  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 (C-TMLE) procedure. The original
implementation/instantiation of the C-TMLE template can be presented as a
greedy forward stepwise C-TMLE algorithm. It does not scale well when the
number $p$ of covariates increases drastically. This motivates the introduction
of a novel instantiation of the C-TMLE template where the covariates are
pre-ordered. Its time complexity is $\mathcal{O}(p)$ as opposed to the original
$\mathcal{O}(p^2)$, 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 implementation/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 $\mathcal{O}(p)$ 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 C-TMLE algorithm is unacceptably slow.
Simulation studies indicate our scalable C-TMLE and SL-C-TMLE algorithms work
well.
