  We study the problem of estimating multiple linear regression equations for
the purpose of both prediction and variable selection. Following recent work on
multi-task learning Argyriou et al. [2008], we assume that the regression
vectors share the same sparsity pattern. This means that the set of relevant
predictor variables is the same across the different equations. This assumption
leads us to consider the Group Lasso as a candidate estimation method. We show
that this estimator enjoys nice sparsity oracle inequalities and variable
selection properties. The results hold under a certain restricted eigenvalue
condition and a coherence condition on the design matrix, which naturally
extend recent work in Bickel et al. [2007], Lounici [2008]. In particular, in
the multi-task learning scenario, in which the number of tasks can grow, we are
able to remove completely the effect of the number of predictor variables in
the bounds. Finally, we show how our results can be extended to more general
noise distributions, of which we only require the variance to be finite.
