MINE: Mutual Information Neural Estimation (1801.04062v5)

Abstract: We argue that the estimation of mutual information between high dimensional continuous random variables can be achieved by gradient descent over neural networks. We present a Mutual Information Neural Estimator (MINE) that is linearly scalable in dimensionality as well as in sample size, trainable through back-prop, and strongly consistent. We present a handful of applications on which MINE can be used to minimize or maximize mutual information. We apply MINE to improve adversarially trained generative models. We also use MINE to implement Information Bottleneck, applying it to supervised classification; our results demonstrate substantial improvement in flexibility and performance in these settings.

• The paper introduces MINE, a neural estimator that uses the Donsker-Varadhan KL-divergence formulation to reliably estimate mutual information in high-dimensional spaces.
• It employs gradient descent and back-propagation, ensuring scalability and consistency for both large sample sizes and high-dimensional data.
• Empirical results demonstrate that integrating MINE in GANs and bi-directional models enhances mode coverage and improves reconstruction quality.

Mutual Information Neural Estimation: A Synopsis

The paper "Mutual Information Neural Estimation" by Mohamed Ishmael Belghazi et al. presents a novel approach to estimating mutual information (MI) between high-dimensional continuous random variables through neural networks. This method, named Mutual Information Neural Estimator (MINE), leverages gradient descent and back-propagation for optimization, promising scalability and consistency.

Key Contributions

1. Mutual Information Neural Estimator (MINE):
   • MINE is designed to estimate MI by employing a dual representation of the Kullback-Leibler (KL) divergence, specifically the Donsker-Varadhan representation, which offers a tighter bound compared to other $f$-divergence representations.
   • The estimator is scalable in terms of both sample size and dimensionality, making it suitable for a wide array of high-dimensional data problems.
2. Applications and Benefits:
   • The paper demonstrates the utility of MINE in several contexts:
     • Generative Adversarial Networks (GANs): Used to address mode collapse by maximizing MI between the samples generated and a code.
     • Bi-directional Adversarial Models: Enhancing inference and improving reconstruction quality in models such as Adversarially Learned Inference (ALI).
     • Information Bottleneck Method: Facilitating continuous application of this method, thereby improving classification tasks on datasets like MNIST.

Theoretical Foundations

MINE’s core innovation lies in leveraging the Donsker-Varadhan representation for the KL-divergence: $KL(P || Q) = \sup_{T} E_P[T] - \log(E_Q[e^T])$ This allows MINE to frame MI estimation as an optimization problem over neural networks, where $T_\theta$ is a network parameterized by $\theta$. The estimator involves maximizing: $$I_\Theta(X; Z) = \sup_{\theta} E_{P_{XZ}}[T_\theta] - \log(E_{P_X \otimes P_Z}[e^{T_\theta}])$$

Empirical Validation

Belghazi et al. validate MINE through several empirical tests:

• Non-linear Dependencies:

MINE effectively captures non-linear relationships, demonstrated by experiments on synthetic datasets where MINE's performance closely aligns with the ground truth.

• Generative Adversarial Networks:

The inclusion of a MI term in the GAN objective helps mitigate mode collapse. For instance, in the Stacked MNIST experiment, MINE significantly improves mode coverage compared to the baseline GAN, capturing all 1000 modes available in the data.

• Bi-directional Models:

By maximizing MI between the data and latent variable distributions, MINE-enhanced ALI models show improved reconstruction quality and sample diversity, outperforming baseline methods.

Implications and Future Directions

The introduction of MINE marks a step forward in the estimation of mutual information for high-dimensional data, a crucial task in various machine learning applications, including generative models and representation learning. The scalability and consistency of MINE open new avenues for applying MI to more complex and higher-dimensional problems.

For future developments in AI, MINE's framework establishes a foundation for integrating robust mutual information estimates into diverse machine learning paradigms. There is potential for further expansion into different types of $f$-divergences and extending MINE’s capabilities to new applications such as causal inference and feature selection.

Conclusion

The paper by Belghazi et al. rigorously establishes a scalable, consistent, and practical approach to mutual information estimation using neural networks. Through theoretical justification and empirical validation, it demonstrates the broad applicability of MINE, suggesting it as a valuable tool for enhancing generative and inferential models within the field of machine learning.