  We propose a robust inferential procedure for assessing uncertainties of
parameter estimation in high-dimensional linear models, where the dimension $p$
can grow exponentially fast with the sample size $n$. Our method combines the
de-biasing technique with the composite quantile function to construct an
estimator that is asymptotically normal. Hence it can be used to construct
valid confidence intervals and conduct hypothesis tests. Our estimator is
robust and does not require the existence of first or second moment of the
noise distribution. It also preserves efficiency in the sense that the worst
case efficiency loss is less than 30\% compared to the square-loss-based
de-biased Lasso estimator. In many cases our estimator is close to or better
than the latter, especially when the noise is heavy-tailed. Our de-biasing
procedure does not require solving the $L_1$-penalized composite quantile
regression. Instead, it allows for any first-stage estimator with desired
convergence rate and empirical sparsity. The paper also provides new proof
techniques for developing theoretical guarantees of inferential procedures with
non-smooth loss functions. To establish the main results, we exploit the local
curvature of the conditional expectation of composite quantile loss and apply
empirical process theories to control the difference between empirical
quantities and their conditional expectations. Our results are established
under weaker assumptions compared to existing work on inference for
high-dimensional quantile regression. Furthermore, we consider a
high-dimensional simultaneous test for the regression parameters by applying
the Gaussian approximation and multiplier bootstrap theories. We also study
distributed learning and exploit the divide-and-conquer estimator to reduce
computation complexity when the sample size is massive. Finally, we provide
empirical results to verify the theory.
