  Sparse optimization problems are ubiquitous in many fields such as
statistics, signal/image processing and machine learning. This has led to the
birth of many iterative algorithms to solve them. A powerful strategy to boost
the performance of these algorithms is known as safe screening: it allows the
early identification of zero coordinates in the solution, which can then be
eliminated to reduce the problem's size and accelerate convergence. In this
work, we extend the existing Gap Safe screening framework by relaxing the
global strong-concavity assumption on the dual cost function. Instead, we
exploit local regularity properties, that is, strong concavity on well-chosen
subsets of the domain. The non-negativity constraint is also integrated to the
existing framework. Besides making safe screening possible to a broader class
of functions that includes beta-divergences (e.g., the Kullback-Leibler
divergence), the proposed approach also improves upon the existing Gap Safe
screening rules on previously applicable cases (e.g., logistic regression). The
proposed general framework is exemplified by some notable particular cases:
logistic function, beta = 1.5 and Kullback-Leibler divergences. Finally, we
showcase the effectiveness of the proposed screening rules with different
solvers (coordinate descent, multiplicative-update and proximal gradient
algorithms) and different data sets (binary classification, hyperspectral and
count data).
