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Orthogonal Statistical Learning


  • Dylan J. Foster
  • Vasilis Syrgkanis


We provide non-asymptotic excess risk guarantees for statistical learning in a setting where the population risk with respect to which we evaluate the target parameter depends on an unknown nuisance parameter that must be estimated from data. We analyze a two-stage sample splitting meta-algorithm that takes as input two arbitrary estimation algorithms: one for the target parameter and one for the nuisance parameter. We show that if the population risk satisfies a condition called Neyman orthogonality, the impact of the nuisance estimation error on the excess risk bound achieved by the meta-algorithm is of second order. Our theorem is agnostic to the particular algorithms used for the target and nuisance and only makes an assumption on their individual performance. This enables the use of a plethora of existing results from statistical learning and machine learning to give new guarantees for learning with a nuisance component. Moreover, by focusing on excess risk rather than parameter estimation, we can give guarantees under weaker assumptions than in previous works and accommodate settings in which the target parameter belongs to a complex nonparametric class. We provide conditions on the metric entropy of the nuisance and target classes such that oracle rates---rates of the same order as if we knew the nuisance parameter---are achieved. We also derive new rates for specific estimation algorithms such as variance-penalized empirical risk minimization, neural network estimation and sparse high-dimensional linear model estimation. We highlight the applicability of our results in four settings of central importance: 1) heterogeneous treatment effect estimation, 2) offline policy optimization, 3) domain adaptation, and 4) learning with missing data.

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  • Dylan J. Foster & Vasilis Syrgkanis, 2019. "Orthogonal Statistical Learning," Papers 1901.09036,, revised Sep 2020.
  • Handle: RePEc:arx:papers:1901.09036

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    References listed on IDEAS

    1. Yingqi Zhao & Donglin Zeng & A. John Rush & Michael R. Kosorok, 2012. "Estimating Individualized Treatment Rules Using Outcome Weighted Learning," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(499), pages 1106-1118, September.
    2. Robinson, Peter M, 1988. "Root- N-Consistent Semiparametric Regression," Econometrica, Econometric Society, vol. 56(4), pages 931-954, July.
    3. Lester Mackey & Vasilis Syrgkanis & Ilias Zadik, 2017. "Orthogonal Machine Learning: Power and Limitations," Papers 1711.00342,, revised Aug 2018.
    4. Miruna Oprescu & Vasilis Syrgkanis & Zhiwei Steven Wu, 2018. "Orthogonal Random Forest for Causal Inference," Papers 1806.03467,, revised Sep 2019.
    5. Xin Zhou & Nicole Mayer-Hamblett & Umer Khan & Michael R. Kosorok, 2017. "Residual Weighted Learning for Estimating Individualized Treatment Rules," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(517), pages 169-187, January.
    6. Denis Nekipelov & Vira Semenova & Vasilis Syrgkanis, 2018. "Regularized Orthogonal Machine Learning for Nonlinear Semiparametric Models," Papers 1806.04823,, revised Oct 2020.
    7. Max H. Farrell & Tengyuan Liang & Sanjog Misra, 2018. "Deep Neural Networks for Estimation and Inference," Papers 1809.09953,, revised Sep 2019.
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