  We wish to infer the value of a parameter at a law from which we sample
independent observations. The parameter is smooth and we can define two
variation-independent features of the law, its $Q$- and $G$-components, such
that estimating them consistently at a fast enough product of rates allows to
build a confidence interval (CI) with a given asymptotic level from a plain
targeted minimum loss estimator (TMLE). Say that the above product is not fast
enough and the algorithm for the $G$-component is fine-tuned by a real-valued
$h$. A plain TMLE with an $h$ chosen by cross-validation would typically not
yield a CI. We construct a collaborative TMLE (C-TMLE) and show under mild
conditions that, if there exists an oracle $h$ that makes a bulky remainder
term asymptotically Gaussian, then the C-TMLE yields a CI. We illustrate our
findings with the inference of the average treatment effect. We conduct a
simulation study where the $G$-component is estimated by the LASSO and $h$ is
the bound on the coefficients' norms. It sheds light on small sample
properties, in the face of low- to high-dimensional baseline covariates, and
possibly positivity violation.
