Consistent Vector-valued Regression on Probability Measures

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Consistent Vector-valued Regression on Probability Measures

Szabo, Z; Sriperumbudur, B; Poczos, B; Gretton, A; (2015) Consistent Vector-valued Regression on Probability Measures. Presented at: Invited talk at Prof. Bernhard Schölkopf's lab, Tübingen. Green open access

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Abstract

I will focus on the distribution regression problem (DRP): our goal is to regress from probability measures to vector-valued outputs, in the two-stage sampled setup when only samples from the distributions are available. The studied DRP framework incorporates several important machine learning and statistical tasks, including multi-instance learning or point estimation problems without analytical solution (such as hyperparameter estimation). Obtaining theoretical guarantees, bounds on the generalization error of the estimated predictor is pretty challenging due to the two-stage sampled characteristic of the task. To the best of our knowledge, among the vast number of heuristic approaches in the literature, the only theoretically justified technique tackling the DRP problem requires that the domain of the distributions be compact Euclidean, and uses density estimation (which often performs poorly in practice). We present a simple, analytically tractable alternative: we embed the probability measures to a reproducing kernel Hilbert space, and perform ridge regression from the embedded distributions to the outputs. We prove that this method is consistent under mild conditions, on separable topological domains endowed with kernels. Specifically, we establish the consistency of the traditional set kernel in regression, which was a 15-year-old open question. We demonstrate the efficiency of our method in supervised entropy learning and aerosol prediction based on multispectral satellite images.

Type:	Conference item (Presentation)
Title:	Consistent Vector-valued Regression on Probability Measures
Event:	Invited talk at Prof. Bernhard Schölkopf's lab
Location:	Tübingen
Dates:	2015-01-14 - 2015-01-18
Open access status:	An open access version is available from UCL Discovery
Publisher version:	http://www.gatsby.ucl.ac.uk/~szabo/talks/invited_t...
Language:	English
Keywords:	consistency, convergence rate, distribution regression, mean embedding, operator-valued kernel, set kernel, two-stage sampling
UCL classification:	UCL UCL > Provost and Vice Provost Offices > School of Life and Medical Sciences UCL > Provost and Vice Provost Offices > School of Life and Medical Sciences > Faculty of Life Sciences UCL > Provost and Vice Provost Offices > School of Life and Medical Sciences > Faculty of Life Sciences > Gatsby Computational Neurosci Unit
URI:	https://discovery.ucl.ac.uk/id/eprint/1460734