CounterFlowNet: From Minimal Changes to Meaningful Counterfactual Explanations

Oleksii Furman; Patryk Marszałek; Jan Masłowski; Piotr Gaiński; Maciej Zięba; Marek Śmieja

cs.LGFeb 19, 2026

CounterFlowNet: From Minimal Changes to Meaningful Counterfactual Explanations

Authors: Oleksii Furman, Patryk Marszałek, Jan Masłowski, Piotr Gaiński, Maciej Zięba, Marek Śmieja

Organizations: Wrocław University of Science and Technology, Wrocław, Poland · Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland · Doctoral School of Exact and Natural Sciences, Jagiellonian University, Kraków, Poland · Tooploox Sp. z o.o., Poland

Abstract

Counterfactual explanations (CFs) provide human-interpretable insights into model's predictions by identifying minimal changes to input features that would alter the model's output. However, existing methods struggle to generate multiple high-quality explanations that (1) affect only a small portion of the features, (2) can be applied to tabular data with heterogeneous features, and (3) are consistent with the user-defined constraints. We propose CounterFlowNet, a generative approach that formulates CF generation as sequential feature modification using conditional Generative Flow Networks (GFlowNet). CounterFlowNet is trained to sample CFs proportionally to a user-specified reward function that can encode key CF desiderata: validity, sparsity, proximity and plausibility, encouraging high-quality explanations. The sequential formulation yields highly sparse edits, while a unified action space seamlessly supports continuous and categorical features. Moreover, actionability constraints, such as immutability and monotonicity of features, can be enforced at inference time via action masking, without retraining. Experiments on eight datasets under two evaluation protocols demonstrate that CounterFlowNet achieves superior trade-offs between validity, sparsity, plausibility, and diversity with full satisfaction of the given constraints.

Figures & tables

Figure 1: CounterFlowNet generates multiple counterfactual explanations by framing feature modification as a sequential decision process. Given an original data point (left), CounterFlowNet samples multiple valid CFs (CF1–CF3) proportional to a composite reward, naturally balancing sparsity, proximity, plausibility, and validity without requiring separate optimization.

Panel of Figure 2 — Figure 2: Overview of the CounterFlowNet generative process. Top: The forward policy $P_{F}$ constructs a counterfactual $\mathbf{x}^{\prime}$ from source instance $\mathbf{x}_{0}$ through sequential feature modifications until a stop action terminates the trajectory. Bottom: Each action is factorized into two stages: sampling a feature index $d_{t}\sim P_{F}(d\mid s_{t},\mathbf{x}_{0},y^{\prime})$ ( Stage 1 , Eq. 13 ), then sampling its new value $v_{t}\sim P_{F}(v\mid d_{t},s_{t},\mathbf{x}_{0},y^{\prime})$ ( Stage 2 , Eq. 15 ).

German Credit (Logistic Regression)
Method	Spars.(%) $\uparrow$	Div.(%) $\uparrow$	H. Mean(%) $\uparrow$	Val.(%) $\uparrow$	Cov.(%) $\uparrow$	Unary(%) $\uparrow$
L2C	61.35	37.31	46.39	100.00	100.00	99.06
DICE-R	88.23	15.29	26.06	100.00	100.00	90.81
DICE-G	43.45	37.56	40.29	62.87	90.24	56.66
COPA	57.88	18.88	28.47	44.00	44.00	84.31
MCCE	28.76	33.40	30.91	48.74	100.00	58.76

Table 1: Results on Protocol A datasets (discretized features,

B=4

). Best in bold , second-best underlined .

Method	Val. $\uparrow$	Prox.-Cont $\downarrow$	Spars.-Cat $\downarrow$	$\epsilon$ -Spars. $\downarrow$	LOF $\downarrow$	Div. $\uparrow$
Adult Income (Neural Network)
DICE	1.00	0.44	0.05	0.50	0.17	0.06
CCHVAE	1.00	0.50	0.06	0.98	0.22	0.03
DiCoFlex	1.00	0.85	0.49	0.97	0.31	0.40
CFN	1.00	0.19	0.27	0.36	0.20	0.21
Bank (Neural Network)

Table 2: Results on Protocol B datasets (discretized features,

B=64

). Best in bold , second-best underlined .

Figure 3: Effect of discretization granularity

B

on Adult dataset.

Panel of Figure 4 — Figure 4: Reward component ablation on the Adult dataset. (a) Increasing plausibility weight $\lambda_{p}$ improves LOF at the cost of proximity. (b-c) Higher proximity weight $\lambda_{d}$ yields sparser, more plausible CFs with lower diversity. All CounterFlowNet configurations consistently dominate the baseline Pareto frontiers.

Adult Income - Unconstrained
CFN	Val. $\uparrow$	Prox.-Cont $\downarrow$	Spars.-Cat $\downarrow$	$\epsilon$ -Spars. $\downarrow$	LOF $\downarrow$	Div. $\uparrow$
Retrained	0.99	0.22	0.28	0.41	0.22	0.23
Adult Income - Constraints Set 1
Retrained	0.99	0.24	0.27	0.45	0.22	0.23
Inference	0.99	0.24	0.27	0.46	0.22	0.23
Adult Income - Constraints Set 2

Table 3: Impact of actionability constraints on Adult. CFs quality under three increasingly strict constraint sets, comparing inference-time constraint action masking to retraining with the same constraints applied during training.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Protocol	$n$	$d_{\text{cont}}$	$d_{\text{cat}}$	Imbalance
German Credit	A	1,000	3	17	30%
Adult	A,B	32,561	4	8	24%
Graduate Adm.	A	500	3	4	34%
Student Perf.	A	649	4	10	33%
Lending Club	B	30,000	8	4	22%
GMC	B	30,000	6	3	7%

Appendix

Table 4: Dataset characteristics. Protocol A: discretized (

B=4

); Protocol B: continuous with generation-time quantization (

B=64

Desideratum	Metric	Description	Formula
Validity ( $\uparrow$ )	Val.(%)	Proportion of valid counterfactuals.	$\displaystyle\frac{1}{N\cdot K}\sum_{i=1}^{N}\sum_{k=1}^{K}\mathbb{1}\!\left[h(\mathbf{x}^{\prime(k)}_{i})=y^{\prime}_{i}\right]$
Validity ( $\uparrow$ )	Cov.(%)	Proportion of instances with at least one valid CF.	$\displaystyle\frac{1}{N}\sum_{i=1}^{N}\mathbb{1}\!\left[\exists\,k:h(\mathbf{x}^{\prime(k)}_{i})=y^{\prime}_{i}\right]$
Sparsity ( $\uparrow$ )	Spars.(%)	Proportion of features that remain unchanged.	$\displaystyle\frac{1}{N\cdot K}\sum_{i=1}^{N}\sum_{k=1}^{K}\frac{1}{D}\sum_{j=1}^{D}\mathbb{1}\!\left[x^{\prime(k)}_{i,j}=x_{i,j}\right]$
Diversity ( $\uparrow$ )	Div.(%)	Mean pairwise Hamming distance.	$\displaystyle\frac{1}{N}\sum_{i=1}^{N}\frac{2}{K(K\!-\!1)}\!\sum_{k<l}\frac{1}{D}\sum_{j=1}^{D}\mathbb{1}\!\left[x^{\prime(k)}_{i,j}\neq x^{\prime(l)}_{i,j}\right]$
Balance ( $\uparrow$ )	H. Mean(%)	Harmonic mean of Diversity and Sparsity.	$\displaystyle\frac{2\cdot\text{Div}\cdot\text{Spars.}}{\text{Div}+\text{Spars.}}$
Constraints ( $\uparrow$ )	Unary(%)	Proportion of CFs satisfying monotonic constraints.	$\displaystyle\frac{1}{N\cdot K}\sum_{i=1}^{N}\sum_{k=1}^{K}\frac{1}{\|\mathcal{F}_{m}\|}\sum_{j\in\mathcal{F}_{m}}\mathbb{1}\!\left[\text{mon}_{j}(\mathbf{x}^{\prime(k)}_{i},\mathbf{x}_{i})\right]$

Appendix

Table 5: Description of quantitative evaluation metrics for Protocol A. For each test instance

i\in\{1,\dots,N\}

, we generate a set of

K

counterfactuals

\mathcal{C}_{i}=\{\mathbf{x}^{\prime(k)}_{i}\}_{k=1}^{K}

. Let

h

denote the classifier,

y^{\prime}_{i}

the desired target class,

D

the total number of features, and

\mathcal{F}_{m}

the set of features subject to monotonic constraints.

Desideratum	Metric	Description	Formula
Validity ( $\uparrow$ )	Val.	Success rate of changing model predictions.	$\displaystyle\frac{1}{N}\sum_{n=1}^{N}\mathbb{1}\!\left[h(\mathbf{x}^{\prime}_{n})=y^{\prime}\right]$
Proximity ( $\downarrow$ )	Prox.-Cont	$L_{1}$ distance on continuous features.	$\displaystyle\frac{1}{N_{\text{val}}}\sum_{n=1}^{N_{\text{val}}}\left\\|\mathbf{x}^{0}_{n,\text{num}}-\mathbf{x}^{\prime\,}_{n,\text{num}}\right\\|_{1}$
Sparsity ( $\downarrow$ )	Spars.-Cat	Proportion of changed categorical features.	$\displaystyle\frac{1}{N_{\text{val}}}\sum_{n=1}^{N_{\text{val}}}\frac{\left\\|\mathbf{x}^{0}_{n,\text{cat}}-\mathbf{x}^{\prime\,}_{n,\text{cat}}\right\\|_{0}}{D_{\text{cat}}}$
Sparsity ( $\downarrow$ )	$\epsilon$ -Spars.	Continuous features changed beyond $\epsilon\cdot V_{j}$ ( $\epsilon{=}0.05$ ).	$\displaystyle\frac{1}{N_{\text{val}}}\sum_{n=1}^{N_{\text{val}}}\frac{1}{D_{\text{num}}}\sum_{j=1}^{D_{\text{num}}}\mathbb{1}\!\left[\|x^{0}_{n,j}-x^{\prime}_{n,j}\|>\epsilon\cdot V_{j}\right]$
Plausibility ( $\downarrow$ )	LOF	Median Local Outlier Factor (log scale).	$\displaystyle\text{median}_{n=1}^{N_{\text{val}}}\;\log\!\left(\text{LOF}_{k}(\mathbf{x}^{\prime}_{n})\right)$
Diversity ( $\uparrow$ )	Div.	Mean pairwise mixed distance between counterfactuals, averaged over groups.	$\displaystyle\frac{1}{G}\sum_{g=1}^{G}\frac{1}{D\binom{K_{g}}{2}}\sum_{i<j}^{K_{g}}\!\left(\left\\|\mathbf{x}^{\prime}_{i,\text{num}}-\mathbf{x}^{\prime\,}_{j,\text{num}}\right\\|_{1}+\left\\|\mathbf{x}^{\prime}_{i,\text{cat}}-\mathbf{x}^{\prime\,}_{j,\text{cat}}\right\\|_{0}\right)$

Appendix

Table 6: Description of quantitative evaluation metrics for Protocol B. We generate

N

counterfactuals

\{\mathbf{x}^{\prime}_{n}\}_{n=1}^{N}

for a test set of

N

instances

\{\mathbf{x}^{0}_{n}\}_{n=1}^{N}

, of which

N_{\text{val}}

are valid. Let

h

denote the classifier,

D_{\text{cat}}

and

D_{\text{num}}

the number of categorical and continuous features,

D=D_{\text{cat}}+D_{\text{num}}

the total,

V_{j}

the range of feature

j

G

the number of unique original instances (groups),

K_{g}

the number of counterfactuals for group

g

, and

\mathcal{N}

\mathcal{C}

the sets of numerical and categorical feature indices.

Panel of Figure 5 — Figure 5: Ablation of $\lambda_{p}$ and $\lambda_{d}$ with fixed weights for other reward weights ( $\lambda_{v}=1.0,\lambda_{s}=0.4$ )

Panel of Figure 6 — Figure 6: Visualization of feature utilization of CounterFlowNet method and other baseline methods on the Adult dataset. The plots show the frequency with which each feature is modified in the generated CFs. CounterFlowNet demonstrates a more balanced feature utilization compared to baselines.

Feature	Original Input	Counterfactual 1	Counterfactual 2
Age	25	25	29
Workclass	Self-emp-not-inc	Federal-gov	Self-emp-not-inc
Education	HS-grad	Prof-school	HS-grad
Marital Status	Never-married	Married-AF-spouse	Married-civ-spouse
Occupation	Sales	Sales	Sales
Relationship	Other-relative	Wife	Wife

Appendix

Table 7: Counterfactual examples for the Adult dataset. The original prediction is "low income," and the counterfactual prediction is "high income."

Feature	Original Input	Counterfactual 1	Counterfactual 2
Revolving Util. of Unsecured Lines	69	69	69
Age	45	21	56
Times 30-59 Days Past Due	1	0	3
Debt Ratio	162.57	162.57	162.57
Monthly Income	5488.28	5488.28	5488.28
Open Credit Lines & Loans	5	0	4

Appendix

Table 8: Counterfactual examples for the GMC dataset. The original prediction is "high risk," and the counterfactual prediction is "low risk."

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FlowCF: Sparse Counterfactual Explanations for Mixed-Type Tabular Data using Flow Matching

Emmanouil Panagiotou, Eirini Ntoutsi

Freie Universität Berlin · University of the Bundeswehr Munich

Flow Matching Explainable Artificial Intelligence Counterfactual Explanations Tabular ML

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In the field of Explainable AI (XAI), counterfactual (CF) explanations interpret a model's decision by suggesting the changes to the input that would lead to a more favourable outcome. To be useful in practice, such an explanation should change few features and change them as little as possible, properties known as sparsity and proximity. We observe that existing methods remain limited in this respect, especially for numerical features, whether they are model-agnostic and amortised, or gradient-based with full access to the model. In this paper, we propose FlowCF, a model-agnostic generative method that frames CF generation as sparse transport from the factual to the target class. We solve this transport with flow matching, which we extend to mixed feature types with a novel mixed flow operator, and exploit the resulting geometry to optimise for sparsity through a gating network that minimises the number of features the transport changes. Extensive experiments on six benchmark datasets demonstrate that FlowCF produces the best numerical sparsity and proximity, changing 29% of the numerical features where the best baseline changes 89%, at 70% smaller displacement, while remaining comparable on the other desiderata.

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