Proximal methods for the latent group lasso penalty
This work addresses a computational bottleneck for researchers and practitioners in machine learning dealing with high-dimensional structured sparsity problems, though it is incremental as it builds on existing proximal methods.
The paper tackles the optimization of regularized least squares problems with overlapping group lasso penalties, which are nonsmooth and difficult to solve, by proposing an accelerated proximal method with an active set strategy. The result is improved computational and prediction performance over state-of-the-art methods, as demonstrated empirically on toy and real data.
We consider a regularized least squares problem, with regularization by structured sparsity-inducing norms, which extend the usual $\ell_1$ and the group lasso penalty, by allowing the subsets to overlap. Such regularizations lead to nonsmooth problems that are difficult to optimize, and we propose in this paper a suitable version of an accelerated proximal method to solve them. We prove convergence of a nested procedure, obtained composing an accelerated proximal method with an inner algorithm for computing the proximity operator. By exploiting the geometrical properties of the penalty, we devise a new active set strategy, thanks to which the inner iteration is relatively fast, thus guaranteeing good computational performances of the overall algorithm. Our approach allows to deal with high dimensional problems without pre-processing for dimensionality reduction, leading to better computational and prediction performances with respect to the state-of-the art methods, as shown empirically both on toy and real data.