A recently introduced technique for a sparse optimization problem called "safe screening" allows us to identify irrelevant variables in the early stage of optimization. In this paper, we first propose a flexible framework for safe screening based on the Fenchel-Rockafellar duality and then derive a strong safe screening rule for norm-regularized least squares by the framework. We call the proposed screening rule for norm-regularized least squares "dynamic Sasvi" because it can be interpreted as a generalization of Sasvi. Unlike the original Sasvi, it does not require the exact solution of a more strongly regularized problem; hence, it works safely in practice. We show that our screening rule can eliminate more features and increase the speed of the solver in comparison with other screening rules both theoretically and experimentally.

Sparse learning techniques have been routinely used for feature selection as the resulting model usually has a small number of non-zero entries. Safe screening, which eliminates the features that are guaranteed to have zero coefficients for a certain value of the regularization parameter, is a technique for improving the computational efficiency. Safe screening is gaining increasing attention since 1) solving sparse learning formulations usually has a high computational cost especially when the number of features is large and 2) one needs to try several regularization parameters to select a suitable model. In this paper, we propose an approach called "Sasvi" (Safe screening with variational inequalities). Sasvi makes use of the variational inequality that provides the sufficient and necessary optimality condition for the dual problem. Several existing approaches for Lasso screening can be casted as relaxed versions of the proposed Sasvi, thus Sasvi provides a stronger safe screening rule. We further study the monotone properties of Sasvi for Lasso, based on which a sure removal regularization parameter can be identified for each feature. Experimental results on both synthetic and real data sets are reported to demonstrate the effectiveness of the proposed Sasvi for Lasso screening.