Convergence of Distributed Stochastic Variance Reduced Methods Without Sampling Extra Data
Stochastic variance reduced methods have gained a lot of interest recently for empirical risk minimization due to its appealing run time complexity. When the data size is large and disjointly stored on different machines, it becomes imperative to distribute the implementation of such variance reduce...
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| Veröffentlicht in: | IEEE transactions on signal processing Jg. 68; S. 3976 - 3989 |
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2020
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| Abstract | Stochastic variance reduced methods have gained a lot of interest recently for empirical risk minimization due to its appealing run time complexity. When the data size is large and disjointly stored on different machines, it becomes imperative to distribute the implementation of such variance reduced methods. In this paper, we consider a general framework that directly distributes popular stochastic variance reduced methods in the master/slave model, by assigning outer loops to the parameter server, and inner loops to worker machines. This framework is natural and friendly to implement, but its theoretical convergence is not well understood. We obtain a comprehensive understanding of algorithmic convergence with respect to data homogeneity by measuring the smoothness of the discrepancy between the local and global loss functions. We establish the linear convergence of distributed versions of a family of stochastic variance reduced algorithms, including those using accelerated and recursive gradient updates, for minimizing strongly convex losses. Our theory captures how the convergence of distributed algorithms behaves as the number of machines and the size of local data vary. Furthermore, we show that when the data are less balanced, regularization can be used to ensure convergence at a slower rate. We also demonstrate that our analysis can be further extended to handle nonconvex loss functions. |
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| AbstractList | Stochastic variance reduced methods have gained a lot of interest recently for empirical risk minimization due to its appealing run time complexity. When the data size is large and disjointly stored on different machines, it becomes imperative to distribute the implementation of such variance reduced methods. In this paper, we consider a general framework that directly distributes popular stochastic variance reduced methods in the master/slave model, by assigning outer loops to the parameter server, and inner loops to worker machines. This framework is natural and friendly to implement, but its theoretical convergence is not well understood. We obtain a comprehensive understanding of algorithmic convergence with respect to data homogeneity by measuring the smoothness of the discrepancy between the local and global loss functions. We establish the linear convergence of distributed versions of a family of stochastic variance reduced algorithms, including those using accelerated and recursive gradient updates, for minimizing strongly convex losses. Our theory captures how the convergence of distributed algorithms behaves as the number of machines and the size of local data vary. Furthermore, we show that when the data are less balanced, regularization can be used to ensure convergence at a slower rate. We also demonstrate that our analysis can be further extended to handle nonconvex loss functions. |
| Author | Chen, Wei Chi, Yuejie Cen, Shicong Liu, Tie-Yan Zhang, Huishuai |
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| References | tang (ref38) 0 ref14 zinkevich (ref21) 0 lin (ref36) 0 lian (ref35) 0 reddi (ref5) 2016 nguyen (ref10) 2019 defazio (ref24) 0 lewis (ref45) 2004; 5 zhou (ref11) 0 shamir (ref7) 0 ref39 zhao (ref12) 0 wang (ref29) 0 hu (ref43) 0 zhang (ref17) 2015 lee (ref6) 2017; 18 li (ref41) 0 bertsekas (ref15) 1989; 23 smith (ref19) 2017; 18 mokhtari (ref40) 2016; 17 goldstein (ref4) 0 recht (ref2) 0 nguyen (ref9) 0 kone?n? (ref1) 2015 ref46 shalev-shwartz (ref25) 2013; 14 wangni (ref37) 0 ref23 ref26 ref42 guyon (ref44) 0 alistarh (ref30) 0 wang (ref18) 0 ref22 alistarh (ref34) 0 johnson (ref3) 0 fan (ref20) 2019 bernstein (ref31) 0 zhao (ref13) 0 lin (ref27) 0 wang (ref8) 0 shamir (ref16) 0 seide (ref32) 0 wen (ref33) 0 fang (ref28) 0 |
| References_xml | – volume: 23 year: 1989 ident: ref15 publication-title: Parallel and Distributed Computation Numerical Methods – start-page: 545 year: 0 ident: ref44 article-title: Result analysis of the nips 2003 feature selection challenge publication-title: Proc Advances Neural Inf Process Syst – start-page: 2406 year: 0 ident: ref29 article-title: SpiderBoost and momentum: Faster variance reduction algorithms publication-title: Proc Advances Neural Inf Process Syst – volume: 5 start-page: 361 year: 2004 ident: ref45 article-title: RCV1: A new benchmark collection for text categorization research publication-title: J Mach Learn Res – start-page: 362 year: 2015 ident: ref17 article-title: DiSCO: Distributed optimization for self-concordant empirical loss publication-title: Int Conf Mach Learn – start-page: 2338 year: 0 ident: ref18 article-title: Giant: Globally improved approximate newton method for distributed optimization publication-title: Proc Advances Neural Inf Process Syst – start-page: 5973 year: 0 ident: ref34 article-title: The convergence of sparsified gradient methods publication-title: Proc Advances Neural Inf Process Syst – start-page: 1000 year: 0 ident: ref16 article-title: Communication-efficient distributed optimization using an approximate Newton-type method publication-title: Proc Int Conf Mach Learn – ident: ref42 doi: 10.1214/17-AOS1637 – ident: ref26 doi: 10.1145/3055399.3055448 – year: 2016 ident: ref5 article-title: Aide: Fast and communication efficient distributed optimization – start-page: 1662 year: 0 ident: ref41 publication-title: Proc Int Conf Artif Intell Statist – ident: ref23 doi: 10.1007/s10107-016-1030-6 – start-page: 1646 year: 0 ident: ref24 article-title: Saga: A fast incremental gradient method with support for non-strongly convex composite objectives publication-title: Proc Advances Neural Inf Process Syst – ident: ref39 doi: 10.1109/TSP.2018.2872003 – volume: 17 start-page: 2165 year: 2016 ident: ref40 article-title: DSA: Decentralized double stochastic averaging gradient algorithm publication-title: J Mach Learn Res – start-page: 2928 year: 0 ident: ref12 article-title: SCOPE: Scalable composite optimization for learning on spark publication-title: Proc 31st AAAI Conf Artif Intell – start-page: 1709 year: 0 ident: ref30 article-title: QSGD: Communication-efficient SGD via gradient quantization and encoding publication-title: Proc Advances Neural Inf Process Syst – volume: 14 start-page: 567 year: 2013 ident: ref25 article-title: Stochastic dual coordinate ascent methods for regularized loss minimization publication-title: J Mach Learn Res – start-page: 2595 year: 0 ident: ref21 article-title: Parallelized stochastic gradient descent publication-title: Proc Advances Neural Inf Process Syst – year: 2019 ident: ref20 article-title: Communication-efficient accurate statistical estimation publication-title: arXiv 1906 04870 – start-page: 559 year: 0 ident: ref31 article-title: SignSDG: Compressed optimisation for non-convex problems publication-title: Proc Int Conf Mach Learn – start-page: 5975 year: 0 ident: ref11 article-title: A simple stochastic variance reduced algorithm with fast convergence rates publication-title: Proc Int Conf Mach Learn – start-page: 687 year: 0 ident: ref28 article-title: Spider: Near-optimal non-convex optimization via stochastic path-integrated differential estimator publication-title: Proc Advances Neural Inf Process Syst – ident: ref22 doi: 10.1109/CDC.2012.6426691 – start-page: 315 year: 0 ident: ref3 article-title: Accelerating stochastic gradient descent using predictive variance reduction publication-title: Proc Advances Neural Inf Process Syst – volume: 18 start-page: 8590 year: 2017 ident: ref19 article-title: CoCoA: A general framework for communication-efficient distributed optimization publication-title: J Mach Learn Res – start-page: 693 year: 0 ident: ref2 article-title: Hogwild: A lock-free approach to parallelizing stochastic gradient descent publication-title: Proc Advances Neural Inf Process Syst – start-page: 4855 year: 0 ident: ref38 article-title: ${D}^{2}$: Decentralized training over decentralized data publication-title: Proc Int Conf Mach Learn – year: 2015 ident: ref1 article-title: Federated optimization: Distributed optimization beyond the datacenter – start-page: 1058 year: 0 ident: ref32 publication-title: Proc Annu Conf Int Speech Commun Assoc – volume: 18 start-page: 4404 year: 2017 ident: ref6 article-title: Distributed stochastic variance reduced gradient methods by sampling extra data with replacement publication-title: J Mach Learn Res – year: 0 ident: ref36 publication-title: Proc Int Conf Learn Representations – year: 2019 ident: ref10 article-title: Finite-sum smooth optimization with SARAH – start-page: 5330 year: 0 ident: ref35 article-title: Can decentralized algorithms outperform centralized algorithms? A case study for decentralized parallel stochastic gradient descent publication-title: Proc Advances Neural Inf Process Syst – start-page: 46 year: 0 ident: ref7 article-title: Without-replacement sampling for stochastic gradient methods publication-title: Proc Advances Neural Inf Process Syst – start-page: 6552 year: 0 ident: ref13 article-title: Proximal SCOPE for distributed sparse learning publication-title: Proc Advances Neural Inf Process Syst – start-page: 1882 year: 0 ident: ref8 article-title: Memory and communication efficient distributed stochastic optimization with minibatch prox publication-title: Proc Conf Learn Theory – start-page: 1299 year: 0 ident: ref37 article-title: Gradient sparsification for communication-efficient distributed optimization publication-title: Proc Advances Neural Inf Process Syst – start-page: 111 year: 0 ident: ref4 article-title: Efficient distributed sgd with variance reduction publication-title: Proc IEEE 16th Int Conf Data Mining – ident: ref14 doi: 10.1561/2200000016 – start-page: 2613 year: 0 ident: ref9 article-title: Sarah: A novel method for machine learning problems using stochastic recursive gradient publication-title: Proc Int Conf Mach Learn – ident: ref46 doi: 10.1137/140961791 – start-page: 3384 year: 0 ident: ref27 article-title: A universal catalyst for first-order optimization publication-title: Proc Advances Neural Inf Process Syst – start-page: 2038 year: 0 ident: ref43 publication-title: Proc Int Conf Mach Learn – start-page: 1509 year: 0 ident: ref33 article-title: Terngrad: Ternary gradients to reduce communication in distributed deep learning publication-title: Proc Advances Neural Inf Process Syst |
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| SubjectTerms | Algorithms Convergence Distributed databases Distributed optimization Empirical analysis Homogeneity master/slave model Optimization Regularization Risk management Servers Signal processing algorithms Smoothness stochastic optimization Stochastic processes variance reduction |
| Title | Convergence of Distributed Stochastic Variance Reduced Methods Without Sampling Extra Data |
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