  Performing statistical inference in high-dimension is an outstanding
challenge. A major source of difficulty is the absence of precise information
on the distribution of high-dimensional estimators. Here, we consider linear
regression in the high-dimensional regime $p\gg n$. In this context, we would
like to perform inference on a high-dimensional parameters vector
$\theta^*\in{\mathbb R}^p$. Important progress has been achieved in computing
confidence intervals for single coordinates $\theta^*_i$. A key role in these
new methods is played by a certain debiased estimator $\hat{\theta}^{\rm d}$
that is constructed from the Lasso. Earlier work establishes that, under
suitable assumptions on the design matrix, the coordinates of
$\hat{\theta}^{\rm d}$ are asymptotically Gaussian provided $\theta^*$ is
$s_0$-sparse with $s_0 = o(\sqrt{n}/\log p )$. The condition $s_0 = o(\sqrt{n}/
\log p )$ is stronger than the one for consistent estimation, namely $s_0 =
o(n/ \log p)$. We study Gaussian designs with known or unknown population
covariance. When the covariance is known, we prove that the debiased estimator
is asymptotically Gaussian under the nearly optimal condition $s_0 = o(n/ (\log
p)^2)$. Note that earlier work was limited to $s_0 = o(\sqrt{n}/\log p)$ even
for perfectly known covariance. The same conclusion holds if the population
covariance is unknown but can be estimated sufficiently well, e.g. under the
same sparsity conditions on the inverse covariance as assumed by earlier work.
For intermediate regimes, we describe the trade-off between sparsity in the
coefficients and in the inverse covariance of the design. We further discuss
several applications of our results to high-dimensional inference. In
particular, we propose a new estimator that is minimax optimal up to a factor
$1+o_n(1)$ for i.i.d. Gaussian designs.
