Alpha-divergence Variational Inference Meets Importance Weighted Auto-Encoders: Methodology and Asymptotics
This work addresses a theoretical gap in variational inference for machine learning practitioners, offering incremental improvements in algorithm stability and analysis.
The paper tackles the lack of theoretical guarantees in alpha-divergence variational inference methods by introducing the VR-IWAE bound, which generalizes the IWAE bound and provides unbiased gradient estimators, leading to improved stochastic gradient descent procedures.
Several algorithms involving the Variational Rényi (VR) bound have been proposed to minimize 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.