Region-free Safe Screening Tests for $\ell_{1}$-penalized Convex Problems

We address the problem of safe screening for $\ell_{1}$-penalized convex regression/classification problems, i.e., the identification of zero coordinates of the solutions. Unlike previous contributions of the literature, we propose a screening methodology which does not require the knowledge of a so-called “safe region”. Our approach does not rely on any other assumption than convexity (in particular, no strong-convexity hypothesis is needed) and therefore applies to a wide family of convex problems. When the Fenchel conjugate of the data-fidelity term is strongly convex, we show that the popular “GAP sphere test” proposed by Fercoq et al. can be recovered as a particular case of our methodology (up to a minor modification). We illustrate numerically the performance of our procedure on the “sparse support vector machine classification” problem.