Learning Representations for Neural Network-Based Classification Using the Information Bottleneck Principle (1802.09766v6)

Abstract: In this theory paper, we investigate training deep neural networks (DNNs) for classification via minimizing the information bottleneck (IB) functional. We show that the resulting optimization problem suffers from two severe issues: First, for deterministic DNNs, either the IB functional is infinite for almost all values of network parameters, making the optimization problem ill-posed, or it is piecewise constant, hence not admitting gradient-based optimization methods. Second, the invariance of the IB functional under bijections prevents it from capturing properties of the learned representation that are desirable for classification, such as robustness and simplicity. We argue that these issues are partly resolved for stochastic DNNs, DNNs that include a (hard or soft) decision rule, or by replacing the IB functional with related, but more well-behaved cost functions. We conclude that recent successes reported about training DNNs using the IB framework must be attributed to such solutions. As a side effect, our results indicate limitations of the IB framework for the analysis of DNNs. We also note that rather than trying to repair the inherent problems in the IB functional, a better approach may be to design regularizers on latent representation enforcing the desired properties directly.

Learning Representations for Neural Network-Based Classification Using the Information Bottleneck Principle

In the paper titled "Learning Representations for Neural Network-Based Classification Using the Information Bottleneck Principle," the authors Rana Ali Amjad and Bernhard C. Geiger provide a theoretical critique of employing the Information Bottleneck (IB) framework for training deep neural networks (DNNs). This paper contributes significantly to understanding the limitations of applying the IB principle in practice, especially for deterministic DNNs used in classification tasks.

Key Challenges Identified

1. Ill-Posed Optimization Problem: The principal challenge identified by the authors is that the direct application of the IB functional in deterministic settings frequently results in an ill-posed optimization problem. The mutual information between the inputs and intermediate representations, when calculated straightforwardly for deterministic networks, can become infinite or piecewise constant. This unpredictability renders the optimization process highly impractical.
2. Inadequate for Desired Properties: The IB framework, inherently focusing on compression and retention of relevant information, fails to ensure other desirable characteristics such as robustness to noise and simplicity of decision boundaries in learned representations. The authors highlight with examples that minimizing the IB functional does not necessarily lead to representations that are robust or facilitate straightforward classification decisions.

Proposed Remedies

To address the challenges above, the authors suggest several modifications:

1. Stochastic Neural Networks: Employing stochastic neural networks, either by introducing noise in the intermediate layers or by redesigning the network to include stochastic components, can help in making the mutual information finite and manageable. This approach also provides generalized data augmentation beneficial for training with better robustness.
2. Including Decision Rules: Incorporating decision rules directly into the formulation can mitigate some of the adverse computational characteristics associated with the IB functional.
3. Modified Cost Functions: Replacing the mutual information terms in the IB framework with quantized, smoothed, or bound alternatives can stabilize computation and better promote desired characteristics. For instance, utilizing variational bounds or noise-induced alterations to mutual information calculations are advantageous and align better with gradient-based optimization techniques.

Empirical Support from Related Work

The paper critically assesses empirical research and corroborates its theoretical findings by drawing from works that have successfully employed modified versions of the IB framework. Studies like those by Alemi et al. and Kolchinsky et al. have adapted the framework by combining the principle with techniques like variational inference and auxiliary noise, yielding better-performing models on classification tasks with enhanced generalization and robustness.

Implications and Future Directions

The implications of this work are profound, revisiting and challenging the prevailing assumptions about the effectiveness of the IB framework in its unmodified form. By dissecting the inherent complexities and proposing pragmatic solutions, the authors open new avenues for developing more robust and efficient DNN architectures. It underscores the necessity for designing representation regularizers that capture practical characteristics of representations directly, rather than relying solely on theoretical constructs like information-theoretic compression.

In conclusion, this paper furnishes a comprehensive theoretical analysis of the IB framework's limitations and provides a scholarly foundation for further innovations in network training methodologies. Future research could focus on empirically validating these proposed solutions across varied datasets and their applications in real-world scenarios, providing more insights into the interplay between theoretical constructs and practical implementation outcomes.