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.
Lounici et al. REF considered the group lasso regularization for MTL and showed that under certain restricted eigenvalue conditions, the effect of the number of predictor variables in the upper bound of sparsity oracle inequalities could be negligible with respect to the number of tasks.
Taking Advantage of Sparsity in Multi-Task Learning
10 pages, 1 figure, Proc. Computational Learning Theory Conference (COLT 2009)
Computer Science Mathematics