f-Divergence Variational Inference

Abstract

This paper introduces the f -divergence variational inference (f -VI) that generalizes variational inference to all f -divergences. Initiated from minimizing a crafty surrogate f -divergence that shares the statistical consistency with the f -divergence, the f -VI framework not only unifies a number of existing VI methods, e.g. Kullback-Leibler VI [1], Rényi's α-VI [2], and χ-VI [3], but offers a standardized toolkit for VI subject to arbitrary divergences from f -divergence family. A general f -variational bound is derived and provides a sandwich estimate of marginal likelihood (or evidence). The development of the f -VI unfolds with a stochastic optimization scheme that utilizes the reparameterization trick, importance weighting and Monte Carlo approximation; a mean-field approximation scheme that generalizes the well-known coordinate ascent variational inference (CAVI) is also proposed for f -VI. Empirical examples, including variational autoencoders and Bayesian neural networks, are provided to demonstrate the effectiveness and the wide applicability of f -VI.

Variational Inference MPC using Tsallis Divergence

“…Although it is not possible to directly solve Eq. (11) in this case, we draw from Expectation Maximization (EM) to derive an iterative update scheme for the k + 1th iteration with the form…”

“…In Wang et al [9], Regli and Silva [10], the authors proposed variants of the α-divergence to improve the performance and robustness of the inference algorithm. Wan et al [11] further extended the VI framework to f -divergence, which is a broad statistical divergence family that recovers the KL, α and χ-divergence as special cases.…”