  To make inference about a group of parameters on high-dimensional data, we
develop the method of estimator augmentation for the block Lasso, which is
defined via the block norm. By augmenting a block Lasso estimator $\hat{\beta}$
with the subgradient $S$ of the block norm evaluated at $\hat{\beta}$, we
derive a closed-form density for the joint distribution of $(\hat{\beta},S)$
under a high-dimensional setting. This allows us to draw from an estimated
sampling distribution of $\hat{\beta}$, or more generally any function of
$(\hat{\beta},S)$, by Monte Carlo algorithms. We demonstrate the application of
estimator augmentation in group inference with the group Lasso and a de-biased
group Lasso constructed as a function of $(\hat{\beta},S)$. Our numerical
results show that importance sampling via estimator augmentation can be orders
of magnitude more efficient than parametric bootstrap in estimating tail
probabilities for significance tests. This work also brings new insights into
the geometry of the sample space and the solution uniqueness of the block
Lasso.
