  We consider the problem of estimating a sparse linear regression vector
$\beta^*$ under a gaussian noise model, for the purpose of both prediction and
model selection. We assume that prior knowledge is available on the sparsity
pattern, namely the set of variables is partitioned into prescribed groups,
only few of which are relevant in the estimation process. This group sparsity
assumption suggests us to consider the Group Lasso method as a means to
estimate $\beta^*$. We establish oracle inequalities for the prediction and
$\ell_2$ estimation errors of this estimator. These bounds hold under a
restricted eigenvalue condition on the design matrix. Under a stronger
coherence condition, we derive bounds for the estimation error for mixed
$(2,p)$-norms with $1\le p\leq \infty$. When $p=\infty$, this result implies
that a threshold version of the Group Lasso estimator selects the sparsity
pattern of $\beta^*$ with high probability. Next, we prove that the rate of
convergence of our upper bounds is optimal in a minimax sense, up to a
logarithmic factor, for all estimators over a class of group sparse vectors.
Furthermore, we establish lower bounds for the prediction and $\ell_2$
estimation errors of the usual Lasso estimator. Using this result, we
demonstrate that the Group Lasso can achieve an improvement in the prediction
and estimation properties as compared to the Lasso.
