Hyperfast second-order local solvers for efficient statistically preconditioned distributed optimization
Statistical preconditioning enables fast methods for distributed large-scale empirical risk minimization problems. In this approach, multiple worker nodes compute gradients in parallel, which are then used by the central node to update the parameter by solving an auxiliary (preconditioned) smaller-s...
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| Veröffentlicht in: | EURO journal on computational optimization Jg. 10; S. 100045 |
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2022
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| Abstract | Statistical preconditioning enables fast methods for distributed large-scale empirical risk minimization problems. In this approach, multiple worker nodes compute gradients in parallel, which are then used by the central node to update the parameter by solving an auxiliary (preconditioned) smaller-scale optimization problem. The recently proposed Statistically Preconditioned Accelerated Gradient (SPAG) method [1] has complexity bounds superior to other such algorithms but requires an exact solution for computationally intensive auxiliary optimization problems at every iteration. In this paper, we propose an Inexact SPAG (InSPAG) and explicitly characterize the accuracy by which the corresponding auxiliary subproblem needs to be solved to guarantee the same convergence rate as the exact method. We build our results by first developing an inexact adaptive accelerated Bregman proximal gradient method for general optimization problems under relative smoothness and strong convexity assumptions, which may be of independent interest. Moreover, we explore the properties of the auxiliary problem in the InSPAG algorithm assuming Lipschitz third-order derivatives and strong convexity. For such problem class, we develop a linearly convergent Hyperfast second-order method and estimate the total complexity of the InSPAG method with hyperfast auxiliary problem solver. Finally, we illustrate the proposed method's practical efficiency by performing large-scale numerical experiments on logistic regression models. To the best of our knowledge, these are the first empirical results on implementing high-order methods on large-scale problems, as we work with data where the dimension is of the order of 3 million, and the number of samples is 700 million.
•Inexact Statistically Preconditioned Accelerated Gradient Method for large-scale distributed convex empirical risk minimization problems.•Hyperfast second-order algorithm for minimizing strongly convex functions with Lipschitz third-order derivatives.•Inexact adaptive accelerated Bregman proximal gradient method for minimization under relative smoothness and strong convexity assumptions.•Empirical evidence for the efficiency of tensor optimization methods for large-scale ERM problems. |
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| AbstractList | Statistical preconditioning enables fast methods for distributed large-scale empirical risk minimization problems. In this approach, multiple worker nodes compute gradients in parallel, which are then used by the central node to update the parameter by solving an auxiliary (preconditioned) smaller-scale optimization problem. The recently proposed Statistically Preconditioned Accelerated Gradient (SPAG) method [1] has complexity bounds superior to other such algorithms but requires an exact solution for computationally intensive auxiliary optimization problems at every iteration. In this paper, we propose an Inexact SPAG (InSPAG) and explicitly characterize the accuracy by which the corresponding auxiliary subproblem needs to be solved to guarantee the same convergence rate as the exact method. We build our results by first developing an inexact adaptive accelerated Bregman proximal gradient method for general optimization problems under relative smoothness and strong convexity assumptions, which may be of independent interest. Moreover, we explore the properties of the auxiliary problem in the InSPAG algorithm assuming Lipschitz third-order derivatives and strong convexity. For such problem class, we develop a linearly convergent Hyperfast second-order method and estimate the total complexity of the InSPAG method with hyperfast auxiliary problem solver. Finally, we illustrate the proposed method's practical efficiency by performing large-scale numerical experiments on logistic regression models. To the best of our knowledge, these are the first empirical results on implementing high-order methods on large-scale problems, as we work with data where the dimension is of the order of 3 million, and the number of samples is 700 million.
•Inexact Statistically Preconditioned Accelerated Gradient Method for large-scale distributed convex empirical risk minimization problems.•Hyperfast second-order algorithm for minimizing strongly convex functions with Lipschitz third-order derivatives.•Inexact adaptive accelerated Bregman proximal gradient method for minimization under relative smoothness and strong convexity assumptions.•Empirical evidence for the efficiency of tensor optimization methods for large-scale ERM problems. Statistical preconditioning enables fast methods for distributed large-scale empirical risk minimization problems. In this approach, multiple worker nodes compute gradients in parallel, which are then used by the central node to update the parameter by solving an auxiliary (preconditioned) smaller-scale optimization problem. The recently proposed Statistically Preconditioned Accelerated Gradient (SPAG) method [1] has complexity bounds superior to other such algorithms but requires an exact solution for computationally intensive auxiliary optimization problems at every iteration. In this paper, we propose an Inexact SPAG (InSPAG) and explicitly characterize the accuracy by which the corresponding auxiliary subproblem needs to be solved to guarantee the same convergence rate as the exact method. We build our results by first developing an inexact adaptive accelerated Bregman proximal gradient method for general optimization problems under relative smoothness and strong convexity assumptions, which may be of independent interest. Moreover, we explore the properties of the auxiliary problem in the InSPAG algorithm assuming Lipschitz third-order derivatives and strong convexity. For such problem class, we develop a linearly convergent Hyperfast second-order method and estimate the total complexity of the InSPAG method with hyperfast auxiliary problem solver. Finally, we illustrate the proposed method's practical efficiency by performing large-scale numerical experiments on logistic regression models. To the best of our knowledge, these are the first empirical results on implementing high-order methods on large-scale problems, as we work with data where the dimension is of the order of 3 million, and the number of samples is 700 million. |
| ArticleNumber | 100045 |
| Author | Dvurechensky, Pavel Lukashevich, Aleksandr Lee, Soomin Ordentlich, Erik Uribe, César A. Kamzolov, Dmitry Gasnikov, Alexander |
| Author_xml | – sequence: 1 givenname: Pavel orcidid: 0000-0003-1201-2343 surname: Dvurechensky fullname: Dvurechensky, Pavel email: pavel.dvurechensky@wias-berlin.de organization: Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany – sequence: 2 givenname: Dmitry orcidid: 0000-0001-8488-9692 surname: Kamzolov fullname: Kamzolov, Dmitry email: kamzolov.dmitry@phystech.edu organization: Moscow Institute of Physics and Technology, Dolgoprudny, Russia – sequence: 3 givenname: Aleksandr orcidid: 0000-0002-4986-9941 surname: Lukashevich fullname: Lukashevich, Aleksandr email: aleksandr.lukashevich@skoltech.ru organization: Center for Energy Science and Technology, Skolkovo Institute of Science and Technology, Moscow, Russia – sequence: 4 givenname: Soomin surname: Lee fullname: Lee, Soomin email: soominl@yahooinc.com organization: Yahoo! Research, Sunnyvale, CA, United States of America – sequence: 5 givenname: Erik surname: Ordentlich fullname: Ordentlich, Erik email: eord@yahooinc.com organization: Yahoo! Research, Sunnyvale, CA, United States of America – sequence: 6 givenname: César A. orcidid: 0000-0002-7080-9724 surname: Uribe fullname: Uribe, César A. email: cauribe@rice.edu organization: Rice University, Houston, TX, United States of America – sequence: 7 givenname: Alexander surname: Gasnikov fullname: Gasnikov, Alexander email: gasnikov.av@mipt.ru organization: Moscow Institute of Physics and Technology, Dolgoprudny, Russia |
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| Keywords | Empirical risk minimization Distributed optimization Tensor optimization methods Statistical preconditioning |
| Language | English |
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