There Was Never a Bottleneck in Concept Bottleneck Models

Deep learning representations are often difficult to interpret, which can hinder their deployment in sensitive applications. Concept Bottleneck Models (CBMs) have emerged as a promising approach to mitigate this issue by learning representations that support target task performance while ensuring that each component predicts a concrete concept from a predefined set. In this work, we argue that CBMs do not impose a true bottleneck: the fact that a component can predict a concept does not guarantee that it encodes only information about that concept. This shortcoming raises concerns regarding interpretability and the validity of intervention procedures. To overcome this limitation, we propose Minimal Concept Bottleneck Models (MCBMs), which incorporate an Information Bottleneck (IB) objective to constrain each representation component to retain only the information relevant to its corresponding concept. This IB is implemented via a variational regularization term added to the training loss. As a result, MCBMs yield more interpretable representations, support principled concept-level interventions, and remain consistent with probability-theoretic foundations.

💡 Research Summary

The paper critically examines Concept Bottleneck Models (CBMs), which aim to make deep neural networks interpretable by forcing each latent dimension z_j to predict a human‑understandable concept c_j. While CBMs ensure that z_j contains enough information to recover c_j (a sufficient statistic), they do not prevent z_j from also encoding unrelated nuisance information n from the raw input x. This “information leakage” undermines two core promises of CBMs: (i) interpretability, because z_j cannot be explained solely by c_j, and (ii) intervenability, because manipulating z_j to change a concept may unintentionally affect other latent factors.

To address this, the authors introduce Minimal Concept Bottleneck Models (MCBMs), which explicitly impose an Information Bottleneck (IB) on each z_j. The IB objective minimizes the conditional mutual information I(Z_j; X | C_j), forcing z_j to be a minimal sufficient statistic of c_j and to discard any information not relevant to c_j. Using a variational formulation, this objective becomes a KL‑divergence term between the stochastic encoder p_θ(z_j | x) and a new “representation head” q_ϕ(ẑ_j | c_j). Under Gaussian assumptions, the KL reduces to a mean‑squared error between the encoder’s mean f_θ(x) and a concept‑dependent prototype g_z_ϕ(c_j). The full training loss combines three components: the standard prediction loss L_y, the concept reconstruction loss L_c, and the IB regularizer L_IB, weighted by hyper‑parameters β and γ.

The paper provides a thorough theoretical analysis. It shows that achieving I(Z_j; X | C_j)=0 is equivalent to satisfying the Markov chain X ↔ C_j ↔ Z_j, which in turn guarantees that p(z_j | c_j)=p(z_j | x). This property makes the conditional distribution p(z_j | c_j) well‑defined, enabling principled interventions: setting c_j to a desired value directly determines the distribution of z_j, without side effects on other concepts.

Empirical evaluation is conducted on synthetic datasets and real‑world medical imaging tasks. The authors measure information leakage via I(Z; N) and demonstrate that MCBMs dramatically reduce this metric compared to vanilla CBMs and related variants (e.g., Hard CBMs, Stochastic CBMs). Visualization of latent spaces shows near‑perfect disentanglement: each z_j correlates almost exclusively with its assigned concept. Intervention experiments—where a specific concept is forced to a target value—show that MCBMs alter the final prediction ŷ exactly as predicted by the causal graph, whereas CBMs exhibit unintended changes due to residual nuisance information. Importantly, predictive accuracy of MCBMs matches or slightly exceeds that of standard CBMs, indicating that the IB regularizer does not sacrifice performance.

The paper also surveys related work on information leakage, concept embedding models, and various extensions of CBMs (autoregressive, energy‑based, hard bottleneck). It positions MCBMs as a principled solution that directly addresses the root cause—lack of an explicit bottleneck—rather than applying ad‑hoc fixes.

In summary, the authors identify a fundamental flaw in existing concept bottleneck approaches, propose a mathematically grounded IB‑augmented architecture (MCBM), and validate that it yields truly bottlenecked, interpretable, and intervenable representations without compromising task performance. This contribution has significant implications for deploying trustworthy AI in high‑stakes domains where both accuracy and human‑understandable reasoning are essential.