  We develop uniformly valid confidence regions for regression coefficients in
a high-dimensional sparse median regression model with homoscedastic errors.
Our methods are based on a moment equation that is immunized against
non-regular estimation of the nuisance part of the median regression function
by using Neyman's orthogonalization. We establish that the resulting
instrumental median regression estimator of a target regression coefficient is
asymptotically normally distributed uniformly with respect to the underlying
sparse model and is semi-parametrically efficient. We also generalize our
method to a general non-smooth Z-estimation framework with the number of target
parameters $p_1$ being possibly much larger than the sample size $n$. We extend
Huber's results on asymptotic normality to this setting, demonstrating uniform
asymptotic normality of the proposed estimators over $p_1$-dimensional
rectangles, constructing simultaneous confidence bands on all of the $p_1$
target parameters, and establishing asymptotic validity of the bands uniformly
over underlying approximately sparse models.
  Keywords: Instrument; Post-selection inference; Sparsity; Neyman's Orthogonal
Score test; Uniformly valid inference; Z-estimation.
