[Paper Review] Stochastic Gradient VB and the Variational Auto-Encoder

Diederik P. Kingma, Max Welling|arXiv (Cornell University)|Dec 20, 2013

Gaussian Processes and Bayesian Inference17 references232 citations

TL;DR

This paper introduces Stochastic Gradient Variational Bayes (SGVB), a scalable algorithm that enables efficient training of variational auto-encoders by reparameterizing the variational lower bound using an independent noise variable. This enables end-to-end backpropagation through stochastic nodes, allowing gradient-based optimization of both generative and recognition models even with intractable posteriors and large datasets.

ABSTRACT

Can we efficiently learn the parameters of directed probabilistic models, in the presence of continuous latent variables with intractable posterior distributions, and large datasets? We introduce an unsupervised on-line learning algorithm that efficiently optimizes the variational lower bound on the marginal likelihood and that, under some mild conditions, even works in the intractable case. The algorithm, Stochastic Gradient Variational Bayes (SGVB), optimizes a probabilistic encoder (also called a recognition model) to approximate the intractable posterior distribution of the latent variables. Crucial is a reparameterization of the variational bound with an independent noise variable, yielding a stochastic objective function which can be jointly optimized w.r.t. variational and generative parameters using standard gradient-based stochastic optimization methods. Theoretical advantages are reflected in experimental results.

Motivation & Objective

• To address the challenge of learning parameters in directed probabilistic models with continuous latent variables and intractable posterior distributions.
• To develop an efficient, on-line learning algorithm suitable for large-scale datasets.
• To enable joint optimization of both generative and variational (recognition) models using gradient-based methods.
• To provide a theoretically grounded method that works even when exact posterior inference is intractable.

Proposed method

• Reparameterize the variational lower bound using an independent noise variable to make the objective function differentiable with respect to model parameters.
• Construct a stochastic objective function that depends on the noise variable and can be optimized via stochastic gradient descent.
• Use backpropagation through the stochastic computation graph to jointly update both the generative model and the recognition model parameters.
• Apply standard stochastic optimization techniques to minimize the variational lower bound on the marginal likelihood.
• Ensure the method remains valid under mild regularity conditions, even when the true posterior is intractable.

Experimental results

Research questions

• RQ1 Can we efficiently optimize the variational lower bound in the presence of intractable posterior distributions for continuous latent variables?
• RQ2 Can we train both the generative and recognition models end-to-end using gradient-based methods?
• RQ3 Does the reparameterization trick enable effective and scalable learning in large-scale settings?
• RQ4 Is the method robust and effective even when exact posterior inference is computationally infeasible?

Key findings

• The SGVB algorithm enables efficient, scalable training of variational auto-encoders using stochastic gradient descent.
• The reparameterization trick allows backpropagation through stochastic nodes, making joint optimization of generative and recognition models feasible.
• The method achieves good performance on large datasets despite intractable posterior distributions.
• Theoretical analysis shows the method is valid under mild conditions, even when exact posterior computation is not possible.