Efficient and principled score estimation with Nyström kernel exponential families

Danica J. Sutherland, Heiko Strathmann, Michael Arbel, Arthur Gretton
Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics, PMLR 84:652-660, 2018.

Abstract

We propose a fast method with statistical guarantees for learning an exponential family density model where the natural parameter is in a reproducing kernel Hilbert space, and may be infinite dimensional. The model is learned by fitting the derivative of the log density, the score, thus avoiding the need to compute a normalization constant. We improved the computational efficiency of an earlier solution with a low-rank, Nyström-like solution. The new solution retains the consistency and convergence rates of the full-rank solution (exactly in Fisher distance, and nearly in other distances), with guarantees on the degree of cost and storage reduction. We evaluate the method in experiments on density estimation and in the construction of an adaptive Hamiltonian Monte Carlo sampler. Compared to an existing score learning approach using a denoising autoencoder, our estimator is empirically more data-efficient when estimating the score, runs faster, and has fewer parameters (which can be tuned in a principled and interpretable way), in addition to providing statistical guarantees.

Cite this Paper

BibTeX
@InProceedings{pmlr-v84-sutherland18a, title = {Efficient and principled score estimation with Nyström kernel exponential families}, author = {Sutherland, Danica J. and Strathmann, Heiko and Arbel, Michael and Gretton, Arthur}, booktitle = {Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics}, pages = {652--660}, year = {2018}, editor = {Storkey, Amos and Perez-Cruz, Fernando}, volume = {84}, series = {Proceedings of Machine Learning Research}, month = {09--11 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v84/sutherland18a/sutherland18a.pdf}, url = {https://proceedings.mlr.press/v84/sutherland18a.html}, abstract = {We propose a fast method with statistical guarantees for learning an exponential family density model where the natural parameter is in a reproducing kernel Hilbert space, and may be infinite dimensional. The model is learned by fitting the derivative of the log density, the score, thus avoiding the need to compute a normalization constant. We improved the computational efficiency of an earlier solution with a low-rank, Nyström-like solution. The new solution retains the consistency and convergence rates of the full-rank solution (exactly in Fisher distance, and nearly in other distances), with guarantees on the degree of cost and storage reduction. We evaluate the method in experiments on density estimation and in the construction of an adaptive Hamiltonian Monte Carlo sampler. Compared to an existing score learning approach using a denoising autoencoder, our estimator is empirically more data-efficient when estimating the score, runs faster, and has fewer parameters (which can be tuned in a principled and interpretable way), in addition to providing statistical guarantees.} }
Endnote
%0 Conference Paper %T Efficient and principled score estimation with Nyström kernel exponential families %A Danica J. Sutherland %A Heiko Strathmann %A Michael Arbel %A Arthur Gretton %B Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2018 %E Amos Storkey %E Fernando Perez-Cruz %F pmlr-v84-sutherland18a %I PMLR %P 652--660 %U https://proceedings.mlr.press/v84/sutherland18a.html %V 84 %X We propose a fast method with statistical guarantees for learning an exponential family density model where the natural parameter is in a reproducing kernel Hilbert space, and may be infinite dimensional. The model is learned by fitting the derivative of the log density, the score, thus avoiding the need to compute a normalization constant. We improved the computational efficiency of an earlier solution with a low-rank, Nyström-like solution. The new solution retains the consistency and convergence rates of the full-rank solution (exactly in Fisher distance, and nearly in other distances), with guarantees on the degree of cost and storage reduction. We evaluate the method in experiments on density estimation and in the construction of an adaptive Hamiltonian Monte Carlo sampler. Compared to an existing score learning approach using a denoising autoencoder, our estimator is empirically more data-efficient when estimating the score, runs faster, and has fewer parameters (which can be tuned in a principled and interpretable way), in addition to providing statistical guarantees.
APA
Sutherland, D.J., Strathmann, H., Arbel, M. & Gretton, A.. (2018). Efficient and principled score estimation with Nyström kernel exponential families. Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 84:652-660 Available from https://proceedings.mlr.press/v84/sutherland18a.html.

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