Learning with Importance Weighted Variational Inference

• ESSEC Business School, France [daudel@essec.edu]

• Télécom Paris, Institut Polytechnique de Paris, France [roueff@telecom-paris.fr]

1 Abstract

Variational inference (VI) methods seek to find the best approximation to an unknown target posterior density within a more tractable family of probability densities $𝒬$ [Jordan et al., 1999, Blei et al., 2017]. A common setting where VI is applied is when one is given a model that depends on a parameter $\theta$ and the goal is to optimize the associated marginal log likelihood, with the posterior density being intractable. Since direct optimization of the marginal log likelihood cannot be carried out, variational bounds involving the variational family $𝒬$ are constructed as surrogate objective functions to the marginal log likelihood that are more amenable to optimization.

While the most traditional variational bound is the Evidence Lower BOund (ELBO), popular alternatives to the ELBO that rely on importance weighting ideas have been proposed to improve on VI in the context of maximum likelihood optimization, such as the Importance Weighted Auto-Encoder (IWAE) in Burda et al. [2016] and the Variational Rényi (VR) bounds in Li and Turner [2016]. The methodology to learn the parameters of interest using these bounds typically amounts to running gradient-based VI algorithms that incorporate the reparameterization trick [Kingma and Welling, 2014]. However, the way the choice of the variational bound impacts the outcome of VI algorithms can be unclear and an active line of research in VI is then concerned with better comprehending this aspect.

Among the existing works, Daudel et al. [2023] introduce and study the VR-IWAE bound, a variational bound that depends on two hyperparameters $(N,\alpha )\in {\mathbb{N}}^{\star }\times [0,1)$ and that unifies the ELBO, IWAE and VR methodologies when the reparameterization trick is available. In particular, Daudel et al. [2023] provide analyses of the VR-IWAE bound that elucidate the role $N$ and $\alpha$ play in this bound. Yet, solely focusing on the behavior of a variational bound is insufficient to assess the effectiveness of algorithms based on this bound at learning the parameters of interest [Rainforth et al., 2018].

In our work [Daudel and Roueff, 2024], we study the role of $N$ and $\alpha$ in two gradient estimators of the VR-IWAE bound that are at the center of the Importance Weighted VI methodology, namely the reparameterized and doubly-reparameterized gradient estimators of the VR-IWAE bound. In doing so, we provide insights that apply to widely-used gradient estimators of the IWAE and VR bounds and more broadly we further advance the understanding of Importance Weighted VI methods. We illustrate our theoretical findings empirically.