Additively blockseparability in the optimization variables. Periodic and eventtriggered communication for distributed continuoustime convex optimization solmaz s. Distributed online convex optimization over jointly. 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. Parallel and distributed successive convex approximation. In this paper, we have studied the problem of distributed optimization of nonsmooth and strongly convex functions. The implementation of the algorithms removes the need for performing the intermediate projections. For any vector,we denote, while is the 2norm in the. We design and analyze a fully distributed algorithm for convex constrained optimization in networks without any consistent naming infrastructure. 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. Abstractwe study diffusion and consensus based optimization of a sum of unknown convex objective functions over distributed networks. Optimal algorithms for nonsmooth distributed optimization.
Pdf optimal algorithms for smooth and strongly convex. Finally, optimal convergence rates for distributed algorithms were investigated in 8 for smooth and stronglyconvex objective functions, and 16, 17 for totally connected networks. In this paper, a distributed convex optimization algorithm under persistent attacks is investigated in the framework of hybrid dynamical systems. M,l strongly convex functions be the set of ail continuously differentiable convex functions f with the properties. Distributed smooth and strongly convex optimization with inexact dual methods mahyar fazlyab, santiago paternain, alejandro ribeiro and victor m. 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. A control perspective for centralized and distributed convex optimization. Olnt on twice differentiable strongly convex functions, where t denotes the time horizon, see, for example. Based on pushsum protocol and dual decomposition, we design a regularized dual gradient distributed algorithm to solve this. We propose a class of distributed stochastic gradient algorithms that solve the problem using only local computation and communication. Pushsum distributed dual averaging for convex optimization konstantinos i.
Eventtriggered quantized communicationbased distributed. Optimal algorithms for smooth and strongly convex distributed optimization in networks 3. Relaxing the nonconvex problem to a convex problem convex neural networks strategy 3. Distributed optimization has recently seen a surge of interest. At each round, each agent in the network commits to a decision 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. In this study, the authors propose a distributed discretetime algorithm for unconstrained optimisation with eventtriggered communication over weightbalanced directed networks. 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.
Contrary to what is known in the consensus literature, where the same dynamics works for both undirected and. Stochastic subgradient algorithms for strongly convex. As a result, many algorithms were recently introduced to minimize the average f 1 n p. Juan xu, kaiqing zhang distributed optimization 7 32. Distributed strongly convex optimization ieee conference.
The existence of attacks may influence the behavior of an algorithm that solves the optimization problem. 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 online convex optimization on timevarying. Optimal algorithms for smooth and strongly convexdistributed. On distributed convex optimization under inequality and equality constraints 153 such that the following supgradient inequality holds for any. The first algorithm recovers the best previously known rate, and our second algorithm attains the optimal convergence rate. We have proposed two efficient noneuclidean algorithms based on mirror descent. Stability analysis of distributed convex optimization. Local nonconvex optimization convexity convergence rates apply escape saddle points using, for example, cubic regularization and saddlefree newton update strategy 2. A lot of effort has been invested into characterizing the convergence rates of gradient based algorithms for nonlinear convex optimization. Of course, many optimization problems are not convex, and it can be di. Distributed nesterov gradient methods over arbitrary graphs.
Distributed subgradient projection algorithm for convex. And so, in the rest of the paper, we rigorously study the attainable performance for distributed stochastic optimization and learning. Communication complexity of distributed convex learning. 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. Optimal convergence rates for convex distributed optimization in. Optimal algorithms for smooth and strongly convex distributed. Optimal algorithms for smooth and strongly convex distributed optimization in networks kevin scaman1 francis bach2 sebastien bubeck. Recently, motivated by large datasets and problems in machine learning, the interest has shifted towards distributed optimization. Distributed nonconvex constrained optimization over time. Distributed convex optimization stanford university. Parallel and distributed blockcoordinate frankwolfe algorithms. Optimal algorithms for smooth and strongly convex distributed optimization in networks such approaches is the. 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.
For strongly convex optimization, we employ a smoothed constraint continue reading. Pushsum distributed dual averaging for convex optimization. Optimal distributed convex optimization on slowly time. Stochastic subgradient algorithms for strongly convex optimization over distributed networks muhammed o. Our main goal is to help the reader develop a working knowledge of convex optimization, i. Harnessing smoothness to accelerate distributed optimization. On the generalization ability of online strongly convex. Theprivatedistributedoptimization problem a private distributed optimization pdop problem p for nagents is speci ed by four parameters. Distributed smooth and strongly convex optimization with. 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. Strongly convex functions on compact domains have unique minima 7. The majority of these works studied distributed strongly convex optimization overundirectedgraphs,with 5 assumingthat all the functions.
Revisiting projectionfree optimization for strongly convex. A series of works on distributed optimization is based on distributed consensus and subgradient methods. This paper presents a class of subgradientpush algorithms for online distributed optimization over timevarying networks. In this paper, we determine the optimal convergence rates for strongly convex and smooth distributed optimization in two settings. Distributed algorithms for robust convex optimization via.
For example, based on the distributed consensus algorithm, a distributed subgradient method under a general communication network was studied in 16. Later in 18, 5, the authors extended these results to nonsmooth problems or non strongly convex problems. Revisiting projectionfree optimization for strongly convex constraint sets. Distributed convex optimization arezou keshavarz, brendan odonoghue, eric chu, and stephen boyd informationsystemslaboratory,electricalengineering,stanforduniversity convex optimization a convex optimization problem is as follows. Distributed strongly convex optimization request pdf. More over, the four regularity assumptions that we investigate in this paper are. Key tradeo pay expensive communication cost to exchange for.
Convex optimization with random pursuit research collection. We revisit the frankwolfe fw optimization under strongly convex constraint. Distributed subgradient projection algorithm for convex optimization s. C where the cost function f is convex obeys jensens inequality. Communicationefficient distributed block minimization for. Distributed subgradientpush online convex optimization on. Damon moskaoyama, tim roughgarden, and devavrat shah abstract. 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. Lipschitz con tinuity, strong convexity, smoothness, and both strong convexity and. 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. 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. 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. Contrary to what is known in the consensus literature, where the same dynamics. Distributed online convex optimization over jointly connected digraphs.
In this work we present a distributed algorithm for strongly convex constrained optimization. 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. Distributed convex optimization algorithm mathematics. Accelerated distributed nesterov gradient descent for. Optimal stochastic approximation algorithms for strongly. Optimal algorithms for smooth and strongly convex distributed optimization in. Online strongly convex programming algorithms sham m. The class of optimization problems along with the key features of the algorithms proposed in these papers are summarized in table 1 and brie. Mainly, it was shown that distributed optimization. In this setting, a private strongly convex objective function is revealed to each agent at each time step. We let the function denote the projection operator onto the nonnegative orthant in. 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. Here, we analyze gradientfree optimization algorithms on convex functions. Blackbox optimization procedures the lower bounds provided hereafter depend on a new notion of blackbox optimization procedures for the problem in eq.
A lot of effort has been invested into characterizing the convergence rates of gradient based algorithms for non. We consider the problem of distributed convex learning and optimization, where a set of mma. Distributed continuoustime convex optimization on weightbalanced digraphs bahman gharesifard jorge cort. 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. 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. Online convex optimization is a sequential paradigm in which at each round, the learner predicts a. In this case, an interesting question is under what conditions the optimal solution can be found. Continuoustime distributed convex optimization on weight. Periodic and eventtriggered communication for distributed. Each node in a network of n computers converges to the optimum of a strongly convex. Optimal distributed stochastic mirror descent for strongly.
1109 1655 833 851 1297 1106 429 1360 836 708 631 1131 1480 304 1594 32 1208 1029 1042 1284 1325 305 116 1395 728 1088 41 649 735 870 273 698 470 532 1333 904 629 487 413 967 790 204 581