- The paper introduces a novel stochastic search method for variational Bayesian inference that directly optimizes the lower bound using control variates.
- It replaces traditional bounding techniques in mean-field variational inference with Monte Carlo gradient approximations to better handle intractable expectations.
- Empirical evaluations on logistic regression and HDP models show that the approach improves inference accuracy and scalability for large datasets.
Variational Bayesian Inference with Stochastic Search
The paper by Paisley, Blei, and Jordan discusses an advanced method for approximate Bayesian inference, particularly focusing on the challenges posed by mean-field variational Bayesian (MFVB) inference. MFVB is an optimization approach used to approximate the full posterior distribution of latent variables in Bayesian models. This method is notable for its adaptability across various problem domains, including mixture modeling and sequential modeling, and its capacity to handle large data sets via online inference.
Overview of Mean-field Variational Inference
Mean-field variational inference approximates a full posterior distribution using a factorized set of distributions. The traditional approach maximizes a lower bound of the marginal likelihood, which is often hindered by intractable expectations. While typical solutions involve introducing bounds to make expectations tractable, this can lead to suboptimal posterior approximations, as the true objective function may not be directly optimized.
Introduction of Stochastic Search Variational Bayes
The authors propose an alternative approach that applies stochastic optimization to directly optimize the variational lower bound. This method utilizes a stochastic approximation of the gradient, based on Monte Carlo integration. A key innovation is the introduction of control variates to reduce variance in the stochastic gradient estimation. Control variates are tractable functions that correlate well with intractable functions, enabling efficient variance reduction without requiring the function to strictly bound the intractable one.
Empirical Evaluation
The methodology is empirically evaluated on two non-conjugate models: logistic regression and the hierarchical Dirichlet process (HDP). The logistic regression implementation utilizes control variates in the form of lower bounds and second-order Taylor expansions to handle intractable expectations. For the HDP model, a piecewise approach incorporating both upper bounds and control variates demonstrated effective variance reduction.
Implications and Future Directions
This stochastic search approach introduces a significant improvement in variational inference flexibility and accuracy, particularly for models where traditional bounds are inefficient or impractical. By circumventing the need for strict lower bounds, this method can potentially lead to more accurate posterior approximations. Future work might explore further extensions of this approach to other complex models and larger datasets, as well as improving the computational efficiency of the control variate selection process.
Conclusion
The proposed stochastic search variational Bayes provides a meaningful advancement in the field of approximate Bayesian inference. By directly optimizing the variational objective and effectively managing the variance of stochastic approximations, this method affords researchers a novel toolset for addressing intractable integrals inherent in Bayesian learning frameworks. The paper positions itself as a valuable resource for both understanding the limitations of current approaches and exploring new methodologies in the field of Bayesian inference.