958 research outputs found
A Smoothed Dual Approach for Variational Wasserstein Problems
Variational problems that involve Wasserstein distances have been recently
proposed to summarize and learn from probability measures. Despite being
conceptually simple, such problems are computationally challenging because they
involve minimizing over quantities (Wasserstein distances) that are themselves
hard to compute. We show that the dual formulation of Wasserstein variational
problems introduced recently by Carlier et al. (2014) can be regularized using
an entropic smoothing, which leads to smooth, differentiable, convex
optimization problems that are simpler to implement and numerically more
stable. We illustrate the versatility of this approach by applying it to the
computation of Wasserstein barycenters and gradient flows of spacial
regularization functionals
Primal-Dual Rates and Certificates
We propose an algorithm-independent framework to equip existing optimization
methods with primal-dual certificates. Such certificates and corresponding rate
of convergence guarantees are important for practitioners to diagnose progress,
in particular in machine learning applications. We obtain new primal-dual
convergence rates, e.g., for the Lasso as well as many L1, Elastic Net, group
Lasso and TV-regularized problems. The theory applies to any norm-regularized
generalized linear model. Our approach provides efficiently computable duality
gaps which are globally defined, without modifying the original problems in the
region of interest.Comment: appearing at ICML 2016 - Proceedings of the 33rd International
Conference on Machine Learning, New York, NY, USA, 2016. JMLR: W&CP volume 4
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