Black Box Variational Inference (1401.0118v1)

Abstract: Variational inference has become a widely used method to approximate posteriors in complex latent variables models. However, deriving a variational inference algorithm generally requires significant model-specific analysis, and these efforts can hinder and deter us from quickly developing and exploring a variety of models for a problem at hand. In this paper, we present a "black box" variational inference algorithm, one that can be quickly applied to many models with little additional derivation. Our method is based on a stochastic optimization of the variational objective where the noisy gradient is computed from Monte Carlo samples from the variational distribution. We develop a number of methods to reduce the variance of the gradient, always maintaining the criterion that we want to avoid difficult model-based derivations. We evaluate our method against the corresponding black box sampling based methods. We find that our method reaches better predictive likelihoods much faster than sampling methods. Finally, we demonstrate that Black Box Variational Inference lets us easily explore a wide space of models by quickly constructing and evaluating several models of longitudinal healthcare data.

• The paper presents a flexible BBVI method that bypasses model-specific derivations by employing stochastic optimization and Monte Carlo sampling.
• The methodology rewrites the ELBO gradient as an expectation, using techniques like control variates and Rao-Blackwellization to reduce variance.
• The approach achieves faster predictive likelihood convergence compared to traditional sampling methods, enhancing scalability and ease of use.

Black Box Variational Inference

Overview

The paper "Black Box Variational Inference" by Rajesh Ranganath, Sean Gerrish, and David M. Blei presents a flexible and efficient approach for variational inference (VI) that reduces the model-specific derivation overhead typically associated with traditional VI methods. Variational inference is a staple method for approximating posterior distributions in complex latent variable models; however, deriving these inference algorithms often necessitates extensive model-specific work, which can be a significant barrier. The authors propose a "black box" variational inference (BBVI) algorithm that leverages stochastic optimization and Monte Carlo sampling to generalize across a wide range of models with minimal additional derivation effort.

Methodology

The proposed BBVI approach hinges on rewriting the gradient of the variational objective, the Evidence Lower Bound (ELBO), as an expectation with respect to the variational distribution. By sampling from this distribution, a Monte Carlo estimate of the gradient is obtained, which is then used in a stochastic optimization framework to update the variational parameters. This method sidesteps the need for complex model-specific derivations.

The ELBO is optimized via:

1. Stochastic Optimization: Utilizing Robbins-Monro's stochastic optimization to handle noisy gradient estimates.
2. Variance Reduction:
   • Rao-Blackwellization: Averaging over some random variables analytically to reduce the variability of the estimate.
   • Control Variates: Applying control variates to further reduce variance.

The stochastic gradient relies only on the log-likelihood function of the model, making it easily implementable in a black-box manner where all model-specific details are abstracted away.

Results

The paper demonstrates the efficacy of BBVI through comparisons with Metropolis-Hastings within Gibbs sampling methods. Specifically, BBVI achieves better predictive likelihoods much faster than these traditional sampling methods. The paper highlights that variance reduction techniques are crucial for the fast convergence and practical implementation of the algorithm.

Practical Implications

The practical implications of BBVI are manifold:

1. Ease of Use: By significantly lowering the model-specific derivation barrier, BBVI allows practitioners to experiment with a wide variety of models without extensive hand-tuning.
2. Scalability: Using adaptive learning rates and stochastic variational inference, BBVI can be scaled to handle massive datasets efficiently.
3. General Applicability: This method can be applied to any model where the log-likelihood can be computed, making it a versatile tool for researchers.

Future Directions

Future enhancements to BBVI could focus on expanding the library of variational families, dynamically adjusting the number of samples needed for the stochastic gradient, and incorporating quasi-Monte Carlo methods to further reduce sampling variance. These improvements could make BBVI even more robust and efficient, broadening its applicability in the landscape of probabilistic modeling and inference.

In conclusion, the "Black Box Variational Inference" methodology provides a pragmatic and efficient alternative to traditional methods by simplifying implementation and accelerating convergence without sacrificing flexibility or generality.