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Convex sparsity-promoting regularizations are ubiquitous in modernstatistical learning. By construction, they yield solutions with few non-zerocoefficients, which correspond to saturated constraints in the dualoptimization formulation. Working set (WS) strategies are generic optimizationtechniques that consist in solving simpler problems that only consider a subsetof constraints, whose indices form the WS. Working set methods thereforeinvolve two nested iterations: the outer loop corresponds to the definition ofthe WS and the inner loop calls a solver for the subproblems. For the Lassoestimator a WS is a set of features, while for a Group Lasso it refers to a setof groups. In practice, WS are generally small in this context so theassociated feature Gram matrix can fit in memory. Here we show that theGauss-Southwell rule (a greedy strategy for block coordinate descenttechniques) leads to fast solvers in this case. Combined with a working setstrategy based on an aggressive use of so-called Gap Safe screening rules, wepropose a solver achieving state-of-the-art performance on sparse learningproblems. Results are presented on Lasso and multi-task Lasso estimators.

From safe screening rules to working sets for faster Lasso-type solvers