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Online Super Learning

  • Mark J. van der Laan
  • David Benkeser
Chapter
Part of the Springer Series in Statistics book series (SSS)

Abstract

We consider the case that the data O n = (O(1), , O(n)) ∼ P0 n is generated sequentially by a conditional probability distribution Pθ, t of O(t), given certain summary measures of the past \(\bar{O}(t - 1) = (O(1),\ldots,O(t - 1))\), and where this t-specific conditional probability distribution Pθ, t is identified by a common parameter θΘ. For example, the experiment at time t generates a new observation O(t) from a probability distribution \(\bar{P}_{0}(\cdot \mid z)\) determined by a fixed dimensional summary measure Z of the past O(1), , O(t − 1), and one would assume that this conditional distribution \(\bar{P}_{0}\) is an element of some semiparametric model. An important special case is that the sample can be viewed as independent and identically distributed observations from a fixed data generating distribution that is known to belong to some semiparametric statistical model, such as the nonparametric model: in this case \(\bar{P}_{0}(\cdot \mid z) =\bar{ P}_{0}(\cdot )\) does not depend on the past. Another important case is that the data is generated by a group sequential adaptive design in which the randomization and or censoring mechanism is a function of summary measures of the observed data on previously sampled groups (van der Laan 2008b; Chambaz and van der Laan 2011a,b,c). More generally, this covers a whole range of time series models.

References

  1. D. Benkeser, S.D. Lendle, J. Cheng, M.J. van der Laan, Online cross-validation-based ensemble learning. Stat. Med. 37(2), 249–260 (2017b)MathSciNetCrossRefGoogle Scholar
  2. L. Bottou, Stochastic gradient descent tricks, in Neural Networks: Tricks of the Trade (Springer, Berlin, 2012), pp. 421–436CrossRefGoogle Scholar
  3. A. Chambaz, M.J. van der Laan, Targeting the optimal design in randomized clinical trials with binary outcomes and no covariate: theoretical study. Int. J. Biostat. 7(1), Article 10 (2011a)Google Scholar
  4. A. Chambaz, M.J. van der Laan, Targeting the optimal design in randomized clinical trials with binary outcomes and no covariate: simulation study. Int. J. Biostat. 7(1), Article 11 (2011b)Google Scholar
  5. A. Chambaz, M.J. van der Laan, TMLE in adaptive group sequential covariate-adjusted RCTs, in Targeted Learning: Causal Inference for Observational and Experimental Data, ed. by M.J. van der Laan, S. Rose (Springer, Berlin Heidelberg, New York, 2011c)Google Scholar
  6. E.C Polley, S. Rose, M.J. van der Laan, Super-learning, in Targeted Learning: Causal Inference for Observational and Experimental Data, ed. by M.J. van der Laan, S. Rose (Springer, Berlin, Heidelberg, New York, 2011)Google Scholar
  7. M.J. van der Laan, The construction and analysis of adaptive group sequential designs. Technical Report 232, Division of Biostatistics, University of California, Berkeley (2008b)Google Scholar
  8. M.J. van der Laan, S. Dudoit, Unified cross-validation methodology for selection among estimators and a general cross-validated adaptive epsilon-net estimator: finite sample oracle inequalities and examples. Technical Report, Division of Biostatistics, University of California, Berkeley (2003)Google Scholar
  9. M.J. van der Laan, E.C. Polley, A.E. Hubbard, Super learner. Stat. Appl. Genet. Mol. 6(1), Article 25 (2007)Google Scholar
  10. M.J. van der Laan, S. Rose, Targeted Learning: Causal Inference for Observational and Experimental Data (Springer, Berlin, Heidelberg, New York, 2011)CrossRefGoogle Scholar
  11. M.J. van der Laan, D.B. Rubin, Targeted maximum likelihood learning. Int. J. Biostat. 2(1), Article 11 (2006)Google Scholar
  12. M.J. van der Laan, S. Dudoit, A.W. van der Vaart. The cross-validated adaptive epsilon-net estimator. Stat. Decis. 24(3), 373–395 (2006)MathSciNetzbMATHGoogle Scholar
  13. A.W. van der Vaart, S. Dudoit, M.J. van der Laan, Oracle inequalities for multi-fold cross-validation. Stat. Decis. 24(3), 351–371 (2006)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Division of Biostatistics and Department of StatisticsUniversity of California, BerkeleyBerkeleyUSA
  2. 2.Department of Biostatistics and BioinformaticsRollins School of Public Health, Emory UniversityAtlantaUSA

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