Statistical Guarantees for Transformation Based Models with Applications to Implicit Variational Inference

• URL: http://arxiv.org/abs/2010.14056v2
• Date: Wed, 4 Nov 2020 20:02:43 GMT
• Title: Statistical Guarantees for Transformation Based Models with Applications to Implicit Variational Inference
• Authors: Sean Plummer, Shuang Zhou, Anirban Bhattacharya, David Dunson, Debdeep Pati
• Abstract summary: We provide theoretical justification for the use of non-linear latent variable models (NL-LVMs) in non-parametric inference. We use the NL-LVMs to construct an implicit family of variational distributions, deemed GP-IVI. To the best of our knowledge, this is the first work on providing theoretical guarantees for implicit variational inference.
• Score: 8.333191406788423
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Transformation-based methods have been an attractive approach in non-parametric inference for problems such as unconditional and conditional density estimation due to their unique hierarchical structure that models the data as flexible transformation of a set of common latent variables. More recently, transformation-based models have been used in variational inference (VI) to construct flexible implicit families of variational distributions. However, their use in both non-parametric inference and variational inference lacks theoretical justification. We provide theoretical justification for the use of non-linear latent variable models (NL-LVMs) in non-parametric inference by showing that the support of the transformation induced prior in the space of densities is sufficiently large in the $L_1$ sense. We also show that, when a Gaussian process (GP) prior is placed on the transformation function, the posterior concentrates at the optimal rate up to a logarithmic factor. Adopting the flexibility demonstrated in the non-parametric setting, we use the NL-LVM to construct an implicit family of variational distributions, deemed GP-IVI. We delineate sufficient conditions under which GP-IVI achieves optimal risk bounds and approximates the true posterior in the sense of the Kullback-Leibler divergence. To the best of our knowledge, this is the first work on providing theoretical guarantees for implicit variational inference.

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