Scalable collaborative targeted learning for high-dimensional data.

Robust inference of a low-dimensional parameter in a large semi-parametric model relies on external estimators of infinite-dimensional features of the distribution of the data. Typically, only one of the latter is optimized for the sake of constructing a well-behaved estimator of the low-dimensional parameter of interest. Optimizing more than one of them for the sake of achieving a better bias-variance trade-off in the estimation of the parameter of interest is the core idea driving the general template of the collaborative targeted minimum loss-based estimation procedure. The original instantiation of the collaborative targeted minimum loss-based estimation template can be presented as a greedy forward stepwise collaborative targeted minimum loss-based estimation algorithm. It does not scale well when the number p of covariates increases drastically. This motivates the introduction of a novel instantiation of the collaborative targeted minimum loss-based estimation template where the covariates are pre-ordered. Its time complexity is O(p) as opposed to the original O(p2) , a remarkable gain. We propose two pre-ordering strategies and suggest a rule of thumb to develop other meaningful strategies. Because it is usually unclear a priori which pre-ordering strategy to choose, we also introduce another instantiation called SL-C-TMLE algorithm that enables the data-driven choice of the better pre-ordering strategy given the problem at hand. Its time complexity is O(p) as well. The computational burden and relative performance of these algorithms were compared in simulation studies involving fully synthetic data or partially synthetic data based on a real world large electronic health database; and in analyses of three real, large electronic health databases. In all analyses involving electronic health databases, the greedy collaborative targeted minimum loss-based estimation algorithm is unacceptably slow. Simulation studies seem to indicate that our scalable collaborative targeted minimum loss-based estimation and SL-C-TMLE algorithms work well. All C-TMLEs are publicly available in a Julia software package.