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There Was Never a Bottleneck in Concept Bottleneck Models

Conference: ICLR 2026
arXiv: 2506.04877
Code: None (according to the paper)
Area: Interpretability / Concept Bottleneck Models
Keywords: Concept Bottleneck Models, Information Bottleneck, Information Leakage, Intervenability, Representation Learning

TL;DR

The paper points out that Concept Bottleneck Models (CBMs) do not actually possess a true "bottleneck"—the fact that a representation variable \(z_j\) can predict concept \(c_j\) does not mean it encodes only information about \(c_j\). It proposes the Minimal Concept Bottleneck Model (MCBM), which uses information bottleneck regularization to constrain each \(z_j\) to retain only information from its corresponding concept, achieving true decoupled representations and reliable concept interventions.

Background & Motivation

  • The Promise of CBMs: Providing interpretability and intervenability by ensuring each component \(z_j\) of the representation predicts an understandable concept \(c_j\).
  • Information Leakage: \(z_j\) predicting \(c_j\) is not equivalent to \(z_j\) only encoding \(c_j\). In extreme cases, \(z_j\) might encode the entire input \(\mathbf{x}\) while still satisfying CBM constraints.
  • Two Consequences:
  • Compromised Interpretability: \(z_j\) cannot be fully explained by \(c_j\).
  • Ineffective Intervention: Modifying \(z_j\) not only changes \(c_j\) but also affects other information encoded within it.
  • Theoretical Flaws in CBM Interventions: There is no directed path from \(c_j\) to \(z_j\) in CBMs; \(p(z_j|c_j)\) is undefined in the graphical model. Existing interventions rely on ad-hoc empirical quantile approximations of the inverse sigmoid function.

Core Difference: CBM vs. MCBM

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

Method

Overall Architecture

MCBM supplements standard CBMs with an omitted constraint: not only must each representation component \(z_j\) predict the corresponding concept \(c_j\), but \(z_j\) must also carry no residual information about the input \(\mathbf{x}\) given \(c_j\). This constraint is added to the training objective in the form of information bottleneck regularization, compressing \(z_j\) into a minimal sufficient statistic of \(c_j\). This establishes a strict correspondence between "modifying \(z_j\)" and "intervening on \(c_j\)."

Key Designs

1. Data Generation Assumption: Explicitly Modeling the Source of Leakage

The paper assumes a generative process \(p(\mathbf{x}, \mathbf{y}, \mathbf{c}, \mathbf{n}) = p(\mathbf{x}|\mathbf{c}, \mathbf{n})\, p(\mathbf{y}|\mathbf{x})\, p(\mathbf{c}, \mathbf{n})\). The input \(\mathbf{x}\) is determined by both labeled concepts \(\mathbf{c}\) and unlabeled nuisances \(\mathbf{n}\). Nuisances are further divided into task-relevant \(\mathbf{n}_y\) and task-irrelevant \(\mathbf{n}_{\bar{y}}\). This breakdown is necessary because "leakage" in CBMs essentially means \(z_j\) encodes these nuisances that should have been blocked; explicitly defining nuisances allows for precise characterization of what to retain and what to discard.

2. Three Information-Theoretic Objectives: Progressing from Sufficient to Minimal

MCBM optimization consists of three complementary information constraints to tighten the representation. The first is task sufficiency (common to all variants), maximizing \(I(Z;Y)\), which corresponds to \(\max_{\theta, \phi} \mathbb{E}_{p(\mathbf{x}, \mathbf{y})}[\mathbb{E}_{p_\theta(\mathbf{z}|\mathbf{x})}[\log q_\phi(\hat{\mathbf{y}}|\mathbf{z})]]\), ensuring the representation can predict labels. The second is concept sufficiency (also in CBM), maximizing \(I(Z_j; C_j)\), written as \(\max_{\theta, \phi} \mathbb{E}_{p(\mathbf{x}, c_j)}[\mathbb{E}_{p_\theta(z_j|\mathbf{x})}[\log q_\phi(\hat{c}_j|z_j)]]\), requiring \(z_j\) to be sufficient to decode \(c_j\). What distinguishes MCBM is the third: minimizing conditional mutual information \(I(Z_j; X | C_j)\), approximated via KL divergence as \(\min_{\theta, \phi} \mathbb{E}_{p(\mathbf{x}, c_j)}[D_{KL}(p_\theta(z_j|\mathbf{x}) \| q_\phi(\hat{z}_j|c_j))]\). This forces \(z_j\) to contain no extra information about \(\mathbf{x}\) once \(c_j\) is known, making the Markov chain \(X \leftrightarrow C_j \leftrightarrow Z_j\) hold. While the first two only ensure \(z_j\) is "sufficient" to encode \(c_j\), the third ensures it "only" encodes \(c_j\), which is the bottleneck missing in CBMs.

3. Theoretically Grounded Intervention: Equating \(z_j\) Modification to \(c_j\) Intervention

In standard CBMs, no directed path exists from \(c_j\) to \(z_j\), making \(p(z_j|c_j)\) undefined. Existing interventions are thus ad-hoc approximations. After optimizing the third objective in MCBM, \(z_j\) only contains information about \(c_j\), allowing interventions to directly follow \(p(z_j|c_j) = q_\phi(z_j|c_j)\). Replacing \(z_j\) strictly corresponds to setting the concept to a target value without inadvertently modifying other secretly encoded information, upgrading intervenability from "approximate" to "exact."

4. Concept-Conditioned Representation Heads and Stochastic Encoders: Trainable KL Terms

To provide a concrete form for \(q_\phi(\hat{z}_j|c_j)\), the paper designs prototypical representation heads: for binary concepts, \(g_\phi^z(c_j) = \lambda\) (\(c_j=1\)) or \(-\lambda\) (\(c_j=0\)); for multi-class, \(g_\phi^z(c_j) = \lambda \cdot \text{one\_hot}(c_j)\); for continuous concepts, \(g_\phi^z(c_j) = \lambda \cdot c_j\). This essentially learns a prototype anchor for each concept value. The encoder uses a stochastic version \(p_\theta(\mathbf{z}|\mathbf{x}) = \mathcal{N}(\mathbf{z}; f_\theta(\mathbf{x}), \sigma_x^2 I)\) trained with the reparameterization trick. Under this Gaussian assumption, the KL term simplifies to a straightforward MSE between the representation and the prototype.

Loss & Training

By merging the three objectives, the total MCBM training goal is:

\[\max_{\theta, \phi} \sum_{k=1}^N \sum_i \log q_\phi(\hat{\mathbf{y}}|f'_\theta(x^{(k)}, \epsilon^{(i)})) + \beta \sum_{j=1}^n \log q_\phi(\hat{c}_j|f'_{\theta,j}(\mathbf{x}^{(k)}, \epsilon^{(i)})) - \gamma \sum_{j=1}^n D_{KL}(p_\theta(z_j|\mathbf{x}^{(k)}) \| q_\phi(\hat{z}_j|c_j^{(k)}))\]

The first term is task prediction loss, the second is concept prediction loss weighted by \(\beta\), and the third is the information bottleneck regularization unique to MCBM, weighted by \(\gamma\). \(\gamma\) controls the strength of decoupling: increasing it pushes nuisance leakage toward zero but may discard task-useful information in \(\mathbf{n}_y\), making it a knob for the trade-off between "concept purity" and "task accuracy."

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 some datasets, leakage is higher than in Vanilla models.
  2. MCBM Eliminates Leakage: Under high γ, nuisance information drops to 0 across all datasets.
  3. No Systematic Advantage for ARCBM and HCBM: They do not control leakage better than standard CBMs.
  4. The Cost: MCBM's task accuracy drops slightly because it excludes task-useful information in \(\mathbf{n}_y\).

Highlights & Insights

  1. Fundamental Concept Critique: Points out that CBMs are misnamed—there was never a true "bottleneck."
  2. Natural Intro of Information Bottleneck: Precisely formalizes "encoding concepts only" using \(I(Z_j; X | C_j) = 0\).
  3. Analysis of Theoretical Flaws in CBM Intervention (Section 5): Proves that CBM intervention assumptions are probabilistically invalid.
  4. Practical KL Regularization: Reduces to simple MSE loss under Gaussian assumptions.
  5. Visualization of Decoupled Representations: Samples with the same concept values cluster tightly in the MCBM representation space.

Limitations & Future Work

  • Inherent trade-off between task accuracy and concept purity: if the concept set is incomplete, excluding nuisances inevitably degrades performance.
  • Requires concept labeling—a limitation shared with all CBM methods.
  • Treatment of continuous concepts relies on Gaussian assumptions.
  • Hyperparameter \(\gamma\) needs tuning to balance decoupling and performance.
  • Not yet validated on larger-scale models or more complex tasks.
  • CBM Variants: CEM (Concept Embedding), 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 re-examination of the CBM field.
  • Technical Depth: ⭐⭐⭐⭐ — Rigorous information-theoretic formalization and clear variational derivation.
  • Experimental Thoroughness: ⭐⭐⭐⭐ — 5 datasets, 8+ methods compared, multi-angle analysis.
  • Value: ⭐⭐⭐⭐ — Provides a principled solution for truly interpretable concept models.