Distributed Stochastic Mirror Descent Algorithm Over Time-varying Network

In this paper, we propose a distributed stochastic mirror descent algorithm for solving distributed general (non-differentiable) convex optimization problem over a time-varying multi-agent network. We adopt Bregman divergence rather than Euclidean distance as the augmented distance measuring function to solve the distributed first-order Lagrangian-based convex optimization problem. With a fixed step-size, our algorithm achieves a convergence rate $O\left (\frac{1}{T}\right)$ with an error bound, which is the best known convergence rate for distributed first-order algorithms. Numerical experiments demonstrate the performance of the proposed algorithm.

[1]  Asuman E. Ozdaglar,et al.  Convergence Rate of Distributed ADMM Over Networks , 2016, IEEE Transactions on Automatic Control.

[2]  Deming Yuan,et al.  Distributed Primal-Dual Subgradient Method for Multiagent Optimization via Consensus Algorithms. , 2011, IEEE transactions on systems, man, and cybernetics. Part B, Cybernetics : a publication of the IEEE Systems, Man, and Cybernetics Society.

[3]  Angelia Nedic,et al.  Stochastic Gradient-Push for Strongly Convex Functions on Time-Varying Directed Graphs , 2014, IEEE Transactions on Automatic Control.

[4]  L. Bregman.  The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming , 1967 .

[5]  Yiguang Hong,et al.  Distributed regression estimation with incomplete data in multi-agent networks , 2018, Science China Information Sciences.

[6]  Qiong Wu,et al.  Distributed Mirror Descent over Directed Graphs , 2014, ArXiv.

[7]  Qing Ling,et al.  EXTRA: An Exact First-Order Algorithm for Decentralized Consensus Optimization , 2014, 1404.6264.

[8]  Stephen P. Boyd,et al.  Fast linear iterations for distributed averaging , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[9]  Martin J. Wainwright,et al.  Dual Averaging for Distributed Optimization: Convergence Analysis and Network Scaling , 2010, IEEE Transactions on Automatic Control.

[10]  Stephen P. Boyd,et al.  Distributed average consensus with least-mean-square deviation , 2007, J. Parallel Distributed Comput..

[11]  Asuman E. Ozdaglar,et al.  Constrained Consensus and Optimization in Multi-Agent Networks , 2008, IEEE Transactions on Automatic Control.

[12]  Han-Fu Chen,et al.  Primal-dual algorithm for distributed constrained optimization , 2015, Syst. Control. Lett..

[13]  Marc Teboulle,et al.  Mirror descent and nonlinear projected subgradient methods for convex optimization , 2003, Oper. Res. Lett..

[14]  Asuman E. Ozdaglar,et al.  Distributed Subgradient Methods for Multi-Agent Optimization , 2009, IEEE Transactions on Automatic Control.

[15]  Daniel W. C. Ho,et al.  Optimal distributed stochastic mirror descent for strongly convex optimization , 2016, Autom..

[16]  Feng Liu,et al.  Initialization-free distributed algorithms for optimal resource allocation with feasibility constraints and application to economic dispatch of power systems , 2015, Autom..

[17]  Shouyang Wang,et al.  Distributed continuous-time approximate projection protocols for shortest distance optimization problems , 2015, Autom..

[18]  G. Hug,et al.  Distributed robust economic dispatch in power systems: A consensus + innovations approach , 2012, 2012 IEEE Power and Energy Society General Meeting.

[19]  Qing Ling,et al.  On the Linear Convergence of the ADMM in Decentralized Consensus Optimization , 2013, IEEE Transactions on Signal Processing.