Hamiltonian Variational Auto-Encoder (1805.11328v2)

Abstract: Variational Auto-Encoders (VAEs) have become very popular techniques to perform inference and learning in latent variable models as they allow us to leverage the rich representational power of neural networks to obtain flexible approximations of the posterior of latent variables as well as tight evidence lower bounds (ELBOs). Combined with stochastic variational inference, this provides a methodology scaling to large datasets. However, for this methodology to be practically efficient, it is necessary to obtain low-variance unbiased estimators of the ELBO and its gradients with respect to the parameters of interest. While the use of Markov chain Monte Carlo (MCMC) techniques such as Hamiltonian Monte Carlo (HMC) has been previously suggested to achieve this [23, 26], the proposed methods require specifying reverse kernels which have a large impact on performance. Additionally, the resulting unbiased estimator of the ELBO for most MCMC kernels is typically not amenable to the reparameterization trick. We show here how to optimally select reverse kernels in this setting and, by building upon Hamiltonian Importance Sampling (HIS) [17], we obtain a scheme that provides low-variance unbiased estimators of the ELBO and its gradients using the reparameterization trick. This allows us to develop a Hamiltonian Variational Auto-Encoder (HVAE). This method can be reinterpreted as a target-informed normalizing flow [20] which, within our context, only requires a few evaluations of the gradient of the sampled likelihood and trivial Jacobian calculations at each iteration.

• The paper introduces HVAE to optimize ELBO estimation using a time-inhomogeneous Hamiltonian Monte Carlo framework that reduces variance.
• It employs deterministic transitions and target-informed flows to eliminate the need for learned reverse kernels, enhancing computational efficiency.
• Empirical results on Gaussian models and MNIST demonstrate robust performance and scalability in high-dimensional inference tasks.

An Overview of Hamiltonian Variational Auto-Encoder

The paper introduces the Hamiltonian Variational Auto-Encoder (HVAE), a novel advancement in variational inference that leverages Hamiltonian Monte Carlo dynamics integrated with variational auto-encoders (VAEs). This research primarily focuses on resolving the variance and scalability issues surrounding the estimation of the evidence lower bound (ELBO) and its gradients, a fundamental task in VAEs.

Background and Motivation

VAEs are powerful techniques that marry latent variable models with neural network parameterizations, providing flexible posterior approximations. Nevertheless, these models often face challenges related to posterior flexibility and computational efficiency, especially when dealing with large datasets. While Markov chain Monte Carlo (MCMC) methods such as Hamiltonian Monte Carlo (HMC) have been suggested to improve the efficiency of ELBO estimators, existing frameworks have struggled with the need for reverse kernels, which influence performance significantly and generally resist optimality.

Methodological Contributions

This work aims to optimize the selection of reverse kernels within MCMC settings, employing a time-inhomogeneous HMC strategy. By using the Hamiltonian Importance Sampling (HIS) approach, the authors successfully construct a framework where HVAE optimizes the necessary parameters using low-variance, unbiased estimators that leverage the reparameterization trick. This setup translates effectively into a target-informed normalizing flow, maintaining high computational scalability and robustness across datasets and dimensions. Notably, this approach circumvents the need for learned reverse kernels, a notable departure from prior methods.

HVAE operates by using deterministic transitions within an extended space composed of position and momentum variables. In effect, this leads to a flow-based model wherein the target distribution actively influences the transformation, preserving desired properties of the dynamics and reducing the complexity of the required Jacobian calculations.

Empirical Analysis and Results

The empirical evaluation of HVAE is two-pronged: initially, the method is tested within a Gaussian model context to gauge its parameter estimation efficacy. Results illustrate the superiority of HVAE over traditional Variational Bayes (VB) methods and Normalizing Flows (NFs), particularly in higher-dimensional settings where standard approaches falter.

The research further extends to the MNIST dataset, a classic case in unsupervised learning with VAEs. Here, HVAE enhances the standard convolutional VAE performance by achieving improved negative log-likelihood (NLL) estimates, thus demonstrating its potential for real-world applications. The findings underscore HVAE's flexibility, where variations in leapfrog step sizes and tempering schemes notably influence model performance, highlighting its robustness.

Theoretical and Practical Implications

Theoretically, this research positions HVAE as a scalable and efficient method for MCMC-based variational inference, blending the robustness of Hamiltonian dynamics with the adaptability of machine learning models. By removing the reliance on reverse dynamics and offering a simplified implementation pathway, HVAE marks a significant advancement in variational methods, fostering more informed posterior approximations.

Practically, HVAE's design reinforces its applicability to large-scale datasets by emphasizing computational scalability. As HVAE optimizes fewer parameters than traditional flows, it becomes an advantageous option in memory-constrained settings where inference needs surpass resource availability.

Future Directions

The paper opens avenues for further refinement, such as exploring alternative deterministic dynamics that might preserve target distributions beyond Hamiltonian dynamics. Joint applications with the controlled variance reduction techniques proposed in other recent works could magnify the practical utility and precision of HVAE, potentially paving the way for even broader adoption in high-dimensional, complex data environments.

In sum, the Hamiltonian Variational Auto-Encoder represents a significant step forward in the toolkit available to researchers working with variational methods, offering enhanced performance, scalability, and theoretical elegance to efficiently tackle the complexity inherent in deep generative models.