Residual Stochastic Concept Bottleneck Models

Stephen Marcou; Mateo Espinosa Zarlenga; Sonia Laguna; Moritz Vandenhirtz; Julia E. Vogt; Mateja Jamnik

cs.LGOct 16, 2026

Residual Stochastic Concept Bottleneck Models

Authors: Stephen Marcou, Mateo Espinosa Zarlenga, Sonia Laguna, Moritz Vandenhirtz, Julia E. Vogt, Mateja Jamnik

Organizations: ETH Zurich · University of Oxford · University of Cambridge

Abstract

Concept-based models (CMs) learn human-interpretable intermediate representations that enable users to understand the model's reasoning and steer its predictions through test-time concept interventions. However, in realistic settings, their performance can be limited by concept incompleteness, where the training concepts fail to capture all task-relevant information. Adding an unconstrained residual pathway to the model's predictive process can overcome the consequences of incompleteness. However, this introduces an inscrutable, leak-prone prediction pathway that may weaken concept interventions, as these often leave residual variables unchanged. To address this limitation, we introduce Residual Stochastic Concept Bottleneck Models (Res-SCBMs), a family of concept-based models that jointly model concept and residual variables through a multivariate Gaussian distribution. Res-SCBMs learn a covariance structure that explicitly captures dependencies between the known concepts and learnt residual variables, allowing interventions to propagate to statistically dependent residual variables through Gaussian conditioning. This extends the intervention capabilities of previous CMs without requiring concept completeness during training. Our results show that, across multiple concept-incomplete benchmarks, Res-SCBMs improve predictive performance over competing baselines while retaining effective test-time concept interventions.

Figures & tables

Figure 1: Overview of Res-SCBMs. Concept and residual variables are jointly modelled through a multivariate Gaussian distribution. At test time, concept interventions propagate to statistically dependent residual variables through Gaussian conditioning, updating the residual representation before prediction.

Method	CUB	CUB Incomplete	AwA2	AwA2 Incomplete
DNN (black box)	$68.60_{\pm 0.44}$	$64.77_{\pm 0.15}$	$92.93_{\pm 0.15}$	$92.50_{\pm 0.10}$
Linear head (GT concepts)	$100.00_{\hskip 25.60747pt}$	$84.91_{\pm 0.72}$	$100.00_{\hskip 25.60747pt}$	$76.20_{\pm 0.03}$
Hard CBM	$69.63_{\pm 0.90}$	$47.87_{\pm 1.00}$	$92.30_{\pm 0.10}$	$68.20_{\pm 0.87}$
CEM	$70.50_{\pm 0.36}$	$64.77_{\pm 1.71}$	$92.63_{\pm 0.25}$	$\mathbf{91.87}_{\bm{\pm 0.06}}$
SCBM	$71.30_{\pm 0.00}$	$49.03_{\pm 0.71}$	$91.87_{\pm 0.15}$	$68.43_{\pm 0.31}$
Res-SCBM w/o $\mathcal{L}_{\mathrm{int}}$ (ours)	$70.87_{\pm 0.15}$	$64.43_{\pm 0.72}$	$92.60_{\pm 0.17}$	$\mathbf{91.87}_{\bm{\pm 0.06}}$

Table 1: Task accuracy (%) before concept interventions. Values are means over seeds, with standard deviations in subscript. Best/second-best concept-based results are bold/underlined. The DNN is reported as a predictive baseline, while the linear head uses ground-truth concepts and is reported as a reference. Residual models outperform non-leaky baselines under concept incompleteness.

Method	CUB	CUB Incomplete	AwA2	AwA2 Incomplete
Hard CBM	$97.83_{\pm 0.12}$	$97.00_{\pm 0.17}$	$99.30_{\pm 0.00}$	$99.10_{\pm 0.00}$
CEM	$97.83_{\pm 0.06}$	$97.13_{\pm 0.15}$	$\mathbf{99.37}_{\bm{\pm 0.06}}$	$\mathbf{99.27}_{\bm{\pm 0.06}}$
SCBM	$\mathbf{97.97}_{\bm{\pm 0.06}}$	$97.17_{\pm 0.06}$	$99.17_{\pm 0.06}$	$99.10_{\pm 0.00}$
Res-SCBM w/o $\mathcal{L}_{\mathrm{int}}$ (ours)	$\mathbf{97.97}_{\bm{\pm 0.06}}$	$97.33_{\pm 0.06}$	$99.00_{\pm 0.00}$	$99.10_{\pm 0.00}$
Res-SCBM (ours)	$\mathbf{97.97}_{\bm{\pm 0.06}}$	$\mathbf{97.53}_{\bm{\pm 0.12}}$	$99.03_{\pm 0.06}$	$99.10_{\pm 0.00}$

Table 2: Mean concept ROC-AUC (%) before concept interventions. Values are means over seeds, with standard deviations in subscript. Best/second-best concept-based results are bold/underlined. Residual models remain competitive with baseline concept prediction performance.

Figure 2: Task accuracy (top) and mean concept ROC-AUC (bottom) as the fraction of corrected concepts increases on CUB , CUB-Incomplete , AwA2 , AwA2-Incomplete . Res-SCBM remains responsive to concept corrections despite its residual pathway.

Figure 3: Effect of intervention-aware training. Task accuracy (left) and mean concept ROC-AUC (right) under concept interventions on CUB-Incomplete , with/without

\mathcal{L}_{\mathrm{int}}

and residual dimensions

R\in\{20,500\}

. The horizontal line shows linear classifier accuracy on ground-truth concepts. Intervention-aware training preserves responsiveness at increased residual capacity.

Figure 4: Learned concept–residual cross-covariance for two AwA2-Incomplete test samples (concepts: rows; residuals: columns). Distinct patterns across residual dimensions and samples reveal input-dependent concept–residual dependencies.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Dataset	Train	Val.	Test	Classes	Concepts	Groups
CUB	4,796	1,198	5,794	200	112	28
CUB-Incomplete	4,796	1,198	5,794	200	22	7
AwA2	22,393	7,464	7,465	50	85	28
AwA2-Incomplete	22,393	7,464	7,465	50	9	6

Appendix

Table 3: High-level properties of the datasets used in our experiments. Train, val., and test indicate the number of samples in each split. All input images have dimensions

3\times 224\times 224

Concept Bottleneck Models

Concept Bottleneck Models route predictions through human-understandable concepts that people can inspect, correct, or intervene on.

See all 88 papers

Jun 5, 2025

Towards Reasonable Concept Bottleneck Models

We propose a novel, flexible, and efficient framework for designing Concept Bottleneck Models (CBMs) that enables practitioners to explicitly encode and extend their prior knowledge and beliefs about the concept-concept (

C-C

) and concept-task (

C \to Y

) relationships within the model's reasoning when making predictions. The resulting

\textbf{C}

oncept

\textbf{REA}

soning

\textbf{M}

odels (CREAMs) architecturally encode arbitrary types of

C-C

relationships such as mutual exclusivity, hierarchical associations, and/or correlations, as well as potentially sparse

C \to Y

relationships. Moreover, CREAM can optionally incorporate a regularized side-channel to complement the potentially {incomplete concept sets}, achieving competitive task performance while encouraging predictions to be concept-grounded. To evaluate CBMs in such settings, we introduce a

C \to Y

agnostic metric that quantifies interpretability when predictions partially rely on the side-channel. In our experiments, we show that, without additional computational overhead, CREAM models support efficient interventions, can avoid concept leakage, and achieve black-box-level performance under missing concepts. We further analyze how an optional side-channel affects interpretability and intervenability. Importantly, the side-channel enables CBMs to remain effective even in scenarios where only a limited number of concepts are available.

Nektarios Kalampalikis, Kavya Gupta, Georgi Vitanov +1

Department of Computer Science Saarland University, Germany

Concept Bottleneck Models Interpretable ML

Ask PDF

Aug 7, 2026

ReCBM: Uncertainty-Gated Relational Reasoning for Concept Bottleneck Models

Concept Bottleneck Models (CBMs) provide an interpretable framework by grounding predictions in human-understandable concepts, enabling semantic inspection and test-time intervention. Recent variants have improved CBMs through richer concept representations, uncertainty estimation, and dependency modeling. However, robust reasoning under unreliable concept states remains underexplored. Without such reasoning, misleading semantic evidence can propagate through the bottleneck, compromising both explanations and downstream predictions. To address this issue, we propose ReCBM, an uncertainty-gated relational reasoning framework for CBMs. ReCBM introduces semantically defined concept relations into the bottleneck and uses uncertainty to guide their refinement. By modeling co-occurrence, implication, and exclusion, ReCBM specifies how evidence is exchanged across concepts, while uncertainty modulates the contribution of each concept during this process. Experiments across diverse datasets showed that ReCBM improved concept and task recovery under missing and flipped concepts, supported uncertainty-aware intervention, and extracted compact task-relevant concept subsets without degrading downstream performance.

An Sui, Yuzhu Li, Fuping Wu +1

School of Data Science, Fudan University, Shanghai, China · Institute of Science and Technology for Brain-Inspired Intelligence, Fudan University, Shanghai, China. · National Heart and Lung Institute, Imperial College London, London, United Kingdom.

Concept Bottleneck Models

Ask PDF

Feb 16, 2026

Concepts' Information Bottleneck Models

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.

Karim Galliamov, Syed M Ahsan Kazmi, Adil Khan +1

University of Amsterdam · University of West of England · University of Hull +1

Concept Bottleneck Models Information Bottleneck Variational Information Bottleneck

Ask PDF

Abstract

Concept Bottleneck Models

Explore similar work

Towards Reasonable Concept Bottleneck Models

ReCBM: Uncertainty-Gated Relational Reasoning for Concept Bottleneck Models

Concepts' Information Bottleneck Models