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Sample size selection in optimization methods for machine learning

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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 \({O(1/\epsilon)}\) 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 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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Authors and Affiliations

  1. Department of Computer Science, University of Colorado, Boulder, CO, USA

    Richard H. Byrd

  2. Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL, USA

    Gillian M. Chin & Jorge Nocedal

  3. Google Inc., Mountain View, CA, USA

    Yuchen Wu

Authors
  1. Richard H. Byrd
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  2. Gillian M. Chin
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  3. Jorge Nocedal
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  4. Yuchen Wu
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Corresponding author

Correspondence to Jorge Nocedal.

Additional information

This work was supported by National Science Foundation grant CMMI 0728190 and grant DMS-0810213, by Department of Energy grant DE-FG02-87ER25047-A004 and grant DE-SC0001774, and by an NSERC fellowship and a Google grant.

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Byrd, R.H., Chin, G.M., Nocedal, J. et al. Sample size selection in optimization methods for machine learning. Math. Program. 134, 127–155 (2012). https://doi.org/10.1007/s10107-012-0572-5

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  • DOI: https://doi.org/10.1007/s10107-012-0572-5

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