Sample size selection in optimization methods for machine learning
This paper presents a methodology for using varying sample sizes in batch-type optimization methods for large-scale machine learning problems. The first part of the paper deals with the delicate issue of dynamic sample selection in the evaluation of the function and gradient. We propose a criterion...
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| Vydáno v: | Mathematical programming Ročník 134; číslo 1; s. 127 - 155 |
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| Hlavní autoři: | , , , |
| Médium: | Journal Article Konferenční příspěvek |
| Jazyk: | angličtina |
| Vydáno: |
Berlin/Heidelberg
Springer-Verlag
01.08.2012
Springer Springer Nature B.V |
| Témata: | |
| ISSN: | 0025-5610, 1436-4646 |
| On-line přístup: | Získat plný text |
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| Abstract | This paper presents a methodology for using varying sample sizes in batch-type optimization methods for large-scale machine learning problems. The first part of the paper deals with the delicate issue of dynamic sample selection in the evaluation of the function and gradient. We propose a criterion for increasing the sample size based on variance estimates obtained during the computation of a batch gradient. We establish an
complexity bound on the total cost of a gradient method. The second part of the paper describes a practical Newton method that uses a smaller sample to compute Hessian vector-products than to evaluate the function and the gradient, and that also employs a dynamic sampling technique. The focus of the paper shifts in the third part of the paper to
L
1
-regularized problems designed to produce sparse solutions. We propose a Newton-like method that consists of two phases: a (minimalistic) gradient projection phase that identifies zero variables, and subspace phase that applies a subsampled Hessian Newton iteration in the free variables. Numerical tests on speech recognition problems illustrate the performance of the algorithms. |
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| AbstractList | Issue Title: Special Issue on ISMP 2012 This paper presents a methodology for using varying sample sizes in batch-type optimization methods for large-scale machine learning problems. The first part of the paper deals with the delicate issue of dynamic sample selection in the evaluation of the function and gradient. We propose a criterion for increasing the sample size based on variance estimates obtained during the computation of a batch gradient. We establish an $${O(1/\epsilon)}$$ complexity bound on the total cost of a gradient method. The second part of the paper describes a practical Newton method that uses a smaller sample to compute Hessian vector-products than to evaluate the function and the gradient, and that also employs a dynamic sampling technique. The focus of the paper shifts in the third part of the paper to L ^sub 1^-regularized problems designed to produce sparse solutions. We propose a Newton-like method that consists of two phases: a (minimalistic) gradient projection phase that identifies zero variables, and subspace phase that applies a subsampled Hessian Newton iteration in the free variables. Numerical tests on speech recognition problems illustrate the performance of the algorithms.[PUBLICATION ABSTRACT] This paper presents a methodology for using varying sample sizes in batch-type optimization methods for large-scale machine learning problems. The first part of the paper deals with the delicate issue of dynamic sample selection in the evaluation of the function and gradient. We propose a criterion for increasing the sample size based on variance estimates obtained during the computation of a batch gradient. We establish an complexity bound on the total cost of a gradient method. The second part of the paper describes a practical Newton method that uses a smaller sample to compute Hessian vector-products than to evaluate the function and the gradient, and that also employs a dynamic sampling technique. The focus of the paper shifts in the third part of the paper to L 1 -regularized problems designed to produce sparse solutions. We propose a Newton-like method that consists of two phases: a (minimalistic) gradient projection phase that identifies zero variables, and subspace phase that applies a subsampled Hessian Newton iteration in the free variables. Numerical tests on speech recognition problems illustrate the performance of the algorithms. This paper presents a methodology for using varying sample sizes in batch-type optimization methods for large-scale machine learning problems. The first part of the paper deals with the delicate issue of dynamic sample selection in the evaluation of the function and gradient. We propose a criterion for increasing the sample size based on variance estimates obtained during the computation of a batch gradient. We establish an $${O(1/\epsilon)}$$ complexity bound on the total cost of a gradient method. The second part of the paper describes a practical Newton method that uses a smaller sample to compute Hessian vector-products than to evaluate the function and the gradient, and that also employs a dynamic sampling technique. The focus of the paper shifts in the third part of the paper to L sub(1)-regularized problems designed to produce sparse solutions. We propose a Newton-like method that consists of two phases: a (minimalistic) gradient projection phase that identifies zero variables, and subspace phase that applies a subsampled Hessian Newton iteration in the free variables. Numerical tests on speech recognition problems illustrate the performance of the algorithms. |
| Author | Nocedal, Jorge Wu, Yuchen Chin, Gillian M. Byrd, Richard H. |
| Author_xml | – sequence: 1 givenname: Richard H. surname: Byrd fullname: Byrd, Richard H. organization: Department of Computer Science, University of Colorado – sequence: 2 givenname: Gillian M. surname: Chin fullname: Chin, Gillian M. organization: Department of Industrial Engineering and Management Sciences, Northwestern University – sequence: 3 givenname: Jorge surname: Nocedal fullname: Nocedal, Jorge email: nocedal@eecs.northwestern.edu organization: Department of Industrial Engineering and Management Sciences, Northwestern University – sequence: 4 givenname: Yuchen surname: Wu fullname: Wu, Yuchen organization: Google Inc |
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| PublicationTitle | Mathematical programming |
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