Decision-Making with Auto-Encoding Variational Bayes (2002.07217v3)

Abstract: To make decisions based on a model fit with auto-encoding variational Bayes (AEVB), practitioners often let the variational distribution serve as a surrogate for the posterior distribution. This approach yields biased estimates of the expected risk, and therefore leads to poor decisions for two reasons. First, the model fit with AEVB may not equal the underlying data distribution. Second, the variational distribution may not equal the posterior distribution under the fitted model. We explore how fitting the variational distribution based on several objective functions other than the ELBO, while continuing to fit the generative model based on the ELBO, affects the quality of downstream decisions. For the probabilistic principal component analysis model, we investigate how importance sampling error, as well as the bias of the model parameter estimates, varies across several approximate posteriors when used as proposal distributions. Our theoretical results suggest that a posterior approximation distinct from the variational distribution should be used for making decisions. Motivated by these theoretical results, we propose learning several approximate proposals for the best model and combining them using multiple importance sampling for decision-making. In addition to toy examples, we present a full-fledged case study of single-cell RNA sequencing. In this challenging instance of multiple hypothesis testing, our proposed approach surpasses the current state of the art.

• The paper demonstrates that using the variational distribution as a posterior surrogate biases risk estimates, necessitating a multi-objective framework.
• The methodology trains models with various objectives and employs iterative sampling, validated on pPCA, MNIST, and single-cell RNA-seq datasets.
• Empirical findings show improved posterior calibration, increased decision accuracy, and better false discovery control in complex tasks.

Decision-Making with Auto-Encoding Variational Bayes

The paper "Decision-Making with Auto-Encoding Variational Bayes" by Romain Lopez et al. investigates the use of auto-encoding variational Bayes (AEVB) for decision-making tasks. It critically examines the common practice of using the variational distribution as a surrogate for the posterior distribution, highlighting its limitations and proposing a novel framework to improve decision quality in variational autoencoders (VAEs).

Overview

The central premise of this research is to address the shortcomings inherent in using the variational distribution for posterior approximation. The authors argue that this approach results in biased estimates of the expected risk, adversely affecting decision-making. There are two primary reasons for this:

1. The model fit by AEVB may not accurately represent the underlying data distribution.
2. The variational distribution may not adequately approximate the posterior distribution under the fitted model.

To mitigate these issues, the paper explores fitting several objective functions besides the traditional Evidence Lower Bound (ELBO) for the variational distribution, while continuing to fit the generative model based on the ELBO. Specifically, it proposes evaluating the suitability of different approximate posteriors via importance sampling error and model parameter bias in probabilistic principal component analysis (pPCA).

Methodology

The authors propose a three-step procedure for improving decision-making with VAEs:

1. Model Training: Train the model using several objective functions (e.g., VAE, IWAE, WW, and $\rchi$-VAE) and select the best model based on a performance metric, such as the IWELBO on held-out data.
2. Approximating the Posterior: Fit several approximate posteriors for the selected model using the same objective functions and iterative sampling methods like annealed importance sampling (AIS).
3. Decision Making: Combine the approximate posteriors with multiple importance sampling to make decisions that minimize the expected loss under the posterior.

Theoretical and Empirical Analysis

The paper conducts a rigorous theoretical analysis using a pPCA model to understand the trade-offs between different posterior approximations. They derive concentration bounds for the log-likelihood ratio and use these to bound the error for importance sampling estimators. This analysis shows that overestimating the posterior variance is generally more favorable than underestimating it.

Empirically, the paper evaluates the proposed methodology on three datasets:

• pPCA synthetic data
• MNIST dataset for classification-based decision theory
• Single-cell RNA sequencing data for multiple hypothesis testing and differential gene expression detection.

Significant findings include:

• pPCA Analysis: IWAE and $\rchi$-VAE deliver better posterior approximation compared to VAE, as indicated by lower mean absolute error (MAE) metrics and improved IWELBO.
• MNIST Experiment: The proposed multi-step approach (IWAE-MIS) not only leads to better classification accuracy but also demonstrates improved decision-making with the reject option.
• Single-cell RNA-seq Data: The $\rchi$-VAE outperforms other methods in terms of controlling the false discovery rate, offering better-calibrated estimates of expected posterior FDR.

Implications and Future Work

The proposed framework holds promising implications for both the theoretical understanding and practical application of VAEs in decision-making tasks. By leveraging different objective functions for fitting the variational distribution, the approach mitigates the bias and high variance issues typically associated with using the variational distribution as a proposal. Practically, this framework can lead to better decision-making in fields such as genomics, image classification, and possibly other domains requiring Bayesian decision theory.

Future developments could further explore hybrid algorithms that merge this framework with recent advancements in loss-calibrated inference and amortized Monte Carlo integration. Moreover, further research is needed to address the computational overhead introduced by fitting multiple models and to ensure the proposed methods' scalability.

Conclusion

This paper presents a robust analysis and a novel methodology that significantly enhance the decision-making capability of auto-encoding variational Bayes. The findings encourage exploration beyond traditional ELBO-based training, advocating for a composite approach using multiple objective functions and multiple importance sampling. This methodology sets a new direction in employing VAEs for more reliable and accurate decision-making in various complex tasks.