You can also prove that the global problem is minimized exactly when the local problems are minimized, although that only tells you that if your distributed optimization process converges, then it has found the global minimum. Here, we analyze gradientfree optimization algorithms on convex functions. Pdf optimal algorithms for smooth and strongly convex. Additively blockseparability in the optimization variables. The class of optimization problems along with the key features of the algorithms proposed in these papers are summarized in table 1 and brie. While the classical stochastic approximation algorithms are asymptotically optimal for solving differentiable and strongly convex problems, the acsa algorithm, when employed with proper stepsize policies, can achieve optimal or nearly optimal rates of convergence for solving different classes of sco problems during a given number of iterations.
Accelerated distributed nesterov gradient descent for smooth and strongly convex functions guannan qu, na li abstract this paper considers the distributed optimization problem over a network, where the objective is to optimize a global function formed by a sum of local functions, using only local computation and communication. Periodic and eventtriggered communication for distributed. Key tradeo pay expensive communication cost to exchange for. We revisit the frankwolfe fw optimization under strongly convex constraint. In this paper we study new stochastic approximation sa type algorithms, namely, the accelerated sa acsa, for solving strongly convex stochastic composite optimization sco problems. Communication complexity of distributed convex learning and optimization yossi arjevani. Distributed nesterov gradient methods over arbitrary graphs.
Periodic and eventtriggered communication for distributed continuoustime convex optimization solmaz s. We design and analyze a fully distributed algorithm for convex constrained optimization in networks without any consistent naming infrastructure. Later in 18, 5, the authors extended these results to nonsmooth problems or non strongly convex problems. Each node in a network of n computers converges to the optimum of a strongly convex, llipchitz continuous, separable objective at a rate olog sqrtn t t where t is the number of iterations. Harnessing smoothness to accelerate distributed optimization guannan qu, na li abstract there has been a growing effort in studying the distributed optimization problem over a network. In this setting, a private strongly convex objective function is revealed to each agent at each time step. Recently, motivated by large datasets and problems in machine learning, the interest has shifted towards distributed optimization. We let the function denote the projection operator onto the nonnegative orthant in. Communication complexity of distributed convex learning. Stochastic subgradient algorithms for strongly convex optimization over distributed networks muhammed o.
Optimal algorithms for smooth and strongly convex distributed optimization in networks 3. Revisiting projectionfree optimization for strongly convex. Based on pushsum protocol and dual decomposition, we design a regularized dual gradient distributed algorithm to solve this. Distributed subgradientpush online convex optimization on.
The existence of attacks may influence the behavior of an algorithm that solves the optimization problem. We have proposed two efficient noneuclidean algorithms based on mirror descent. Mainly, it was shown that distributed optimization. More over, the four regularity assumptions that we investigate in this paper are.
Eventtriggered quantized communicationbased distributed. In this study, the authors propose a distributed discretetime algorithm for unconstrained optimisation with eventtriggered communication over weightbalanced directed networks. The implementation of the algorithms removes the need for performing the intermediate projections. The first algorithm recovers the best previously known rate, and our second algorithm attains the optimal convergence rate. Contrary to what is known in the consensus literature, where the same dynamics works for both undirected and. A lot of effort has been invested into characterizing the convergence rates of gradient based algorithms for nonlinear convex optimization. A series of works on distributed optimization is based on distributed consensus and subgradient methods. References 9, 10 consider distributed first order strongly convex optimization for static networks, assuming that the data distributions that underlie each nodes local cost function are equal. With the interest in decentralized architectures and motivated by the problem of distributed convex optimization, a distributed version of online optimization is proposed in 15 and 16. M,l strongly convex functions be the set of ail continuously differentiable convex functions f with the properties. Local nonconvex optimization convexity convergence rates apply escape saddle points using, for example, cubic regularization and saddlefree newton update strategy 2.
Distributed subgradient projection algorithm for convex. Introduction distributed optimization nds many applications in machine learning, for example when the data set is large and training is achieved using a cluster of computing units. Lipschitz con tinuity, strong convexity, smoothness, and both strong convexity and. This paper investigates a distributed optimization problem over a cooperative multiagent timevarying network, where each agent has its own decision variables that should be set so as to minimize its individual objective subjected to global coupled constraints. Stability analysis of distributed convex optimization. Distributed strongly convex optimization ieee conference. Convex optimization with random pursuit research collection. Contrary to what is known in the consensus literature, where the same dynamics. In this paper, we have studied the problem of distributed optimization of nonsmooth and strongly convex functions. Finally, optimal convergence rates for distributed algorithms were investigated in 8 for smooth and stronglyconvex objective functions, and 16, 17 for totally connected networks. We consider the problem of distributed convex learning and optimization, where a set of mma. C where the cost function f is convex obeys jensens inequality. In this paper, we determine the optimal convergence rates for strongly convex and smooth distributed optimization in two settings.
Theprivatedistributedoptimization problem a private distributed optimization pdop problem p for nagents is speci ed by four parameters. Optimal algorithms for smooth and strongly convex distributed optimization in networks such approaches is the. Optimal algorithms for nonsmooth distributed optimization. Harnessing smoothness to accelerate distributed optimization. For strongly convex optimization, we employ a smoothed constraint continue reading. On distributed convex optimization under inequality and equality constraints 153 such that the following supgradient inequality holds for any. Pushsum distributed dual averaging for convex optimization.
Distributed online convex optimization over jointly. The majority of these works studied distributed strongly convex optimization overundirectedgraphs,with 5 assumingthat all the functions. Preciado abstractin this paper, we consider a class of decentralized convex optimization problems in which a network of agents aims to minimize a global objective function that is a sum of. Each node in a network of n computers converges to the optimum of a strongly convex. In this paper, a distributed convex optimization algorithm under persistent attacks is investigated in the framework of hybrid dynamical systems. Optimal convergence rates for convex distributed optimization in. We propose a class of distributed stochastic gradient algorithms that solve the problem using only local computation and communication. A control perspective for centralized and distributed convex optimization. Relaxing the nonconvex problem to a convex problem convex neural networks strategy 3. Our main goal is to help the reader develop a working knowledge of convex optimization, i. Online strongly convex programming algorithms sham m. A lot of effort has been invested into characterizing the convergence rates of gradient based algorithms for non.
Distributed algorithms for robust convex optimization via the scenario approach keyou you, member, ieee, roberto tempo, fellow, ieee, and pei xie abstractthis paper proposes distributed algorithms to solve robust convex optimization rco when the constraints are affected by nonlinear uncertainty. For example, based on the distributed consensus algorithm, a distributed subgradient method under a general communication network was studied in 16. In this case, an interesting question is under what conditions the optimal solution can be found. Distributed online convex optimization over jointly connected digraphs. Parallel and distributed successive convex approximation. Distributed strongly convex optimization request pdf. And so, in the rest of the paper, we rigorously study the attainable performance for distributed stochastic optimization and learning. Continuoustime distributed convex optimization on weight. As a result, many algorithms were recently introduced to minimize the average f 1 n p. Distributed algorithms for robust convex optimization via. Abstractwe study diffusion and consensus based optimization of a sum of unknown convex objective functions over distributed networks.
Distributed smooth and strongly convex optimization with inexact dual methods mahyar fazlyab, santiago paternain, alejandro ribeiro and victor m. The idea of tracking the gradient averages through the use of consensus coupled with distributed optimization was independently introduced in 12,14 next framework for constrained, nonsmooth, nonconvex instances of p over timevarying graphs and in for the case of strongly convex, unconstrained, smooth optimization over static. Strongly convex functions on compact domains have unique minima 7. Olnt on twice differentiable strongly convex functions, where t denotes the time horizon, see, for example.
Distributed optimization has recently seen a surge of interest. In this work we present a distributed algorithm for strongly convex constrained optimization. Parallel and distributed blockcoordinate frankwolfe algorithms. For any vector,we denote, while is the 2norm in the. Juan xu, kaiqing zhang distributed optimization 7 32. Online convex optimization is a sequential paradigm in which at each round, the learner predicts a. Wainwright, senior member, ieee abstractthe goal of decentralized optimization over a network is to optimize a global objective formed by a sum of local possibly nonsmooth convex functions using only local. Distributed nonconvex constrained optimization over time. Distributed online convex optimization on timevarying. Optimal stochastic approximation algorithms for strongly.
Optimal algorithms for smooth and strongly convexdistributed. Distributed convex optimization arezou keshavarz, brendan odonoghue, eric chu, and stephen boyd informationsystemslaboratory,electricalengineering,stanforduniversity convex optimization a convex optimization problem is as follows. Of course, many optimization problems are not convex, and it can be di. Distributed convex optimization algorithm mathematics. Accelerated distributed nesterov gradient descent for. Optimal algorithms for smooth and strongly convex distributed optimization in. Optimal algorithms for smooth and strongly convex distributed optimization in networks kevin scaman1 francis bach2 sebastien bubeck. Distributed smooth and strongly convex optimization with.
They consider a multiagent system where each agent has a state and an auxiliary variable for the estimates of the optimal solution and the average gradient of the entire cost function. Optimal distributed stochastic mirror descent for strongly. Damon moskaoyama, tim roughgarden, and devavrat shah abstract. Distributed subgradient projection algorithm for convex optimization s. In the next time step, this agent makes a decision about its state using this knowledge, along with the information gathered only from its neighboring. Distributed continuoustime convex optimization on weightbalanced digraphs bahman gharesifard jorge cort. On the generalization ability of online strongly convex. References 7, 8 consider distributed strongly convex optimization for static networks, assuming that the data distributions that underlie each nodes local cost function are equal reference. Revisiting projectionfree optimization for strongly convex constraint sets. Optimal algorithms for smooth and strongly convex distributed. Stochastic subgradient algorithms for strongly convex. At each round, each agent in the network commits to a decision and. Blackbox optimization procedures the lower bounds provided hereafter depend on a new notion of blackbox optimization procedures for the problem in eq. Communicationefficient distributed block minimization for.
507 555 744 1320 166 935 1500 649 971 1585 1148 1265 370 824 1398 764 394 1352 359 1207 1349 1168 877 191 1372 153 1079 1466 1471 1171 402