  In this article, we aim to provide a general and complete understanding of
semi-supervised (SS) causal inference for treatment effects. Specifically, we
consider two such estimands: (a) the average treatment effect and (b) the
quantile treatment effect, as prototype cases, in an SS setting, characterized
by two available data sets: (i) a labeled data set of size $n$, providing
observations for a response and a set of high dimensional covariates, as well
as a binary treatment indicator; and (ii) an unlabeled data set of size $N$,
much larger than $n$, but without the response observed. Using these two data
sets, we develop a family of SS estimators which are ensured to be: (1) more
robust and (2) more efficient than their supervised counterparts based on the
labeled data set only. Beyond the 'standard' double robustness results (in
terms of consistency) that can be achieved by supervised methods as well, we
further establish root-n consistency and asymptotic normality of our SS
estimators whenever the propensity score in the model is correctly specified,
without requiring specific forms of the nuisance functions involved. Such an
improvement of robustness arises from the use of the massive unlabeled data, so
it is generally not attainable in a purely supervised setting. In addition, our
estimators are shown to be semi-parametrically efficient as long as all the
nuisance functions are correctly specified. Moreover, as an illustration of the
nuisance estimators, we consider inverse-probability-weighting type kernel
smoothing estimators involving unknown covariate transformation mechanisms, and
establish in high dimensional scenarios novel results on their uniform
convergence rates, which should be of independent interest. Numerical results
on both simulated and real data validate the advantage of our methods over
their supervised counterparts with respect to both robustness and efficiency.
