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Oracle inequalities for multi-fold cross validation

Author

Listed:
  • Vaart Aad W. van der
  • Dudoit Sandrine
  • Laan Mark J. van der

Abstract

We consider choosing an estimator or model from a given class by cross validation consisting of holding a nonneglible fraction of the observations out as a test set. We derive bounds that show that the risk of the resulting procedure is (up to a constant) smaller than the risk of an oracle plus an error which typically grows logarithmically with the number of estimators in the class. We extend the results to penalized cross validation in order to control unbounded loss functions. Applications include regression with squared and absolute deviation loss and classification under Tsybakov’s condition.

Suggested Citation

  • Vaart Aad W. van der & Dudoit Sandrine & Laan Mark J. van der, 2006. "Oracle inequalities for multi-fold cross validation," Statistics & Risk Modeling, De Gruyter, vol. 24(3), pages 1-21, December.
    • Handle: RePEc:bpj:strimo:v:24:y:2006:i:3:p:21:n:3
    DOI: 10.1524/stnd.2006.24.3.351
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    References listed on IDEAS

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    1. Andrews, Donald W. K., 1991. "Asymptotic optimality of generalized CL, cross-validation, and generalized cross-validation in regression with heteroskedastic errors," Journal of Econometrics, Elsevier, vol. 47(2-3), pages 359-377, February.
    2. Laan Mark J. van der & Dudoit Sandrine & Vaart Aad W. van der, 2006. "The cross-validated adaptive epsilon-net estimator," Statistics & Risk Modeling, De Gruyter, vol. 24(3), pages 1-23, December.
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    Cited by:

    1. Chakraborty, Chiranjit & Joseph, Andreas, 2017. "Machine learning at central banks," Bank of England working papers 674, Bank of England.
    2. Zihao Li & Hui Lan & Vasilis Syrgkanis & Mengdi Wang & Masatoshi Uehara, 2024. "Regularized DeepIV with Model Selection," Papers 2403.04236, arXiv.org.
    3. van der Laan Mark J., 2010. "Targeted Maximum Likelihood Based Causal Inference: Part I," The International Journal of Biostatistics, De Gruyter, vol. 6(2), pages 1-45, February.
    4. Díaz Muñoz Iván & van der Laan Mark J., 2011. "Super Learner Based Conditional Density Estimation with Application to Marginal Structural Models," The International Journal of Biostatistics, De Gruyter, vol. 7(1), pages 1-20, October.
    5. I Díaz & O Savenkov & K Ballman, 2018. "Targeted learning ensembles for optimal individualized treatment rules with time-to-event outcomes," Biometrika, Biometrika Trust, vol. 105(3), pages 723-738.
    6. Verhagen, Mark D., 2021. "Identifying and Improving Functional Form Complexity: A Machine Learning Framework," SocArXiv bka76, Center for Open Science.
    7. Fahimeh Hadavimoghaddam & Mehdi Ostadhassan & Ehsan Heidaryan & Mohammad Ali Sadri & Inna Chapanova & Evgeny Popov & Alexey Cheremisin & Saeed Rafieepour, 2021. "Prediction of Dead Oil Viscosity: Machine Learning vs. Classical Correlations," Energies, MDPI, vol. 14(4), pages 1-16, February.
    8. Mahmood Zafar & Khan Salahuddin, 2009. "On the Use of K-Fold Cross-Validation to Choose Cutoff Values and Assess the Performance of Predictive Models in Stepwise Regression," The International Journal of Biostatistics, De Gruyter, vol. 5(1), pages 1-21, July.
    9. van der Laan Mark J. & Gruber Susan, 2010. "Collaborative Double Robust Targeted Maximum Likelihood Estimation," The International Journal of Biostatistics, De Gruyter, vol. 6(1), pages 1-71, May.
    10. Zhang, Yongli & Yang, Yuhong, 2015. "Cross-validation for selecting a model selection procedure," Journal of Econometrics, Elsevier, vol. 187(1), pages 95-112.
    11. Laan Mark J. van der & Dudoit Sandrine & Vaart Aad W. van der, 2006. "The cross-validated adaptive epsilon-net estimator," Statistics & Risk Modeling, De Gruyter, vol. 24(3), pages 1-23, December.

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