Unbiased Implicit Variational Inference (1808.02078v3)

Abstract: We develop unbiased implicit variational inference (UIVI), a method that expands the applicability of variational inference by defining an expressive variational family. UIVI considers an implicit variational distribution obtained in a hierarchical manner using a simple reparameterizable distribution whose variational parameters are defined by arbitrarily flexible deep neural networks. Unlike previous works, UIVI directly optimizes the evidence lower bound (ELBO) rather than an approximation to the ELBO. We demonstrate UIVI on several models, including Bayesian multinomial logistic regression and variational autoencoders, and show that UIVI achieves both tighter ELBO and better predictive performance than existing approaches at a similar computational cost.

• The paper introduces USIVI, a novel framework that directly optimizes the ELBO using an unbiased gradient estimator without density ratio estimation.
• It employs a semi-implicit variational distribution with neural network mappings to flexibly capture complex posterior distributions.
• Experimental results show that USIVI achieves improved ELBO and predictive log-likelihood on both synthetic and high-dimensional real-world datasets.

Unbiased Implicit Variational Inference: An Essay

The paper on "Unbiased Implicit Variational Inference" (USIVI) presents an advancement in the field of variational inference (VI), a technique often used for approximate Bayesian inference. It addresses critical challenges associated with traditional VI methods by introducing an implicit approach to broaden the applicability of VI techniques through the formulation of a more expressive variational family. The paper outlines a sophisticated approach that integrates variational autoencoders (VAEs) and Bayesian models such as Bayesian multinomial logistic regression.

Core Contributions

The central contribution of this research is the development of USIVI, a novel variational inference framework. USIVI introduces an implicit distribution modeled with hierarchical variational distributions that can be flexibly parameterized using neural networks. Unlike previous approaches that rely on approximations to the evidence lower bound (ELBO), USIVI directly optimizes the ELBO using an unbiased estimator of its gradient. This is achieved without resorting to density ratio estimation, which is often problematic in high-dimensional spaces, by exploiting the properties of the variational distribution and Monte Carlo estimation techniques.

Methodological Innovations

USIVI's methodology is characterized by two essential features:

1. Semi-Implicit Variational Distribution (SIVI): USIVI adopts a semi-implicit variational distribution method to enhance the flexibility of the approximating distribution. The distribution is composed of a reparameterizable conditional distribution but maintains flexibility via dependence on an implicit distribution using neural network mappings. This method effectively captures complex posteriors which are often unattainable by conventional mean-field approximations.
2. Unbiased Gradient Estimator: The paper introduces an unbiased estimator for the gradient of the ELBO that avoids density ratio estimation. This is achieved using a reparameterized variational family wherein the expectation is expressed as a function of the variational distribution. The methodology involves utilizing Markov Chain Monte Carlo (MCMC) techniques to estimate required expectations, significantly improving both the tightness of the ELBO and predictive accuracy over competing methods such as Stochastic Variational Inference (SVI).

Experimental Evaluations and Results

The experimental evaluations carried out on both synthetic datasets and real-world models, such as VAEs and Bayesian multinomial logistic regression, indicate that USIVI achieves better ELBO and predictive log-likelihood compared to existing VI frameworks. Notably, the computational cost of USIVI is comparable to alternative techniques, rendering it a feasible solution for practical applications.

On toy datasets, USIVI was effective in capturing the intrinsic complexities of the target distributions, corroborating the efficacy of implicit methods. The evaluations on real datasets like MNIST and Fashion-MNIST augur well for USIVI’s applicability to higher-dimensional data spaces. The model consistently demonstrated improved test log-likelihood, indicating its capacity for enhanced generalization and robustness in prediction tasks.

Implications and Future Directions

The introduction of USIVI has substantial implications for the future development of probabilistic inference methods in AI. By enabling more expressive variational distributions, USIVI widens the horizon for applying VI methods to a broader array of complex models and datasets. The success of USIVI highlights the potential for implicit methods to address limitations inherent in traditional VI and SVI.

Future research could focus on exploring improvements in sampling efficiency, potentially reducing computation overheads without compromising on inference precision. There is also latitude for extending USIVI’s framework to other probabilistic graphical models and exploring its integration with alternative machine learning paradigms.

In summary, the paper presents significant advancements in variational inference by developing a robust methodology that embraces the complexity of real-world data. Its contributions lie in laying a foundation for future research aimed at refining and expanding the application domains of VI methods beyond current boundaries.