Distributed mirror descent method for saddle point problems over directed graphs

Li, Jueyou; Chen, Guo; Dong, Zhaoyang; Wu, Zhiyou; Yao, Minghai

doi:10.1002/cplx.21794

Cited by 5 publications

(3 citation statements)

References 42 publications

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“…Of particular relevance to this work is [28], where decentralized mirror descent has been developed for when agents receive the gradients with a delay. More recently, the application of mirror descent to saddle point problems is studied in [29]. Moreover, Rabbat in [30] proposes a decentralized mirror descent for stochastic composite optimization problems and provide guarantees for strongly convex regularizers.…”

Section: A Related Literaturementioning

confidence: 99%

Distributed Online Optimization in Dynamic Environments Using Mirror Descent

Shahrampour

Jadbabaie

2018

IEEE Trans. Automat. Contr.

266

209

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This work addresses decentralized online optimization in non-stationary environments. A network of agents aim to track the minimizer of a global time-varying convex function. The minimizer evolves according to a known dynamics corrupted by an unknown, unstructured noise. At each time, the global function can be cast as a sum of a finite number of local functions, each of which is assigned to one agent in the network. Moreover, the local functions become available to agents sequentially, and agents do not have a prior knowledge of the future cost functions. Therefore, agents must communicate with each other to build an online approximation of the global function. We propose a decentralized variation of the celebrated Mirror Descent, developed by Nemirovksi and Yudin. Using the notion of Bregman divergence in lieu of Euclidean distance for projection, Mirror Descent has been shown to be a powerful tool in large-scale optimization. Our algorithm builds on Mirror Descent, while ensuring that agents perform a consensus step to follow the global function and take into account the dynamics of the global minimizer. To measure the performance of the proposed online algorithm, we compare it to its offline counterpart, where the global functions are available a priori. The gap between the two is called dynamic regret. We establish a regret bound that scales inversely in the spectral gap of the network, and more notably it represents the deviation of minimizer sequence with respect to the given dynamics. We then show that our results subsume a number of results in distributed optimization. We demonstrate the application of our method to decentralized tracking of dynamic parameters and verify the results via numerical experiments.

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Section: A Related Literaturementioning

confidence: 99%

Distributed Online Optimization in Dynamic Environments Using Mirror Descent

Shahrampour

Jadbabaie

2018

IEEE Trans. Automat. Contr.

266

209

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“…In general, there are two ways to choose such that it satisfies the requirement. One way is by solving the unconstrained optimization problem defined in (12), and the other one is by heuristic. In order to imitate the time-varying weight matrix, a pool of 50 weight matrices from connected random graphs are generated, in which each weight matrix is satisfied (Assumption 1).…”

Section: Numerical Simulationsmentioning

confidence: 99%

“…In [6], Nedic et al generalized the distributed method in [5] to solve constrained convex optimization. Later, many researchers proposed various extensions based on primal (sub)gradient-based methods in, for example, [7][8][9][10][11][12]. In [13], Duchi et al extended the centralized dual averaging algorithm to the distributed setting and then proposed a distributed dual averaging algorithm.…”

Section: Introductionmentioning

confidence: 99%

Distributed Optimization Methods for Nonconvex Problems with Inequality Constraints over Time-Varying Networks

et al. 2017

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Network-structured optimization problems are found widely in engineering applications. In this paper, we investigate a nonconvex distributed optimization problem with inequality constraints associated with a time-varying multiagent network, in which each agent is allowed to locally access its own cost function and collaboratively minimize a sum of nonconvex cost functions for all the agents in the network. Based on successive convex approximation techniques, we first approximate locally the nonconvex problem by a sequence of strongly convex constrained subproblems. In order to realize distributed computation, we then exploit the exact penalty function method to transform the sequence of convex constrained subproblems into unconstrained ones. Finally, a fully distributed method is designed to solve the unconstrained subproblems. The convergence of the proposed algorithm is rigorously established, which shows that the algorithm can converge asymptotically to a stationary solution of the problem under consideration. Several simulation results are illustrated to show the performance of the proposed method.

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Distributed mirror descent algorithm over unbalanced digraphs based on gradient weighting technique

Shi,

Yang

2023

Journal of the Franklin Institute

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Distributed mirror descent method for saddle point problems over directed graphs

Cited by 5 publications

References 42 publications

Distributed Online Optimization in Dynamic Environments Using Mirror Descent

Distributed Online Optimization in Dynamic Environments Using Mirror Descent

Distributed Optimization Methods for Nonconvex Problems with Inequality Constraints over Time-Varying Networks

Distributed mirror descent algorithm over unbalanced digraphs based on gradient weighting technique

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