Refined $\alpha$-Divergence Variational Inference via Rejection Sampling

TL;DR

This paper introduces a hybrid inference method combining R\'enyi ivergence variational inference and rejection sampling, leading to more accurate approximations of complex distributions.

Contribution

The paper proposes a novel two-stage hybrid inference algorithm that integrates RDVI with rejection sampling, improving approximation accuracy over existing methods.

Findings

The method achieves significantly better approximation accuracy than RDVI alone.

Experiments demonstrate improved sampling efficiency and distribution fidelity.

The approach is effective on both synthetic and real datasets.

Abstract

We present an approximate inference method, based on a synergistic combination of R\'enyi $\alpha$-divergence variational inference (RDVI) and rejection sampling (RS). RDVI is based on minimization of R\'enyi $\alpha$-divergence $D_\alpha(p||q)$ between the true distribution $p(x)$ and a variational approximation $q(x)$; RS draws samples from a distribution $p(x) = \tilde{p}(x)/Z_{p}$ using a proposal $q(x)$, s.t. $Mq(x) \geq \tilde{p}(x), \forall x$. Our inference method is based on a crucial observation that $D_\infty(p||q)$ equals $\log M(\theta)$ where $M(\theta)$ is the optimal value of the RS constant for a given proposal $q_\theta(x)$. This enables us to develop a \emph{two-stage} hybrid inference algorithm. Stage-1 performs RDVI to learn $q_\theta$ by minimizing an estimator of $D_\alpha(p||q)$, and uses the learned $q_\theta$ to find an (approximately) optimal $\tilde{M}(\theta)$. Stage-2 performs RS using the constant $\tilde{M}(\theta)$ to improve the approximate distribution $q_\theta$ and obtain a sample-based approximation. We prove that this two-stage method allows us to learn considerably more accurate approximations of the target distribution as compared to RDVI. We demonstrate our method's efficacy via several experiments on synthetic and real datasets.