Concepts' Information Bottleneck Models

TL;DR

This work addresses the fidelity and interpretability gaps in Concept Bottleneck Models (CBMs) by introducing a Concepts' Information Bottleneck (CIBM) regularizer applied to the concept layer. By explicitly minimizing $I(X;C)$ while preserving $I(C;Y)$, the approach yields minimal-sufficient concepts and reduces concept leakage without architectural changes, via two practical implementations: a bound-based method $\text{IB}_B$ and an estimator-based method $\text{IB}_E$. The authors provide theoretical justifications—including a PAC-Bayes generalization bound—and validate the methods across six CBM variants and three datasets, showing improved end-to-end accuracy, stronger interventions, and lower leakage, supported by information-plane analyses. Overall, CIBMs offer a theoretically grounded, generalizable path to more faithful, intervenable CBMs with practical impact for explanations and debugging in real-world systems.

Abstract

Concept Bottleneck Models (CBMs) aim to deliver interpretable predictions by routing decisions through a human-understandable concept layer, yet they often suffer reduced accuracy and concept leakage that undermines faithfulness. We introduce an explicit Information Bottleneck regularizer on the concept layer that penalizes $I(X;C)$ while preserving task-relevant information in $I(C;Y)$, encouraging minimal-sufficient concept representations. We derive two practical variants (a variational objective and an entropy-based surrogate) and integrate them into standard CBM training without architectural changes or additional supervision. Evaluated across six CBM families and three benchmarks, the IB-regularized models consistently outperform their vanilla counterparts. Information-plane analyses further corroborate the intended behavior. These results indicate that enforcing a minimal-sufficient concept bottleneck improves both predictive performance and the reliability of concept-level interventions. The proposed regularizer offers a theoretic-grounded, architecture-agnostic path to more faithful and intervenable CBMs, resolving prior evaluation inconsistencies by aligning training protocols and demonstrating robust gains across model families and datasets.

Figures (9)

• Figure 1: Our proposed CIBMs pipeline. The image is encoded through $p(z \:\lvert\:x)$, which in turn encodes the concepts with $q(c \:\lvert\:z)$, and the labels are predicted through $q(y \:\lvert\:c)$. These modules are implemented as neural networks. We introduced the IB regularization as mutual information optimizations over the variables as shown in dashed lines.
• Figure 2: Our generative model $p(y \:\lvert\:x)p(c \:\lvert\:x)p(z \:\lvert\:x)p(x)$ (solid lines), and its variational approximation $q(y \:\lvert\:c)q(c \:\lvert\:z)q(z \:\lvert\:x)q(x)$ (dashed lines).
• Figure 3: Change in target prediction accuracy after intervening on concept groups following the random strategy as described in Section \ref{['sec:interventions']}. (TTI stands for Test-Time Interventions, and NR for non-regularized.) We show expanded plots, with less clutter, in Fig. \ref{['fig:expanded_interventions']}.
• Figure : Our proposed CIBMs pipeline. The image is encoded through $p(z \:\lvert\:x)$, which in turn encodes the concepts with $q(c \:\lvert\:z)$, and the labels are predicted through $q(y \:\lvert\:c)$. These modules are implemented as neural networks. We introduced the IB regularization as mutual information optimizations over the variables as shown in dashed lines.
• Figure C.1: Calculation of the empirical entropy of the concepts, $H(C)$, during training for CBM (SJ) on CUB.

Theorems & Definitions (6)

• proposition 1: PAC-Bayes CBM
• proof
• theorem 1: PAC-Bayes CIBM
• proof
• theorem 2: Generalization Advantage of CIBM
• proof