81 research outputs found
Optimization with Sparsity-Inducing Penalties
Sparse estimation methods are aimed at using or obtaining parsimonious
representations of data or models. They were first dedicated to linear variable
selection but numerous extensions have now emerged such as structured sparsity
or kernel selection. It turns out that many of the related estimation problems
can be cast as convex optimization problems by regularizing the empirical risk
with appropriate non-smooth norms. The goal of this paper is to present from a
general perspective optimization tools and techniques dedicated to such
sparsity-inducing penalties. We cover proximal methods, block-coordinate
descent, reweighted -penalized techniques, working-set and homotopy
methods, as well as non-convex formulations and extensions, and provide an
extensive set of experiments to compare various algorithms from a computational
point of view
Screening for a Reweighted Penalized Conditional Gradient Method
The conditional gradient method (CGM) is widely used in large-scale sparse
convex optimization, having a low per iteration computational cost for
structured sparse regularizers and a greedy approach to collecting nonzeros. We
explore the sparsity acquiring properties of a general penalized CGM (P-CGM)
for convex regularizers and a reweighted penalized CGM (RP-CGM) for nonconvex
regularizers, replacing the usual convex constraints with gauge-inspired
penalties. This generalization does not increase the per-iteration complexity
noticeably. Without assuming bounded iterates or using line search, we show
convergence of the gap of each subproblem, which measures distance to
a stationary point. We couple this with a screening rule which is safe in the
convex case, converging to the true support at a rate where
measures how close the problem is to degeneracy. In the
nonconvex case the screening rule converges to the true support in a finite
number of iterations, but is not necessarily safe in the intermediate iterates.
In our experiments, we verify the consistency of the method and adjust the
aggressiveness of the screening rule by tuning the concavity of the
regularizer
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