  The exclusive lasso (also known as elitist lasso) regularization has become
popular recently due to its superior performance on structured sparsity. Its
complex nature poses difficulties for the computation of high-dimensional
machine learning models involving such a regularizer. In this paper, we propose
an adaptive sieving (AS) strategy for generating solution paths of machine
learning models with the exclusive lasso regularizer, wherein a sequence of
reduced problems with much smaller sizes need to be solved. In order to solve
these reduced problems, we propose a highly efficient dual Newton method based
proximal point algorithm (PPDNA). As important ingredients, we systematically
study the proximal mapping of the weighted exclusive lasso regularizer and the
corresponding generalized Jacobian. These results also make popular first-order
algorithms for solving exclusive lasso models practical. Various numerical
experiments for the exclusive lasso models have demonstrated the effectiveness
of the AS strategy for generating solution paths and the superior performance
of the PPDNA.
