Towards Reasonable Concept Bottleneck Models

Nektarios Kalampalikis; Kavya Gupta; Georgi Vitanov; Isabel Valera

cs.LGJun 5, 2025

Towards Reasonable Concept Bottleneck Models

Authors: Nektarios Kalampalikis, Kavya Gupta, Georgi Vitanov, Isabel Valera

Organizations: Department of Computer Science Saarland University, Germany

Abstract

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.

Figures & tables

Panel of Figure 1 — Figure 1: Reasoning graph for FMNIST from ( Seo & Shin, 2019 ) . We show the incomplete concept set used in iFMNIST as well as the additional season-related concepts from cFMNIST . Concepts and classes within the boxes are mutually exclusive. Edges between concepts represent C-C relationships, while edges from concepts to classes represent C→Y dependencies.

Panel of Figure 2 — Figure 2: Partial illustration of CUB’s reasoning graph. Concepts are represented as nodes, with numbers in parentheses indicating the cardinality for each concept. Nodes within the same group are mutually exclusive and disconnected, while edges are one-to-one between nodes from different groups. Bidirected edges indicate statistical dependencies between concepts.

Panel of Figure 3 — Figure 3: Reasoning graph for predicting “Smiling” in CelebA ( Liu et al., 2015 ) , showing the most correlated facial concepts ( $\mathit{C}$ ) that directly influence the prediction ( $\mathit{Y}$ ).

Model	Relationships	$A_{C}$
CBM	Independent	$\mathbbm{I}_{K}$
ACBM	Autoregressive	$(\mathbbm{1}_{i<j})_{i,j=1}^{K}$
SCBM	Correlations	$A_{C}=A_{C}^{T}$
C 2 BM	DAG	sparse $(\mathbbm{1}_{i<j})_{i,j=1}^{K}$
CGM	Causal graph	$A_{C}$

Table 1: Implementation of prior approaches’ C-C relationships in CREAM. For CGM we leave it at

A_{C}

, as it best describes a PDAG.

Figure 4: Sketch of CREAM’s framework. The backbone’s output is split into concept ( $\mathbf{z}_{C}$ ) and side-channel ( $\mathbf{z}_{\mathit{Y}}$ ) representations for concept and task prediction, respectively. The Concept-Concept block models relationships between concepts, while the Concept-Task block uses both $\hat{\mathbf{z}}_{\mathit{Y}}$ and the concepts to predict the task label based on embedded relationships.

Figure 5: Illustration of the simplest case of propagating interventions in CREAM, corresponding to $d=0$ . For deeper graphs, the backward and forward passes is applied recursively.

Model	iFMNIST		cFMNIST		CUB		CelebA
Model	$ACC_{Y}$	$ACC_{C}$	$ACC_{Y}$	$ACC_{C}$	$ACC_{Y}$	$ACC_{C}$	$ACC_{Y}$	$ACC_{C}$
Black-box	$92.68_{0.00}$	-	$92.68_{0.00}$	-	$74.91_{0.02}$	-	$78.47_{0.30}$	-
$C_{true}\!\to\!Y$	$60.00$	-	$100.00$	-	$100.00$	-	$84.51$	-
CBM	${91.19}_{0.40}$	$96.74_{0.36}$	$92.00_{0.17}$	$97.33_{0.06}$	$73.76_{0.32}$	$80.90_{0.08}$	${79.28}_{0.39}$	${79.77}_{0.17}$
ACBM	$52.25_{5.18}$	$98.94_{0.01}$	$90.68_{0.09}$	$98.03_{0.02}$	$66.98_{0.43}$	$\textbf{94.15}_{0.01}$	$81.40_{0.46}$	$80.57_{0.53}$
SCBM	$57.68_{0.63}$	$98.86_{0.02}$	$90.80_{0.17}$	$97.54_{0.06}$	$70.55_{0.19}$	$90.28_{0.04}$	$76.63_{0.47}$	$80.54_{0.11}$

Table 2: Task and concept accuracy (%). We report

\text{mean}_{\text{std}}

over multiple runs. Bold indicates the best-performing method. Overall, CREAM achieves a strong balance between predictive performance and interpretability. See Table 1 for the reasoning structure associated with each model.

Model	iFMNIST		cFMNIST		CUB		CelebA
Model	Time	Peak Memory	Time	Peak Memory	Time	Peak Memory	Time	Peak Memory
CBM+SC	x $1.50_{0.14}$	x $1.00_{0.00}$	x $1.63_{0.21}$	x $1.00_{0.00}$	x $1.02_{0.03}$	x $1.00_{0.00}$	x $1.00_{0.00}$	x $1.00_{0.00}$
ACBM	x $5.30_{0.54}$	x $1.04_{0.00}$	x $7.90_{0.90}$	x $1.04_{0.00}$	x $14.13_{0.14}$	x $1.25_{0.00}$	x $3.27_{0.05}$	x $1.07_{0.00}$
SCBM	x $27.54_{2.82}$	x $1.05_{0.00}$	x $27.99_{3.72}$	x $1.05_{0.00}$	x $9.21_{0.26}$	x $1.30_{0.00}$	x $3.27_{0.05}$	x $1.07_{0.00}$
CREAM w/o SC	x $\textbf{1.59}_{0.10}$	x $\textbf{1.00}_{0.00}$	x $\textbf{1.81}_{0.24}$	x $\textbf{1.00}_{0.00}$	x $\textbf{2.38}_{0.12}$	x $\textbf{1.00}_{0.00}$	x $\textbf{1.00}_{0.01}$	x $\textbf{1.00}_{0.00}$
$\textbf{CGM}_{CD}$	x $9.15_{1.03}$	x $361.87_{2.06}$	x $20.24_{3.08}$	x $1139.22_{3.01}$	-	-	x $2.21_{0.07}$	x $82.57_{2.61}$

Table 3: Comparison of training time and peak GPU memory usage relative to CBM. For CGM, results are reported for CPU execution, along with overall system peak memory usage. All values indicate multiplicative factors relative to CBM (e.g., CREAM requires 1.82× the training time of CBM in iFMNIST).

Figure 6: Impact of individual interventions on task accuracy. The baseline model is

C_{true}\!\to\!Y

. CREAM’s accuracy improves with increasing number of interventions up to the number of

\mathit{C}_{direct}

. For cFMNIST, the inset axes show a zoomed-out view, to account for

\text{CGM}_{CD}

Figure 8: Left: Impact of dropout rates on concept channel importance. Models with complete concept sets need less dropout. Right: Interventions in CREAM on the cFMNIST dataset. Higher

CCI

values, corresponding to higher dropout rates, reflect improved intervenability.

Figure 9: Left: Task accuracy and CCI for varying number of concepts in CUB. We lower the output size of the splitter, while keeping

d_{C}=4

. Right: Concept and task accuracy vs dropout rate (

p

). Values are presented as mean

\pm

standard deviation across 5 seeds. Increased

p

may lead to drops in task accuracy.

Appendix figures & tables21 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Model	$CCI\uparrow$	$PCI\uparrow$	$PSI\downarrow$
iFMNIST	H-CREAM	$0.80_{0.01}$	$0.66_{0.06}$	$0.35_{0.00}$
	S-CREAM	$0.80_{0.02}$	$0.59_{0.06}$	$0.35_{0.00}$
cFMNIST	H-CREAM	$0.88_{0.02}$	$0.66_{0.03}$	$0.07_{0.04}$
	S-CREAM	$0.94_{0.02}$	$0.72_{0.03}$	$0.03_{0.01}$
CUB	S-CREAM	$0.96_{0.00}$	$0.72_{0.00}$	$0.01_{0.00}$
CelebA	S-CREAM	$0.92_{0.11}$	$0.30_{0.02}$	$0.01_{0.01}$

Appendix

Table 5: Interpretability metrics: CREAM variants (S-CREAM for soft concepts, H-CREAM for hard concepts) show higher concept-channel importance relative to the side-channel’s across all datasets. Also, S-CREAM exhibits larger CCI values compared to H-CREAM in FMNIST.

Figure 10: Mean values

\pm

standard deviation of the permutation feature importance of both channels, in all datasets for CREAM, averaged over 5 seeds. Increasing

p

leads to an increase in PCI and decreases PSI.

Dataset	Number of Classes ( $L$ )	Number of Concepts ( $K$ )	$\mathit{C}_{direct}$	Mutex Groups	Effective Group Interventions
iFMNIST	10	8	6	2	1
cFMNIST	10	11	9	3	2
CUB	200	112	112	27	27
CelebA	2	7	7	7	7

Appendix

Table 6: Summary of each dataset. A concept group consists of a mutually exclusive group of concepts.

\mathit{C}_{direct}

refers to the number of concepts directly connected to any task, which coincides with the upper bound of interventions. The reported number of effective group interventions refers to the number of directly connected groups of concepts instead.

Dataset	iFMNIST	cFMNIST	CUB	CelebA
Dropout ( $p)$	$\{0.2,0.5,0.8,0.9\}$	$\{0.2,0.5,0.8\}$	$\{0.1,0.2,0.5,0.8\}$	$\{0.2,0.5,0.8\}$
$d_{C}K+d_{Y}$	$\{8,18,76,78,128\}$	$\{11,21,22,32,43,128\}$	$\{424,512,648,848\}$	$\{7,8,10,36,70,75,256,512\}$
$d_{Y}$	$\{0,10,20,30,64\}$	$\{0,10,40,62\}$	$\{64,176,200,400\}$	$\{0,1,3,5,123,162\}$
$\lambda$	1	$1$	$1$	$\{0.25,0.75,1\}$
$d$	$\{0,2\}$	$\{0,2\}$	$\{0,3\}$	$\{0,5\}$

Appendix

Table 7: Hyperparameter search space explored for each dataset.

Dataset	Model	$\lambda$	lr	$p$	$d_{C}K+d_{Y}$	$d_{Y}$
iFMNIST	H-CREAM	1	1e-3	0.9	78	30
	S-CREAM w/o SC	1	1e-3	-	128	0
	S-CREAM	1	1e-3	0.9	76	20
cFMNIST	H-CREAM	1	1e-3	0.8	43	10
	S-CREAM w/o SC	1	1e-3	-	22	0
	S-CREAM	1	1e-3	0.8	128	40

Appendix

Table 8: Configurations for the best-performing models in each dataset. Note that

d_{C}K+d_{Y}

represents the dimensionality of latent space that is split, and

d_{Y}

the dimensionality of the input to the side-channel. H-CREAM and S-CREAM refers to the models with hard and soft concept representations respectively. The best models have

d=0

number of hidden layers in the Concept-Concept Block.

Figure 11: Expanded version of the illustration of CREAM found in the main text, providing additional details for clarity and completeness.

Model	iFMNIST		cFMNIST		CUB		CelebA
Model	Time	Peak Memory	Time	Peak Memory	Time	Peak Memory	Time	Peak Memory
CBM	x $1.000_{0.000}$	x $1.000_{0.000}$	x $1.000_{0.000}$	x $1.000_{0.000}$	x $1.000_{0.000}$	x $1.000_{0.000}$	x $1.000_{0.000}$	x $1.000_{0.000}$
CBM+SC	x $1.496_{0.137}$	x $1.001_{0.000}$	x $1.631_{0.208}$	x $1.001_{0.000}$	x $1.017_{0.026}$	x $1.004_{0.000}$	x $1.004_{0.003}$	x $1.001_{0.000}$
ACBM	x $5.304_{0.535}$	x $1.044_{0.000}$	x $7.896_{0.900}$	x $1.044_{0.000}$	x $14.125_{0.140}$	x $1.248_{0.000}$	x $3.269_{0.050}$	x $1.072_{0.000}$
SCBM	x $27.535_{2.817}$	x $1.050_{0.000}$	x $27.985_{3.721}$	x $1.053_{0.000}$	x $9.210_{0.258}$	x $1.303_{0.000}$	x $3.269_{0.050}$	x $1.072_{0.000}$
C 2 BM	x $12.194_{0.182}$	x $1.003_{0.000}$	x $15.143_{1.430}$	x $1.005_{0.000}$	-	-	x $2.846_{0.092}$	x $1.938_{0.000}$

Appendix

Table 9: Comparison of Training Time and GPU Peak Memory Usage relative to CBM. Values indicate the factor difference for models (e.g., CREAM takes 1.8× longer to train than the baseline).

Model	iFMNIST		cFMNIST		CelebA
Model	Time	Peak Memory	Time	Peak Memory	Time	Peak Memory
CBM	x $1.000_{0.000}$	x $1.000_{0.000}$	x $1.000_{0.000}$	x $1.000_{0.000}$	x $1.000_{0.000}$	x $1.000_{0.000}$
$\textbf{CGM}_{CD}$	x $9.150_{1.033}$	x $361.874_{2.061}$	x $20.238_{3.084}$	x $1139.224_{3.005}$	x $2.209_{0.070}$	x $82.571_{2.607}$
$\textbf{CGM}_{prior}$	x $8.471_{3.195}$	x $36.138_{0.405}$	x $8.465_{0.969}$	x $40.627_{0.351}$	x $2.227_{0.035}$	x $18.635_{1.404}$
CREAM	x $5.387_{1.437}$	x $1.259_{0.009}$	x $6.609_{2.438}$	x $1.259_{0.042}$	x $1.963_{0.021}$	x $1.122_{0.003}$

Appendix

Table 10: Comparison of Training Time (CPU) and overall system Peak Memory Usage relative to CBM. Values indicate the factor difference for models (e.g., CREAM takes 1.9× longer to train than the baseline).

Figure 12: Impact of individual interventions on task accuracy. We reach similar conclusions as the ones in the main text.

Figure 13: Impact of group interventions compared to individual ones, on task accuracy. Group interventions lead to the same performance gains with lower number of interventions. The number of relevant group interventions in CREAM for each dataset are, from left to right, 1, 2 and 27. Note the baseline’s performance is not visualized in iFMNIST, as it is comparatively too low.

Figure 14: Propagating interventions via invertibility in CREAM. We intervene on

\mathit{C}_{indirect}

first. In the other case, we intervene directly on

\mathit{C}_{direct}

Figure 15: Absolute correlation matrix for the CREAM models reported in Table 5.2 . For each concept we draw a box around its assigned $d_{C}$ dimensions. Note that CUB’s exogenous are too small to plot. We notice that C-C reasoning leads block-like correlations, both within the exogenous of each concepts, and between their exogenous, revealing the hierarchical structure.

Figure 16: Absolute correlation matrix for the CREAM models reported in Table 5.2 . For each concept we draw a box around its assigned

d_{C}

dimensions. We notice that C-C reasoning leads block-like correlations, both within the exogenous of each concepts, and between their exogenous, revealing the hierarchical structure.

Figure 17: Covariance matrix of the outputs of the side-channel for CREAM. On the left, we report the corresponding class each node is assigned to. CUB has too many classes to clearly visualize.

Panel of Figure 18 — Figure 18: Reasoning graph taken from one the $\text{CGM}_{CD}$ runs Dominici et al. (2025) for predicting “Smiling” in CelebA ( Liu et al., 2015 ) .

Figure 19: Concept and Task accuracies across different

d_{C}

values for the exogenous of the concepts. Values are presented as mean

\pm

standard deviation. Increasing

d_{C}

, generally improves performance.

Figure 20: Concept and Task accuracies across different

d

values in the Concept-Concept block. Values are presented as mean

\pm

standard deviation. Increasing

d

of the C-C block degrades performance.

Dataset	Model	Test $ACC_{Y}$	Test $ACC_{C}$
iFMNIST	S-CREAM	$\textbf{92.43}_{0.23}$	$\textbf{99.07}_{0.03}$
	H-CREAM	$92.15_{0.17}$	$98.98_{0.01}$
cFMNIST	S-CREAM	$\textbf{92.38}_{0.16}$	$\textbf{98.08}_{0.06}$
	H-CREAM	$91.29_{0.73}$	$96.70_{0.17}$

Appendix

Table 11: Performance comparison between CREAM variants. Reported values represent the mean and standard deviation over five seeds. S-CREAM outperforms H-CREAM performance and interpretability in both FMNIST variants. The best-performing model variant is highlighted in bold .

Figure 21: Intervenability comparison of H-CREAM and S-CREAM. The latter slightly outperforms the latter. Both models’ task accuracy peaks after 6 and 9 interventions in iFMNIST and cFMNIST respectively.

Figure 22: Absolute correlation matrices of the output of the representation splitter ( z ) for H-CREAM. We notice that more concept exogenous variables have lowered correlations compared to the soft models.

Dataset
iFMNIST	$\text{Clothes}\leftarrow\textbf{z}_{Clothes}$
	$\text{Goods}\leftarrow\textbf{z}_{Goods}$
	$\text{Clothes}\sqcap\text{Goods}\sqsubseteq\bot$
	$\text{Tops}\sqcap\text{Bottoms}\sqsubseteq\bot$
	$\text{Tops}\sqcap\text{Dresses}\sqsubseteq\bot$
	$\vdots$

Appendix

Table 12: Some logic rules based on description logic for CREAM. The z variables correspond to the concept and side-channel, and

\bot

denotes the empty set.

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Concepts' Information Bottleneck Models

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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)

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I(C;Y)

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