A Spectral Approach to Gradient Estimation for Implicit Distributions


Jiaxin Shi, Shengyang Sun, Jun Zhu ;
Proceedings of the 35th International Conference on Machine Learning, PMLR 80:4644-4653, 2018.


Recently there have been increasing interests in learning and inference with implicit distributions (i.e., distributions without tractable densities). To this end, we develop a gradient estimator for implicit distributions based on Stein’s identity and a spectral decomposition of kernel operators, where the eigenfunctions are approximated by the Nystr{ö}m method. Unlike the previous works that only provide estimates at the sample points, our approach directly estimates the gradient function, thus allows for a simple and principled out-of-sample extension. We provide theoretical results on the error bound of the estimator and discuss the bias-variance tradeoff in practice. The effectiveness of our method is demonstrated by applications to gradient-free Hamiltonian Monte Carlo and variational inference with implicit distributions. Finally, we discuss the intuition behind the estimator by drawing connections between the Nystr{ö}m method and kernel PCA, which indicates that the estimator can automatically adapt to the geometry of the underlying distribution.

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