Targeted Learning in Data Science pp 303-315 | Cite as

# Online Super Learning

## Abstract

We consider the case that the data *O*^{ n } = (*O*(1), *…*, *O*(*n*)) ∼ *P*_{0}^{ n } is generated sequentially by a conditional probability distribution *P*_{θ, 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 *P*_{θ, t} is identified by a common parameter *θ* ∈ *Θ*. 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

- 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 - L. Bottou, Stochastic gradient descent tricks, in
*Neural Networks: Tricks of the Trade*(Springer, Berlin, 2012), pp. 421–436CrossRefGoogle Scholar - 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 - 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 - 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 - 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 - 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
- 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
- M.J. van der Laan, E.C. Polley, A.E. Hubbard, Super learner. Stat. Appl. Genet. Mol.
**6**(1), Article 25 (2007)Google Scholar - M.J. van der Laan, S. Rose,
*Targeted Learning: Causal Inference for Observational and Experimental Data*(Springer, Berlin, Heidelberg, New York, 2011)CrossRefGoogle Scholar - M.J. van der Laan, D.B. Rubin, Targeted maximum likelihood learning. Int. J. Biostat.
**2**(1), Article 11 (2006)Google Scholar - 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)MathSciNetMATHGoogle Scholar - 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)MathSciNetMATHGoogle Scholar