Skip to content

There Was Never a Bottleneck in Concept Bottleneck Models

Conference: ICLR 2026 arXiv: 2506.04877 Code: None (per paper description) Area: Interpretability / Concept Bottleneck Models Keywords: Concept Bottleneck Models, Information Bottleneck, Information Leakage, Interventionability, Representation Learning

TL;DR

This paper identifies that Concept Bottleneck Models (CBMs) do not enforce a true "bottleneck" — the fact that a representation variable \(z_j\) can predict concept \(c_j\) does not imply it encodes only the information of \(c_j\). The paper proposes MCBM (Minimal Concept Bottleneck Model), which applies information bottleneck regularization to constrain each \(z_j\) to retain only the information of its corresponding concept, thereby achieving genuinely disentangled representations and reliable concept interventions.

Background & Motivation

  • The Promise of CBMs: By training each component \(z_j\) of the representation to predict an interpretable concept \(c_j\), CBMs aim to provide both interpretability and interventionability.
  • Information Leakage: The ability of \(z_j\) to predict \(c_j\) does not imply that \(z_j\) encodes only \(c_j\). In the extreme case, \(z_j\) may encode the entire input \(\mathbf{x}\) while still satisfying the CBM constraint.
  • Two Consequences:
  • Impaired Interpretability: \(z_j\) cannot be fully explained by \(c_j\).
  • Ineffective Interventions: Modifying \(z_j\) alters not only \(c_j\) but also any other information encoded therein.
  • Theoretical Flaw in CBM Interventions: There is no directed path from \(c_j\) to \(z_j\) in CBMs, leaving \(p(z_j|c_j)\) undefined in the graphical model. Existing interventions rely on an ad-hoc approximation via empirical quantiles of the sigmoid inverse.

Core Distinction: VM vs. CBM vs. MCBM

VM CBM MCBM
\(z_j\) encodes all of \(c_j\)? ✗ ✓ ✓
\(z_j\) encodes only \(c_j\)? ✗ ✗ ✓

Method

1. Data Generating Process

  • Input \(\mathbf{x}\) is determined by concepts \(\mathbf{c}\) and nuisance \(\mathbf{n}\): \(p(\mathbf{x}, \mathbf{y}, \mathbf{c}, \mathbf{n}) = p(\mathbf{x}|\mathbf{c}, \mathbf{n}) p(\mathbf{y}|\mathbf{x}) p(\mathbf{c}, \mathbf{n})\)
  • Nuisance is decomposed into task-relevant \(\mathbf{n}_y\) and task-irrelevant \(\mathbf{n}_{\bar{y}}\) components.

2. Three Information-Theoretic Objectives

Objective 1 (shared by VM/CBM/MCBM): Maximize \(I(Z; Y)\) — representation predicts the target.

\[\max_{\theta, \phi} \mathbb{E}_{p(\mathbf{x}, \mathbf{y})} \left[\mathbb{E}_{p_\theta(\mathbf{z}|\mathbf{x})} \left[\log q_\phi(\hat{\mathbf{y}}|\mathbf{z})\right]\right]\]

Objective 2 (CBM/MCBM): Maximize \(I(Z_j; C_j)\) — \(z_j\) is a sufficient statistic for \(c_j\).

\[\max_{\theta, \phi} \mathbb{E}_{p(\mathbf{x}, c_j)} \left[\mathbb{E}_{p_\theta(z_j|\mathbf{x})} \left[\log q_\phi(\hat{c}_j|z_j)\right]\right]\]

Objective 3 (MCBM only): Minimize \(I(Z_j; X | C_j)\) — information bottleneck.

\[\min_{\theta, \phi} \mathbb{E}_{p(\mathbf{x}, c_j)} \left[D_{KL}\left(p_\theta(z_j|\mathbf{x}) \| q_\phi(\hat{z}_j|c_j)\right)\right]\]

This ensures that \(z_j\) retains no additional information about \(\mathbf{x}\) beyond what is captured by \(c_j\), enforcing the Markov chain \(X \leftrightarrow C_j \leftrightarrow Z_j\).

3. MCBM Training Objective

\[\max_{\theta, \phi} \sum_{k=1}^N \sum_i \log q_\phi(\hat{\mathbf{y}}|f'_\theta(x^{(k)}, \epsilon^{(i)})) + \beta \sum_{j=1}^n \log q_\phi(\hat{c}_j|f'_{\theta,j}(\mathbf{x}^{(k)}, \epsilon^{(i)})) - \gamma \sum_{j=1}^n D_{KL}(p_\theta(z_j|\mathbf{x}^{(k)}) \| q_\phi(\hat{z}_j|c_j^{(k)}))\]
  • First term: task prediction loss.
  • Second term (\(\beta\)-weighted): concept prediction loss.
  • Third term (\(\gamma\)-weighted): information bottleneck regularization (exclusive to MCBM).

4. Intervention Mechanism

Interventions in MCBM are theoretically grounded: $\(p(z_j|c_j) = q_\phi(z_j|c_j)\)$

Since optimizing Objective 3 ensures \(z_j\) encodes only the information of \(c_j\), modifying \(z_j\) strictly corresponds to an intervention on \(c_j\).

5. Representation Head Design

  • Binary concepts: \(g_\phi^z(c_j) = \lambda\) if \(c_j=1\), else \(-\lambda\)
  • Multi-class concepts: \(g_\phi^z(c_j) = \lambda \cdot \text{one\_hot}(c_j)\) (prototype learning)
  • Continuous concepts: \(g_\phi^z(c_j) = \lambda \cdot c_j\)

The encoder uses a stochastic formulation \(p_\theta(\mathbf{z}|\mathbf{x}) = \mathcal{N}(\mathbf{z}; f_\theta(\mathbf{x}), \sigma_x^2 I)\), trained via the reparameterization trick.

Key Experimental Results

Information Leakage Metric: URR (Uncertainty Reduction Ratio)

Measures how much nuisance information beyond the concept set is encoded in \(z\) (lower is better).

Task-Relevant Nuisance Leakage

Method MPI3D Shapes3D CIFAR-10 CUB AwA2
Vanilla 35.0 45.5 19.8 3.8 1.5
CBM 28.1 18.1 18.5 3.8 1.4
CEM 43.2 15.8 27.2 3.9 1.1
ECBM 25.2 47.1 18.1 4.5 1.1
MCBM (high γ) 0.0 0.0 17.6 2.4 0.7

Task-Irrelevant Nuisance Leakage

Method MPI3D Shapes3D
Vanilla 11.3 42.7
CBM 7.4 20.6
CEM 15.5 40.9
MCBM (any γ) 0.0 0.0

Key Findings

  1. CEM and ECBM exacerbate leakage: On certain datasets, leakage exceeds that of the Vanilla model.
  2. MCBM eliminates leakage entirely: Under high \(\gamma\), nuisance information is reduced to 0 across all datasets.
  3. ARCBM and HCBM offer no systematic advantage: Neither consistently outperforms standard CBM in controlling leakage.
  4. Trade-off: MCBM incurs a slight decrease in task accuracy, as task-relevant nuisance \(\mathbf{n}_y\) is also excluded.

Highlights & Insights

  1. Fundamental critique of CBMs: The paper demonstrates that CBMs are, in a principled sense, misnamed — a true bottleneck has never existed in these models.
  2. Natural introduction of the information bottleneck: The condition \(I(Z_j; X | C_j) = 0\) precisely formalizes the requirement that \(z_j\) encodes only its corresponding concept.
  3. Theoretical analysis of CBM intervention failures (Section 5): The paper proves that the intervention assumptions underlying CBMs are probabilistically ill-founded.
  4. Practical KL divergence regularization: Under Gaussian assumptions, the regularization reduces to a simple MSE loss.
  5. Visualization of disentangled representations: In MCBM's representation space, samples sharing the same concept value cluster tightly together.

Limitations & Future Work

  • There is an inherent trade-off between task accuracy and concept purity: when the concept set is incomplete, excluding nuisance necessarily reduces performance.
  • Concept annotations are required — a limitation shared with all CBM-based methods.
  • Handling of continuous concepts relies on Gaussian assumptions.
  • The hyperparameter \(\gamma\) requires tuning to balance disentanglement and task performance.
  • Validation on larger-scale models and more complex tasks remains to be conducted.
  • CBM Variants: CEM (concept embeddings), HCBM (hard bottleneck), ARCBM (autoregressive), SCBM (stochastic)
  • Information Leakage Analysis: Margeloiu et al. 2021; Parisini et al. 2025
  • Information Bottleneck: Tishby et al. 2000; Alemi et al. 2016 (Variational Information Bottleneck)

Rating

  • Novelty: ⭐⭐⭐⭐⭐ — A fundamental reexamination of the CBM paradigm.
  • Technical Depth: ⭐⭐⭐⭐ — Rigorous information-theoretic formalization with clear variational derivations.
  • Experimental Thoroughness: ⭐⭐⭐⭐ — Five datasets, 8+ competing methods, and multi-faceted analysis.
  • Value: ⭐⭐⭐⭐ — Provides a principled solution toward genuinely interpretable concept-based models.