Abstract
In this paper, we consider a distributed nonsmooth optimization problem over a computational multi-agent network. We first extend the (centralized) Nesterov’s random gradient-free algorithm and Gaussian smoothing technique to the distributed case. Then, the convergence of the algorithm is proved. Furthermore, an explicit convergence rate is given in terms of the network size and topology. Our proposed method is free of gradient, which may be preferred by practical engineers. Since only the cost function value is required, our method may suffer a factor up to \(d\) (the dimension of the agent) in convergence rate over that of the distributed subgradient-based methods in theory. However, our numerical simulations show that for some nonsmooth problems, our method can even achieve better performance than that of subgradient-based methods, which may be caused by the slow convergence in the presence of subgradient.
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References
Tsitsiklis, J.N.: Problems in decentralized decision making and computation. Ph.D. Thesis, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology (1984)
Tsitsiklis, J.N., Bertsekas, D.P., Athans, M.: Distributed asynchronous deterministic and stochastic gradient optimization algorithms. IEEE Trans. Autom. Control 31(9), 803–812 (1986)
Bertsekas, D.P., Tsitsiklis, J.N.: Parallel and Distributed Computation: Numerical Methods. Athena Scientific, Belmont (1997)
Xiao, L., Boyd, S., Kim, S.J.: Distributed average consensus with least-mean-square deviation. J. Parallel Distrib. Comput. 67(1), 33–46 (2007)
Nedić, A., Bertsekas, D.P.: Incremental subgradient methods for nondifferentiabl optimization. SIAM J. Optim. 12(1), 109–138 (2001)
Bertsekas, D.P.: Incremental proximal methods for large scale convex optimization. Math. Program. 129(2), 163–195 (2011)
Johansson, B., Rabi, M., Johansson, M.: A randomized incremental subgradient method for distributed optimization in networked systems. SIAM J. Optim. 20(3), 1157–1170 (2009)
Nedić, A., Ozdaglar, A., Parrilo, P.A.: Constrained consensus and optimization in multi-agent networks. IEEE Trans. Autom. Control 55(4), 922–938 (2010)
Nedić, A., Ozdaglar, : Distributed subgradient methods for multi-agent optimization. IEEE Trans. Autom. Control 54(1), 48–61 (2009)
Duchi, J., Agarwal, A., Wainwright, M.: Dual averaging for distributed optimization: convergence analysis and network scaling. IEEE Trans. Autom. Control 57(3), 592–606 (2012)
Nesterov, Y.: Primal-dual subgradient methods for convex problems. Math. Program. A 120(1), 261–283 (2009)
Xiao, L.: Dual averaging methods for regularized stochastic learning and online optimization. J. Mach. Learn. Res. 11, 2543–2596 (2010)
Nedić, A., Ozdaglar, A.: Convergence rate for consensus with delays. J. Glob. Optim. 47(3), 437–456 (2010)
Boyd, S., Ghosh, A., Prabhakar, B., Shah, D.: Randomized gossip algorithms. IEEE Trans. Inf. Theory 52(6), 2508–2530 (2006)
Ram, S.S., Nedić, A., Veeravalli, V.V.: Distributed stochastic subgradient projection algorithms for convex optimization. J. Optim. Theory Appl. 147(3), 516–545 (2011)
Spall, J.C.: Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control. Wiley, Hoboken, NJ (2003)
Bagirov, A.M., Ugon, J.: Piecewise partially separable functions and a derivative-free algorithm for large scale nonsmooth optimization. J. Glob. Optim. 35(2), 163–195 (2006)
Belitz, P., Bewley, T.: New horizons in sphere-packing theory, part II: lattice-based derivative-free optimization via global surrogates. J. Glob. Optim. 56, 61–91 (2013). doi:10.1007/s10898-012-9866-7
Nesterov, Y.: Random gradient-free minimization of convex functions, Technical report, Center for Operations Research and Econometrics (CORE). Catholic University of Louvain (2011)
Shao, C.S., Richard, B., Elizabeth, E., Robert, B.S.: Global optimization for molecular clusters using a new smoothing approach. J. Glob. Optim. 16(2), 167–196 (2000)
Duchi, J.C., Bartlett, P.L., Wainwrighr, M.J.: Randomized smoothing for stochastic optimization. SIAM J. Optim. 22(2), 674–701 (2012)
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)
Polyak, B.T., Tsypkin, J.: Robust identification. Automatica 16, 53–63 (1980)
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This research was partially supported by Natural Science Foundation of Chongqing: cstc2013jjB00001 and cstc2011jjA00010, by Chongqing Municipal Education Commission under Grant: KJ120616, by Postgraduate Scholarship of Federation University Australia, by NSFC11001288, by Key Project of Chinese Ministry of Education: 210179, by Natural Science Foundation of Chongqing: cstc2013jjB0149 and cstc2013jcyjA1338, and by CMEC under Grant: KJ090802.
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Li, J., Wu, C., Wu, Z. et al. Gradient-free method for nonsmooth distributed optimization. J Glob Optim 61, 325–340 (2015). https://doi.org/10.1007/s10898-014-0174-2
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DOI: https://doi.org/10.1007/s10898-014-0174-2