A Comparison of Variational Bounds for the Information Bottleneck Functional

Area of Science:

• Machine Learning
• Information Theory
• Optimization

Background:

• The Information Bottleneck (IB) and Conditional Entropy Bottleneck (CEB) are key information-theoretic frameworks for understanding representation learning.
• Variational bounds for IB and CEB functionals have been proposed, with CEB bounds empirically linked to better generalization and adversarial robustness.
• The theoretical relationship and practical implications of optimizing these bounds remain an active area of research.

Purpose of the Study:

• To investigate the relationship between variational bounds for the Information Bottleneck (IB) and Conditional Entropy Bottleneck (CEB) functionals.
• To understand why optimizing CEB bounds empirically yields superior generalization and adversarial robustness compared to IB bounds.
• To elucidate the conditions under which an ordering between these variational bounds can be established.

Main Methods:

• Relating the variational bounds proposed by Alemi et al. (2017) for IB and Fischer (2020) for CEB.
• Analyzing the general setting to determine if a consistent ordering exists between the variational bounds.
• Investigating the effect of restricting feasible sets on the optimization of these bounds.

Main Results:

• In the most general setting, no definitive ordering can be established between the variational bounds of the IB and CEB functionals.
• An ordering between the bounds can be enforced by restricting the feasible sets over which the optimization occurs.
• The absence of a general ordering suggests the CEB variational bound may be inherently more amenable to optimization or a valuable cost function independently.

Conclusions:

• The empirical success of CEB bounds in improving generalization and robustness might stem from their optimization properties rather than a strict theoretical hierarchy over IB bounds.
• Future research should explore the specific characteristics of the feasible sets that enable ordering and further investigate the CEB bound as a standalone optimization objective.
• This work provides theoretical insights into the practical performance differences observed when optimizing IB and CEB variational bounds in machine learning tasks.