  We study asymptotically normal estimation and confidence regions for
low-dimensional parameters in high-dimensional sparse models. Our approach is
based on the $\ell_1$-penalized M-estimator which is used for construction of a
bias corrected estimator. We show that the proposed estimator is asymptotically
normal, under a sparsity assumption on the high-dimensional parameter,
smoothness conditions on the expected loss and an entropy condition. This leads
to uniformly valid confidence regions and hypothesis testing for
low-dimensional parameters. The present approach is different in that it allows
for treatment of loss functions that we not sufficiently differentiable, such
as quantile loss, Huber loss or hinge loss functions. We also provide new
results for estimation of the inverse Fisher information matrix, which is
necessary for the construction of the proposed estimator. We formulate our
results for general models under high-level conditions, but investigate these
conditions in detail for generalized linear models and provide mild sufficient
conditions. As particular examples, we investigate the case of quantile loss
and Huber loss in linear regression and demonstrate the performance of the
estimators in a simulation study and on real datasets from genome-wide
association studies. We further investigate the case of logistic regression and
illustrate the performance of the estimator on simulated and real data.
