Stochastic Strongly Convex Optimization via Distributed Epoch Stochastic Gradient Algorithm
This article considers the problem of stochastic strongly convex optimization over a network of multiple interacting nodes. The optimization is under a global inequality constraint and the restriction that nodes have only access to the stochastic gradients of their objective functions. We propose an...
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| Vydáno v: | IEEE transaction on neural networks and learning systems Ročník 32; číslo 6; s. 2344 - 2357 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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IEEE
01.06.2021
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 2162-237X, 2162-2388, 2162-2388 |
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| Abstract | This article considers the problem of stochastic strongly convex optimization over a network of multiple interacting nodes. The optimization is under a global inequality constraint and the restriction that nodes have only access to the stochastic gradients of their objective functions. We propose an efficient distributed non-primal-dual algorithm, by incorporating the inequality constraint into the objective via a smoothing technique. We show that the proposed algorithm achieves an optimal <inline-formula> <tex-math notation="LaTeX">\mathcal {O}((1)/(T)) </tex-math></inline-formula> (<inline-formula> <tex-math notation="LaTeX">T </tex-math></inline-formula> is the total number of iterations) convergence rate in the mean square distance from the optimal solution. In particular, we establish a high probability bound for the proposed algorithm, by showing that with a probability at least <inline-formula> <tex-math notation="LaTeX">1-\delta </tex-math></inline-formula>, the proposed algorithm converges at a rate of <inline-formula> <tex-math notation="LaTeX">\mathcal {O}(\ln (\ln (T)/\delta)/ T) </tex-math></inline-formula>. Finally, we provide numerical experiments to demonstrate the efficacy of the proposed algorithm. |
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| AbstractList | This article considers the problem of stochastic strongly convex optimization over a network of multiple interacting nodes. The optimization is under a global inequality constraint and the restriction that nodes have only access to the stochastic gradients of their objective functions. We propose an efficient distributed non-primal-dual algorithm, by incorporating the inequality constraint into the objective via a smoothing technique. We show that the proposed algorithm achieves an optimal O((1)/(T)) ( T is the total number of iterations) convergence rate in the mean square distance from the optimal solution. In particular, we establish a high probability bound for the proposed algorithm, by showing that with a probability at least 1-δ , the proposed algorithm converges at a rate of O(ln(ln(T)/δ)/ T) . Finally, we provide numerical experiments to demonstrate the efficacy of the proposed algorithm.This article considers the problem of stochastic strongly convex optimization over a network of multiple interacting nodes. The optimization is under a global inequality constraint and the restriction that nodes have only access to the stochastic gradients of their objective functions. We propose an efficient distributed non-primal-dual algorithm, by incorporating the inequality constraint into the objective via a smoothing technique. We show that the proposed algorithm achieves an optimal O((1)/(T)) ( T is the total number of iterations) convergence rate in the mean square distance from the optimal solution. In particular, we establish a high probability bound for the proposed algorithm, by showing that with a probability at least 1-δ , the proposed algorithm converges at a rate of O(ln(ln(T)/δ)/ T) . Finally, we provide numerical experiments to demonstrate the efficacy of the proposed algorithm. This article considers the problem of stochastic strongly convex optimization over a network of multiple interacting nodes. The optimization is under a global inequality constraint and the restriction that nodes have only access to the stochastic gradients of their objective functions. We propose an efficient distributed non-primal-dual algorithm, by incorporating the inequality constraint into the objective via a smoothing technique. We show that the proposed algorithm achieves an optimal [Formula Omitted] ([Formula Omitted] is the total number of iterations) convergence rate in the mean square distance from the optimal solution. In particular, we establish a high probability bound for the proposed algorithm, by showing that with a probability at least [Formula Omitted], the proposed algorithm converges at a rate of [Formula Omitted]. Finally, we provide numerical experiments to demonstrate the efficacy of the proposed algorithm. This article considers the problem of stochastic strongly convex optimization over a network of multiple interacting nodes. The optimization is under a global inequality constraint and the restriction that nodes have only access to the stochastic gradients of their objective functions. We propose an efficient distributed non-primal-dual algorithm, by incorporating the inequality constraint into the objective via a smoothing technique. We show that the proposed algorithm achieves an optimal <inline-formula> <tex-math notation="LaTeX">\mathcal {O}((1)/(T)) </tex-math></inline-formula> (<inline-formula> <tex-math notation="LaTeX">T </tex-math></inline-formula> is the total number of iterations) convergence rate in the mean square distance from the optimal solution. In particular, we establish a high probability bound for the proposed algorithm, by showing that with a probability at least <inline-formula> <tex-math notation="LaTeX">1-\delta </tex-math></inline-formula>, the proposed algorithm converges at a rate of <inline-formula> <tex-math notation="LaTeX">\mathcal {O}(\ln (\ln (T)/\delta)/ T) </tex-math></inline-formula>. Finally, we provide numerical experiments to demonstrate the efficacy of the proposed algorithm. |
| Author | Ho, Daniel W. C. Xu, Shengyuan Yuan, Deming |
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| References_xml | – volume: 57 start-page: 151 year: 2012 ident: ref27 article-title: On distributed convex optimization under inequality and equality constraints publication-title: IEEE Trans Autom Control doi: 10.1109/TAC.2011.2167817 – ident: ref11 doi: 10.1109/TAC.2011.2160020 – ident: ref32 doi: 10.1109/TAC.2017.2688452 – ident: ref30 doi: 10.1016/j.sysconle.2004.02.022 – start-page: 122 year: 2016 ident: ref16 article-title: Optimal stochastic strongly convex optimization with a logarithmic number of projections publication-title: Proc 32nd Conf Uncertainty Artif Intell (UAI) – ident: ref8 doi: 10.1109/TNNLS.2016.2549566 – ident: ref26 doi: 10.1109/TCYB.2015.2464255 – ident: ref24 doi: 10.1109/CDC.2018.8619228 – ident: ref22 doi: 10.1109/TAC.2016.2529285 – ident: ref33 doi: 10.1016/j.sysconle.2015.06.006 – ident: ref13 doi: 10.1109/TCYB.2017.2681119 – ident: ref25 doi: 10.1109/TAC.2014.2308612 – ident: ref5 doi: 10.1109/TSMC.2016.2531649 – volume: 15 start-page: 2489 year: 2014 ident: ref14 article-title: Beyond the regret minimization barrier: An optimal algorithm for stochastic strongly-convex optimization publication-title: J Mach Learn Res – start-page: 503 year: 2012 ident: ref18 article-title: Stochastic gradient descent with only one projection publication-title: Proc Adv Neural Inf Process Syst – ident: ref19 doi: 10.1137/15M1048896 – volume: 3176 start-page: 208 year: 2004 ident: ref29 publication-title: Concentration Inequalities – ident: ref10 doi: 10.1109/TAC.2013.2278132 – ident: ref12 doi: 10.1109/TCYB.2015.2453167 – ident: ref3 doi: 10.1109/TAC.2011.2161027 – ident: ref31 doi: 10.1109/TAC.2015.2504962 – ident: ref20 doi: 10.1007/s10957-010-9737-7 – ident: ref9 doi: 10.1109/TAC.2013.2275671 – ident: ref1 doi: 10.1109/TAC.2008.2009515 – ident: ref23 doi: 10.1016/j.automatica.2017.12.053 – ident: ref7 doi: 10.1109/TAC.2014.2309261 – start-page: 3901 year: 2017 ident: ref17 article-title: A richer theory of convex constrained optimization with reduced projections and improved rates publication-title: Proc 34th Int Conf Mach Learn (ICML) – ident: ref28 doi: 10.1109/CDC.2008.4738860 – ident: ref6 doi: 10.1109/TNNLS.2014.2336806 – ident: ref2 doi: 10.1109/TAC.2015.2416927 – ident: ref4 doi: 10.1109/TAC.2010.2041686 – ident: ref21 doi: 10.1109/Allerton.2012.6483272 – start-page: 449 year: 2012 ident: ref15 article-title: Making gradient descent optimal for strongly convex stochastic optimization publication-title: Proc 29th Int Conf Mach Learn |
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| SubjectTerms | Algorithms Approximation algorithms Computational geometry Convergence Convergence rate Convex analysis Convex functions Convexity Distributed algorithms distributed stochastic strongly optimization epoch gradient descent inequality constraint Linear programming multiagent systems Nodes Optimization Probability theory Stochastic processes Stochasticity |
| Title | Stochastic Strongly Convex Optimization via Distributed Epoch Stochastic Gradient Algorithm |
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