  In a high-dimensional setting, sparse model has shown its power in
computational and statistical efficiency. We consider variables selection
problem with a broad class of simultaneous sparsity regularization, enforcing
both feature-wise and group-wise sparsity at the same time. The analysis
leverages an introduction of $\epsilon q$-norm in vector space, which is proved
to has close connection with the mixture regularization and naturally leads to
a dual formulation. Properties of primal/dual optimal solution and optimal
values are discussed, which motivates the design of screening rules. We several
fast safe screening rules in the general framework, rules that discard inactive
features/groups at an early stage that are guaranteed to be inactive in the
exact solution, leading to a significant gain in computation speed.
