19,268 research outputs found
Adaptive Bound Optimization for Online Convex Optimization
We introduce a new online convex optimization algorithm that adaptively
chooses its regularization function based on the loss functions observed so
far. This is in contrast to previous algorithms that use a fixed regularization
function such as L2-squared, and modify it only via a single time-dependent
parameter. Our algorithm's regret bounds are worst-case optimal, and for
certain realistic classes of loss functions they are much better than existing
bounds. These bounds are problem-dependent, which means they can exploit the
structure of the actual problem instance. Critically, however, our algorithm
does not need to know this structure in advance. Rather, we prove competitive
guarantees that show the algorithm provides a bound within a constant factor of
the best possible bound (of a certain functional form) in hindsight.Comment: Updates to match final COLT versio
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
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