Probabilistic Circuits for Variational Inference in Discrete Graphical Models
This addresses the problem of high bias or variance in gradient estimation for variational inference in discrete graphical models, offering a more efficient and accurate approach for researchers and practitioners in machine learning.
The paper tackles the difficulty of variational inference in discrete graphical models by proposing a method using selective Sum Product Networks (SPNs) to compute ELBO gradients exactly without sampling, achieving competitive or better lower bounds than existing approximations like mean-field and Loopy Belief Propagation on models such as Ising models and Latent Dirichlet Allocation.
Inference in discrete graphical models with variational methods is difficult because of the inability to re-parameterize gradients of the Evidence Lower Bound (ELBO). Many sampling-based methods have been proposed for estimating these gradients, but they suffer from high bias or variance. In this paper, we propose a new approach that leverages the tractability of probabilistic circuit models, such as Sum Product Networks (SPN), to compute ELBO gradients exactly (without sampling) for a certain class of densities. In particular, we show that selective-SPNs are suitable as an expressive variational distribution, and prove that when the log-density of the target model is a polynomial the corresponding ELBO can be computed analytically. To scale to graphical models with thousands of variables, we develop an efficient and effective construction of selective-SPNs with size $O(kn)$, where $n$ is the number of variables and $k$ is an adjustable hyperparameter. We demonstrate our approach on three types of graphical models -- Ising models, Latent Dirichlet Allocation, and factor graphs from the UAI Inference Competition. Selective-SPNs give a better lower bound than mean-field and structured mean-field, and is competitive with approximations that do not provide a lower bound, such as Loopy Belief Propagation and Tree-Reweighted Belief Propagation. Our results show that probabilistic circuits are promising tools for variational inference in discrete graphical models as they combine tractability and expressivity.