[Paper Review] Variational Bayesian Inference with Stochastic Search

John Paisley, David M. Blei+1 more|arXiv (Cornell University)|Jun 27, 2012

Bayesian Methods and Mixture ModelsComputer Science271 citations

TL;DR

This paper proposes a stochastic optimization approach for variational Bayesian inference that directly maximizes the variational lower bound using stochastic search with control variates to reduce gradient variance. The method enables efficient inference in non-conjugate models like logistic regression and HDP approximations without relying on closed-form integrals or additional lower bounds.

ABSTRACT

Mean-field variational inference is a method for approximate Bayesian posterior inference. It approximates a full posterior distribution with a factorized set of distributions by max-imizing a lower bound on the marginal likeli-hood. This requires the ability to integrate a sum of terms in the log joint likelihood using this factorized distribution. Often not all in-tegrals are in closed form, which is typically handled by using a lower bound. We present an alternative algorithm based on stochastic optimization that allows for direct optimiza-tion of the variational lower bound. This method uses control variates to reduce the variance of the stochastic search gradient, in which existing lower bounds can play an im-portant role. We demonstrate the approach on two non-conjugate models: logistic regres-sion and an approximation to the HDP. 1.

Motivation & Objective

• Address the challenge of intractable integrals in mean-field variational inference, which often require approximations or lower bounds.
• Develop a method that bypasses the need for closed-form integrals by using stochastic optimization to directly optimize the variational lower bound.
• Enable efficient and scalable Bayesian inference in complex, non-conjugate models such as logistic regression and hierarchical Dirichlet processes.
• Reduce gradient variance in stochastic optimization using control variates, improving convergence and stability.
• Integrate existing lower bounds as tools to enhance the performance of the stochastic search framework.

Proposed method

• Formulate the variational inference problem as a stochastic optimization task over the variational parameters.
• Use Monte Carlo sampling to estimate the gradient of the variational lower bound, enabling optimization without closed-form integration.
• Apply control variates to reduce the variance of the stochastic gradient estimator, improving convergence speed and stability.
• Leverage existing lower bounds as auxiliary tools to inform or improve the control variate construction.
• Iteratively update variational parameters using stochastic gradient ascent on the lower bound, with variance reduction.
• Integrate the method into non-conjugate models such as logistic regression and HDP approximations, where standard mean-field methods fail due to intractable terms.

Experimental results

Research questions

• RQ1 Can stochastic optimization with variance reduction be effectively applied to variational Bayesian inference in non-conjugate models?
• RQ2 How does the use of control variates improve the convergence and accuracy of stochastic variational inference?
• RQ3 To what extent can existing lower bounds be used to enhance the performance of stochastic search in variational inference?
• RQ4 How does the proposed method compare to traditional mean-field variational inference in terms of scalability and accuracy for complex models?
• RQ5 Can the method be successfully applied to real-world non-conjugate models such as logistic regression and HDP approximations?

Key findings

• The proposed stochastic optimization framework enables direct maximization of the variational lower bound without requiring closed-form integrals.
• The use of control variates significantly reduces gradient variance, leading to faster and more stable convergence during optimization.
• The method achieves competitive inference quality on logistic regression and HDP approximation tasks, outperforming standard mean-field approaches in non-conjugate settings.
• Existing lower bounds can be effectively integrated into the stochastic search framework to improve gradient estimation and optimization performance.
• The approach demonstrates scalability and practicality for Bayesian inference in complex models where traditional variational methods fail due to intractable expectations.