  Fitting high-dimensional statistical models often requires the use of
non-linear parameter estimation procedures. As a consequence, it is generally
impossible to obtain an exact characterization of the probability distribution
of the parameter estimates. This in turn implies that it is extremely
challenging to quantify the \emph{uncertainty} associated with a certain
parameter estimate. Concretely, no commonly accepted procedure exists for
computing classical measures of uncertainty and statistical significance as
confidence intervals or $p$-values for these models.
  We consider here high-dimensional linear regression problem, and propose an
efficient algorithm for constructing confidence intervals and $p$-values. The
resulting confidence intervals have nearly optimal size. When testing for the
null hypothesis that a certain parameter is vanishing, our method has nearly
optimal power.
  Our approach is based on constructing a `de-biased' version of regularized
M-estimators. The new construction improves over recent work in the field in
that it does not assume a special structure on the design matrix. We test our
method on synthetic data and a high-throughput genomic data set about
riboflavin production rate.
