Deep Variational Information Bottleneck (1612.00410v7)

Abstract: We present a variational approximation to the information bottleneck of Tishby et al. (1999). This variational approach allows us to parameterize the information bottleneck model using a neural network and leverage the reparameterization trick for efficient training. We call this method "Deep Variational Information Bottleneck", or Deep VIB. We show that models trained with the VIB objective outperform those that are trained with other forms of regularization, in terms of generalization performance and robustness to adversarial attack.

• The paper introduces a variational framework for the Information Bottleneck problem that balances compression and predictiveness.
• The paper leverages neural networks and the reparameterization trick to derive a tractable lower bound for efficient SGD optimization.
• The paper demonstrates that Deep VIB models outperform conventional techniques with lower error rates and enhanced adversarial robustness.

Essay: Deep Variational Information Bottleneck

The paper "Deep Variational Information Bottleneck" by Alemi, Fischer, Dillon, and Murphy introduces an innovative approach to the Information Bottleneck (IB) problem, leveraging variational inference techniques and neural networks for effective and efficient implementation. The authors propose a method they term Deep Variational Information Bottleneck (Deep VIB), which aims to learn representations that are both compact and informative of the target variable, while minimizing the information about the input data.

Concept and Objective

The essence of the IB principle, as formulated originally by Tishby et al., centers around balancing two competing objectives: maximizing the mutual information between the internal representation and the target variable, $I(Z; Y)$, and minimizing the mutual information between the internal representation and the input data, $I(Z; X)$. The constraint form of the objective can be reformulated using a Lagrange multiplier $\beta$, yielding the objective function $R_{IB} = I(Z; Y) - \beta I(Z; X)$. Here, $\beta$ controls the trade-off between representation compression and predictiveness.

Variational Approach

Direct computation of mutual information is often infeasible due to high computational costs, especially for complex, high-dimensional data. Consequently, the authors employ a variational approximation to derive a lower bound on the IB objective. By parameterizing the encoder and decoder using neural networks and applying the reparameterization trick, the authors enable stochastic gradient descent-based optimization. This approach is innovative as it circumvents the reliance on iterative algorithms like Blahut-Arimoto, which are impractical for deep neural networks.

Practical Benefits

The paper demonstrates that models trained with the VIB objective generalize better than those using conventional regularization techniques. Moreover, VIB models exhibit notable robustness to adversarial attacks, a crucial requirement for practical deployments. The stochastic nature of the internal representation learned by VIB inherently provides a buffer against small perturbations, which adversarial attacks typically exploit.

Experimental Validation

MNIST Evaluation

The authors conducted extensive experiments on the MNIST dataset, employing a simple multi-layer perceptron. They compared VIB against several baselines, including Dropout and Confidence Penalty methods. Notably, the VIB model achieved a test error rate of 1.13%, outperforming other regularization techniques. Furthermore, the robustness of VIB to adversarial attacks was confirmed through experiments utilizing both the Fast Gradient Sign (FGS) and $L_2$ optimization attack methods. VIB models required significantly larger perturbations to fool the network compared to deterministic counterparts.

High-Dimensional Data

For higher-dimensional data, such as images from the ImageNet dataset, the VIB framework again proved effective. Using a pre-trained Inception ResNet V2 model, the authors extracted features and replaced the final layers with a VIB architecture. The results reaffirmed VIB's ability to maintain high classification accuracy while significantly enhancing robustness to adversarial examples.

Implications and Future Directions

The Deep VIB method presents several practical and theoretical implications. Practically, it provides a robust and generalizable framework for training neural networks, particularly beneficial for applications where adversarial robustness and interpretability are paramount. Theoretically, it strengthens our understanding of the IB principle by presenting a tractable and scalable implementation.

Future research could extend VIB to sequence-to-sequence tasks, leveraging the IB principle to learn optimal representations over time. Additionally, integrating VIB objectives at multiple network layers could provide deeper insights and potentially further enhance performance and robustness. Exploring richer parametric forms for approximating the marginal distributions can also refine the current framework.

In conclusion, the Deep Variational Information Bottleneck model offers a sophisticated and practical solution to the IB problem, enhancing both performance and robustness of neural networks. This work undoubtedly opens new avenues for both theoretical exploration and applied machine learning advancements.