April 2024 I

Date: 4PM Friday 12th April 2024

Alpha-divergence Variational Inference Meets Importance Weighted Auto-Encoders: Methodology and Asymptotics

Kamélia Daudel ESSEC Business School

Variational Inference methods are optimization-based methods that have generated a lot of attention in Bayesian Statistics due to their applicability to high-dimensional machine learning problems. In particular, several algorithms involving the Variational Rényi (VR) bound have been proposed to optimize an alpha-divergence between a target posterior distribution and a variational distribution. Despite promising empirical results, those algorithms resort to biased stochastic gradient descent procedures and thus lack theoretical guarantees. In this paper, we formalize and study the VR-IWAE bound, a generalization of the Importance Weighted Auto-Encoder (IWAE) bound. We show that the VR-IWAE bound enjoys several desirable properties and notably leads to the same stochastic gradient descent procedure as the VR bound in the reparameterized case, but this time by relying on unbiased gradient estimators. We then provide two complementary theoretical analyses of the VR-IWAE bound and thus of the standard IWAE bound. Those analyses shed light on the benefits or lack thereof of these bounds. Lastly, we illustrate our theoretical claims over toy and real-data examples.

Reference: K. Daudel, J. Benton, Y. Shi and A. Doucet (2023). Alpha-divergence Variational Inference Meets Importance Weighted Auto-Encoders: Methodology and Asymptotics. Journal of Machine Learning Research, 24(243):1−83.

Sampling from multimodal distributions with stochastic localization

Maxence Noble CMAP, École Polytechnique

Building upon score-based learning, new interest in stochastic localization techniques has recently emerged. In these models, one seeks to noise a sample from the data distribution through a stochastic process, called observation process, and progressively learns a denoiser associated to this dynamics. Apart from specific applications, the use of stochastic localization for the problem of sampling from an unnormalized target density has not been explored extensively. This work contributes to fill this gap. We consider a general stochastic localization framework and introduce an explicit class of observation processes, associated with flexible denoising schedules. We provide a complete methodology, Stochastic Localization via Iterative Posterior Sampling (SLIPS), to obtain approximate samples of these dynamics, and as a by-product, samples from the target distribution. Our scheme is based on a Markov chain Monte Carlo estimation of the denoiser and comes with detailed practical guidelines. We illustrate the benefits and applicability of SLIPS on the challenging setting of sampling from multimodal distributions.