  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, R\'{e}nyi's
$\alpha$-VI, and $\chi$-VI, 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.
