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KALE Flow: A Relaxed KL Gradient Flow for Probabilities with Disjoint Support
Pierre Glaser · Michael Arbel · Arthur Gretton

Fri Dec 10 08:30 AM -- 10:00 AM (PST) @ None #None

We study the gradient flow for a relaxed approximation to the Kullback-Leibler (KL) divergencebetween a moving source and a fixed target distribution.This approximation, termed theKALE (KL approximate lower-bound estimator), solves a regularized version ofthe Fenchel dual problem defining the KL over a restricted class of functions.When using a Reproducing Kernel Hilbert Space (RKHS) to define the functionclass, we show that the KALE continuously interpolates between the KL and theMaximum Mean Discrepancy (MMD). Like the MMD and other Integral ProbabilityMetrics, the KALE remains well defined for mutually singulardistributions. Nonetheless, the KALE inherits from the limiting KL a greater sensitivity to mismatch in the support of the distributions, compared with the MMD. These two properties make theKALE gradient flow particularly well suited when the target distribution is supported on a low-dimensional manifold. Under an assumption of sufficient smoothness of the trajectories, we show the global convergence of the KALE flow. We propose a particle implementation of the flow given initial samples from the source and the target distribution, which we use to empirically confirm the KALE's properties.

Author Information

Pierre Glaser (University College London)
Michael Arbel (UCL)
Arthur Gretton (Gatsby Unit, UCL)

Arthur Gretton is a Professor with the Gatsby Computational Neuroscience Unit at UCL. He received degrees in Physics and Systems Engineering from the Australian National University, and a PhD with Microsoft Research and the Signal Processing and Communications Laboratory at the University of Cambridge. He previously worked at the MPI for Biological Cybernetics, and at the Machine Learning Department, Carnegie Mellon University. Arthur's recent research interests in machine learning include the design and training of generative models, both implicit (e.g. GANs) and explicit (high/infinite dimensional exponential family models), nonparametric hypothesis testing, and kernel methods. He has been an associate editor at IEEE Transactions on Pattern Analysis and Machine Intelligence from 2009 to 2013, an Action Editor for JMLR since April 2013, an Area Chair for NeurIPS in 2008 and 2009, a Senior Area Chair for NeurIPS in 2018, an Area Chair for ICML in 2011 and 2012, and a member of the COLT Program Committee in 2013. Arthur was program chair for AISTATS in 2016 (with Christian Robert), tutorials chair for ICML 2018 (with Ruslan Salakhutdinov), workshops chair for ICML 2019 (with Honglak Lee), program chair for the Dali workshop in 2019 (with Krikamol Muandet and Shakir Mohammed), and co-organsier of the Machine Learning Summer School 2019 in London (with Marc Deisenroth).

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