint set over a time-varying multi-agent network. We propose two efficient non-Euclidean stochastic subgradient descent algorithms based on the Bregman divergence as distance-measuring function rather than the Euclidean distances that were employed by the standard distributed stochastic projected subgradient algorithms. For distributed optimization of nonsmooth and strongly convex functions whose only stochastic subgradients are available, the first algorithm …
Two distributed algorithms are described that enable all users connected over a network to cooperatively solve the problem of minimizing the sum of all users' objective functions over the intersection of all users' constraint sets, where each user has its own private nonsmooth convex objective function and closed convex constraint set, which is the intersection of a number of simple, closed convex sets. One algorithm enables each user to adjust its estimate by using a proximity operator of its objective function and the metric projection onto one set randomly selected from the simple, closed c…
This paper studies the distributed optimization problem when the objective functions might be nondifferentiable and subject to heterogeneous set constraints. Unlike existing subgradient methods, we focus on the case when the exact subgradients of the local objective functions can not be accessed by the agents. To solve this problem, we propose a projected primal-dual dynamics using only the objective function's approximate subgradients. We first prove that the formulated optimization problem can generally be solved with an error depending upon the accuracy of the available subgradients. Then, …
In this work, we first consider distributed convex constrained optimization problems where the objective function is encoded by multiple local and possibly nonsmooth objectives privately held by a group of agents, and propose a distributed subgradient method with double averaging (abbreviated as ${\rm DSA_2}$) that only requires peer-to-peer communication and local computation to solve the global problem. The algorithmic framework builds on dual methods and dynamic average consensus; the sequence of test points is formed by iteratively minimizing a local dual model of the overall objective whe…
A wider selection of step sizes is explored for the distributed subgradient algorithm for multi-agent optimization problems, for both time-invariant and time-varying communication topologies. The square summable requirement of the step sizes commonly adopted in the literature is removed. The step sizes are only required to be positive, vanishing and non-summable. It is proved that in both unconstrained and constrained optimization problems, the agents' estimates reach consensus and converge to the optimal solution with the more general choice of step sizes. The idea is to show that a weighted …
In this paper, we revisit a well-known distributed projected subgradient algorithm which aims to minimize a sum of cost functions with a common set constraint. In contrast to most of existing results, weight matrices of the time-varying multi-agent network are assumed to be more general, i.e., they are only required to be row stochastic instead of doubly stochastic. We focus on analyzing convergence properties of this algorithm under general graphs. We first show that there generally exists a graph sequence such that the algorithm is not convergent when the network switches freely within finit…
The stochastic subgradient method is a widely-used algorithm for solving large-scale optimization problems arising in machine learning. Often these problems are neither smooth nor convex. Recently, Davis et al. [1-2] characterized the convergence of the stochastic subgradient method for the weakly convex case, which encompasses many important applications (e.g., robust phase retrieval, blind deconvolution, biconvex compressive sensing, and dictionary learning). In practice, distributed implementations of the projected stochastic subgradient method (stoDPSM) are used to speed-up risk minimizati…
We consider whether distributed subgradient methods can achieve a linear speedup over a centralized subgradient method. While it might be hoped that distributed network of $n$ nodes that can compute $n$ times more subgradients in parallel compared to a single node might, as a result, be $n$ times faster, existing bounds for distributed optimization methods are often consistent with a slowdown rather than speedup compared to a single node. We show that a distributed subgradient method has this "linear speedup" property when using a class of square-summable-but-not-summable step-sizes which incl…
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 de…
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…
We propose graph-dependent implicit regularisation strategies for distributed stochastic subgradient descent (Distributed SGD) for convex problems in multi-agent learning. Under the standard assumptions of convexity, Lipschitz continuity, and smoothness, we establish statistical learning rates that retain, up to logarithmic terms, centralised statistical guarantees through implicit regularisation (step size tuning and early stopping) with appropriate dependence on the graph topology. Our approach avoids the need for explicit regularisation in decentralised learning problems, such as adding con…
We study distributed algorithms for solving global optimization problems in which the objective function is the sum of local objective functions of agents and the constraint set is given by the intersection of local constraint sets of agents. We assume that each agent knows only his own local objective function and constraint set, and exchanges information with the other agents over a randomly varying network topology to update his information state. We assume a state-dependent communication model over this topology: communication is Markovian with respect to the states of the agents and the p…
We study the problem of minimizing the sum of potentially non-differentiable convex cost functions with partially overlapping dependences in an asynchronous manner, where communication in the network is not coordinated. We study the behavior of an asynchronous algorithm based on dual decomposition and block coordinate subgradient methods under assumptions weaker than those used in the literature. At the same time, we allow different agents to use local stepsizes with no global coordination. Sufficient conditions are provided for almost sure convergence to the solution of the optimization probl…
We consider a multi-agent optimization problem where agents subject to local, intermittent interactions aim to minimize a sum of local objective functions subject to a global inequality constraint and a global state constraint set. In contrast to previous work, we do not require that the objective, constraint functions, and state constraint sets to be convex. In order to deal with time-varying network topologies satisfying a standard connectivity assumption, we resort to consensus algorithm techniques and the Lagrangian duality method. We slightly relax the requirement of exact consensus, and …
Dual decomposition has been successfully employed in a variety of distributed convex optimization problems solved by a network of computing and communicating nodes. Often, when the cost function is separable but the constraints are coupled, the dual decomposition scheme involves local parallel subgradient calculations and a global subgradient update performed by a master node. In this paper, we propose a consensus-based dual decomposition to remove the need for such a master node and still enable the computing nodes to generate an approximate dual solution for the underlying convex optimizatio…
Distributed consensus optimization has received considerable attention in recent years; several distributed consensus-based algorithms have been proposed for (nonsmooth) convex and (smooth) nonconvex objective functions. However, the behavior of these distributed algorithms on {\it nonconvex, nonsmooth and stochastic} objective functions is not understood. This class of functions and distributed setting are motivated by several applications, including problems in machine learning and signal processing. This paper presents the first convergence analysis of the decentralized stochastic subgradie…
We study a class of bilevel convex optimization problems where the goal is to find the minimizer of an objective function in the upper level, among the set of all optimal solutions of an optimization problem in the lower level. A wide range of problems in convex optimization can be formulated using this class. An important example is the case where an optimization problem is ill-posed. In this paper, our interest lies in addressing the bilevel problems, where the lower level objective is given as a finite sum of separate nondifferentiable convex component functions. This is the case in a varie…
We consider minimizing a sum of non-smooth objective functions with set constraints in a distributed manner. As to this problem, we propose a distributed algorithm with an exponential convergence rate for the first time. By the exact penalty method, we reformulate the problem equivalently as a standard distributed one without consensus constraints. Then we design a distributed projected subgradient algorithm with the help of differential inclusions. Furthermore, we show that the algorithm converges to the optimal solution exponentially for strongly convex objective functions.
This technical note considers a distributed convex optimization problem with nonsmooth cost functions and coupled nonlinear inequality constraints. To solve the problem, we first propose a modified Lagrangian function containing local multipliers and a nonsmooth penalty function. Then we construct a distributed continuous-time algorithm by virtue of a projected primal-dual subgradient dynamics. Based on the nonsmooth analysis and Lyapunov function, we obtain the existence of the solution to the nonsmooth algorithm and its convergence.
This paper considers a distributed convex optimization problem over a time-varying multi-agent network, where each agent has its own decision variables that should be set so as to minimize its individual objective subject to local constraints and global coupling equality constraints. Over directed graphs, a distributed algorithm is proposed that incorporates the push-sum protocol into dual subgradient methods. Under the convexity assumption, the optimality of primal and dual variables, and constraint violations is first established. Then the explicit convergence rates of the proposed algorithm…
We consider the problem of minimizing the sum of non-smooth convex functions in non-Euclidean spaces, e.g., probability simplex, via only local computation and communication on an undirected graph. We propose two algorithms motivated by mass-spring-damper dynamics. The first algorithm uses an explicit update that computes subgradients and Bregman projections only, and matches the convergence behavior of centralized mirror descent. The second algorithm uses an implicit update that solves a penalized subproblem at each iteration, and achieves the iteration complexity of \(\mathcal{O}(1/K)\). The…
This paper deals with an optimization problem over a network of agents, where the cost function is the sum of the individual 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…
We study the problem of the decentralized computation of entropy-regularized semi-discrete Wasserstein barycenters over a network. Building upon recent primal-dual approaches, we propose a sampling gradient quantization scheme that allows efficient communication and computation of approximate barycenters where the factor distributions are stored distributedly on arbitrary networks. The communication and algorithmic complexity of the proposed algorithm are shown, with explicit dependency on the size of the support, the number of distributions, and the desired accuracy. Numerical results validat…
We consider a network of agents that are cooperatively solving a global optimization problem, where the objective function is the sum of privately known local objective functions of the agents and the decision variables are coupled via linear constraints. Recent literature focused on special cases of this formulation and studied their distributed solution through either subgradient based methods with O(1/sqrt(k)) rate of convergence (where k is the iteration number) or Alternating Direction Method of Multipliers (ADMM) based methods, which require a synchronous implementation and a globally kn…
In this paper, we focus on an aggregative optimization problem under the communication bottleneck. The aggregative optimization is to minimize the sum of local cost functions. Each cost function depends on not only local state variables but also the sum of functions of global state variables. The goal is to solve the aggregative optimization problem through distributed computation and local efficient communication over a network of agents without a central coordinator. Using the variable tracking method to seek the global state variables and the quantization scheme to reduce the communication …
We consider a distributed optimal power flow formulated as an optimization problem that maximizes a nondifferentiable concave function. Solving such a problem by the existing distributed algorithms can lead to data privacy issues because the solution information exchanged within the algorithms can be utilized by an adversary to infer the data. To preserve data privacy, in this paper we propose a differentially private projected subgradient (DP-PS) algorithm that includes a solution encryption step. We show that a sequence generated by DP-PS converges in expectation, in probability, and with pr…
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. This imperfect communication poses a fundamental challenge, and this imperfect communication, if not properly accounted for, prevents the convergence of these algorithms. Our first contribution in this paper is to propose a modified consensus-based gradient method for solving such problems using random (dithered) quantization. This algorithm can be interpreted as a distributed variant of a well-known two-tim…
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…
This paper studies distributed algorithms for the extended monotropic optimization problem, which is a general convex optimization problem with a certain separable structure. The considered objective function is the sum of local convex functions assigned to agents in a multi-agent network, with private set constraints and affine equality constraints. Each agent only knows its local objective function, local constraint set, and neighbor information. We propose two novel continuous-time distributed subgradient-based algorithms with projected output feedback and derivative feedback, respectively,…
In this paper we study the effect of stochastic errors on two constrained incremental sub-gradient algorithms. We view the incremental sub-gradient algorithms as decentralized network optimization algorithms as applied to minimize a sum of functions, when each component function is known only to a particular agent of a distributed network. We first study the standard cyclic incremental sub-gradient algorithm in which the agents form a ring structure and pass the iterate in a cycle. We consider the method with stochastic errors in the sub-gradient evaluations and provide sufficient conditions o…
This paper is concerned with multi-agent optimization problem. A distributed randomized gradient-free mirror descent (DRGFMD) method is developed by introducing a randomized gradient-free oracle in the mirror descent scheme where the non-Euclidean Bregman divergence is used. The classical gradient descent method is generalized without using subgradient information of objective functions. The proposed algorithm is the first distributed non-Euclidean zeroth-order method which achieves an $O(1/\sqrt{T})$ convergence rate, recovering the best known optimal rate of distributed compact constrained c…
We consider centralized and distributed mirror descent algorithms over a finite-dimensional Hilbert space, and prove that the problem variables converge to an optimizer of a possibly nonsmooth function when the step sizes are square summable but not summable. Prior literature has focused on the convergence of the function value to its optimum. However, applications from distributed optimization and learning in games require the convergence of the variables to an optimizer, which is generally not guaranteed without assuming strong convexity of the objective function. We provide numerical simula…
This paper considers a distributed optimization problem in the presence of Byzantine agents capable of introducing untrustworthy information into the communication network. A resilient distributed subgradient algorithm is proposed based on graph redundancy and objective redundancy. It is shown that the algorithm causes all non-Byzantine agents' states to asymptotically converge to the same optimal point under appropriate assumptions. A partial convergence rate result is also provided.
We consider the distributed optimization problem for the sum of convex functions where the underlying communications network connecting agents at each time is drawn at random from a collection of directed graphs. Building on an earlier work [15], where a modified version of the subgradient-push algorithm is shown to be almost surely convergent to an optimizer on sequences of random directed graphs, we find an upper bound of the order of $\sim O(\frac{1}{\sqrt{t}})$ on the convergence rate of our proposed algorithm, establishing the first convergence bound in such random settings.
Distributed adaptive signal processing has attracted much attention in the recent decade owing to its effectiveness in many decentralized real-time applications in networked systems. Because many natural signals are highly sparse with most entries equal to zero, several decentralized sparse adaptive algorithms have been proposed recently. Most of them is focused on the single task estimation problems, in which all nodes receive data associated with the same unknown vector and collaborate to estimate it. However, many applications are inherently multitask oriented and each node has its own unkn…
