An Efficient Method To Estimate Bagging‘s Generalization Error

  • Authors:
  • David H. Wolpert;William G. Macready

  • Affiliations:
  • NASA Ames Research Center, Caelum Research, MS 269-1, Moffett Field, CA 94035. dhw@ptolemy.arc.nasa.gov;Bios Group, LP, 317 Paseo de Peralta, Santa Fe, NM 87501. wgm@biosgroup.com

  • Venue:
  • Machine Learning
  • Year:
  • 1999

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Abstract

Bagging (Breiman, 1994a) is a technique that tries toimprove a learning algorithm‘s performance by using bootstrapreplicates of the training set (Efron & Tibshirani, 1993, Efron, 1979). The computational requirements for estimating the resultantgeneralization error on a test set by means of cross-validation areoften prohibitive, for leave-one-out cross-validation one needs totrain the underlying algorithm on the order of mν times, wherem is the size of the training set and ν is the number ofreplicates. This paper presents several techniques for estimatingthe generalization error of a bagged learning algorithm withoutinvoking yet more training of the underlying learning algorithm(beyond that of the bagging itself), as is required bycross-validation-based estimation. These techniques all exploit thebias-variance decomposition (Geman, Bienenstock & Doursat, 1992, Wolpert, 1996). The best of ourestimators also exploits stacking (Wolpert, 1992). In a set ofexperiments reported here, it was found to be more accurate than boththe alternative cross-validation-based estimator of the baggedalgorithm‘s error and the cross-validation-based estimator of theunderlying algorithm‘s error. This improvement was particularlypronounced for small test sets. This suggests a novel justificationfor using bagging—more accurate estimation of the generalizationerror than is possible without bagging.