DANCE: Diverse, Actionable, Knowledge-Constrained Explanations

• The paper introduces DANCE, a framework that generates counterfactual explanations by jointly optimizing diversity, actionability, and knowledge-constrained plausibility.
• It employs methods like DPP-based diversity, Pareto optimization, and Bayesian techniques to ensure feasible, minimally intrusive, and realistic modifications.
• Empirical evaluations on benchmark datasets demonstrate DANCE’s strong performance in balancing recourse quality with practical constraints and domain-specific causal relationships.

Diverse, Actionable, and kNowledge-Constrained Explanations (DANCE) comprise a unified methodological family for constructing counterfactual explanations of black-box machine learning predictions that jointly maximize (1) diversity of explanations, (2) actionability via feasible, realistic modifications, and (3) knowledge-constrained plausibility as informed by causal or domain relationships. DANCE advances the counterfactual explanation paradigm by integrating constraint-rich optimization, explicit diversity objectives, and encodeable domain/causal priors, thereby ensuring that proposed recommendations are minimally intrusive, comprehensible, feasible, and in accord with real-world regularities (Mothilal et al., 2019, Rasouli et al., 2021, Bobek et al., 25 Nov 2025).

1. Problem Setting and Core Objectives

Given a predictive model $M:\mathbb{R}^M\to\Delta^C$, an input $x\in\mathbb{R}^M$, and a class $d\ne \operatorname{arg\,max}_c M_c(x)$, the DANCE framework seeks a set of $N$ counterfactuals $\{\text{cf}^{(i)}\}_{i=1}^N$ for which $M(\text{cf}^{(i)})=d$. Each counterfactual $cf$ minimizes a weighted composite loss

$$L = L_\text{cls} + \lambda \cdot (\omega_r L_\text{prox} + \omega_s L_\text{spar}+ \omega_p L_\text{plau}+ \omega_d L_\text{div})$$

where:

• $L_\text{cls}$: margin loss for target output.
• $L_\text{prox}$: proximity between $x$ and $cf$ (minimal changes).
• $L_\text{spar}$: sparsity (few features altered).
• $L_\text{plau}$: plausibility under known feature dependencies.
• $L_\text{div}$: diversity among counterfactual set.

The framework targets local post-hoc explanations, where the counterfactuals provide actionable “recourse” and offer interpretable insight into how a decision might differ with minimal, plausible modifications. Critically, DANCE encodes feasibility via constraints, including those that capture domain and causal knowledge (Bobek et al., 25 Nov 2025, Mothilal et al., 2019).

2. Diversity Mechanisms

DIVERSITY is essential for user trust and coverage of the actionable decision boundary. DANCE formalizes diversity either via explicit penalization in the objective function (e.g., log determinant of DPP kernel) or by Pareto-based spread across multi-objective trade-offs.

Two principal diversity mechanisms are employed:

• Determinantal Point Processes (DPPs): For a batch of counterfactuals $\{x'_i\}_{i=1}^m$, the DPP kernel $$L_{ij} = \exp\left(-\frac{\|x'_i - x'_j\|^2}{2\sigma^2}\right)$$ yields $\det L$ as a measure of volumetric diversity. Maximizing $\log\det L$ in the joint objective directly enforces a spread (Mothilal et al., 2019, Bobek et al., 25 Nov 2025).
• Pareto Front (NSGA-III): Multi-objective optimization architectures such as NSGA-III are leveraged to generate diverse, non-dominated counterfactuals along trade-off surfaces, with reference points in objective space ensuring variety in both feature(s) changed and their values (Rasouli et al., 2021).

Post-hoc metrics such as average pairwise distance or feature-set Jaccard diversity validate the output diversity (Mothilal et al., 2019, Rasouli et al., 2021).

3. Feasibility, Actionability, and Constraint Encoding

ACTIONABILITY in DANCE is realized by restricting counterfactual moves to those permitted by user or domain/prior constraints. The constraints can be enforced as hard restrictions or as penalty terms. Forms include:

• Linear Constraints: $C_k(x') = a_k^\top x' - b_k \le 0$, e.g., immutable features, bounded change, physical or regulatory limits (Mothilal et al., 2019, Rasouli et al., 2021, Bobek et al., 25 Nov 2025).
• Nonlinear/Monotonic/Logical Constraints: Addressed via, for example, $C_k(x') = (x'_p - x'_q)(x'_r-x'_s)-\delta\le0$ or logical implications (“if-then” rules) (Mothilal et al., 2019).
• Knowledge Constraints: Any dependency or rule (e.g., causal, ontological, must-link/cannot-link), often represented as a DAG $G=(V,E,W)$ where edges encode conditional independence or functional relations. These are learned via data-driven techniques (such as DirectLiNGAM, NOTEARS, CPD estimation) or specified by domain experts (Bobek et al., 25 Nov 2025).
• Penalty-Based Soft Constraints: Penalized via, e.g., $\sum_k \rho\max\{0,C_k(x')\}$ when strict feasibility is not required (Mothilal et al., 2019).

For plausibility, a key term is

$$L_\text{plau}(cf;G) = \frac{1}{|V|}\sum_{i=1}^{|V|} \left(cf_i - \sum_{p\in Pa(v_i)} w_{p\to i}cf_p\right)^2$$

where $Pa(v_i)$ denotes the parents of $v_i$ in $G$ (Bobek et al., 25 Nov 2025).

4. Optimization and Algorithmic Approaches

The DANCE family applies mixed optimization strategies to navigate the non-convex, constraint-laden counterfactual landscape:

• Greedy/Batch DPP Sampling: For diversity maximization, DPPs are sampled or optimized jointly with proximity and feasibility objectives (Mothilal et al., 2019).
• Multi-objective Genetic Algorithms (NSGA-III): Used in the CARE variant, where population-based search evolves a diverse set along the Pareto front subject to constraints, crossover, and mutation (Rasouli et al., 2021).
• Bayesian Optimization (TPE): Tree-structured Parzen Estimators efficiently search the feasible space, leveraging priors from dependency graphs and data-derived CPDs (Bobek et al., 25 Nov 2025).

Search space initialization and candidate generation reflect the encoded knowledge and constraints, with sampling done from data manifold-aware distributions or as implied by the DAG structure and CPDs (Bobek et al., 25 Nov 2025).

5. Evaluation Metrics

DANCE relies on a multidimensional battery of metrics to assess counterfactuals:

Metric	Purpose	Typical Formula
Diversity	Breadth of generated CFs	$\det L$ (DPP), pairwise distance, or feature-value Jaccard (Mothilal et al., 2019)
Proximity	Similarity to original instance	$\frac{1}{M}\sum \omega_i cf_i x_i$ or normalized distance (Bobek et al., 25 Nov 2025)
Sparsity	Number of modified features	$\frac{1}{M}\sum 1\{cf_i\ne x_i\}$
Plausibility	Violation of knowledge graph	See $L_{plau}$, above; or 0-valued CPDs
Feasibility	Satisfaction of constraints	Fraction $cf$ with all $C_k(cf)\le0$
Outcome Fidelity	Target class achieved	$\max_{c\ne d} P_c(cf) - P_d(cf) < 0$ for desired class $d$

Large-scale benchmarks (over 140 datasets) confirm that DANCE methods, particularly those enforcing domain/causal constraints, attain high rankings in plausibility, proximity, and sparsity, albeit sometimes at a modest diversity cost due to constraint restrictiveness (Bobek et al., 25 Nov 2025).

6. Empirical Outcomes and Case Studies

DANCE variants have demonstrated efficacy across real-world and benchmark scenarios:

• Freshmail Case Study (Bobek et al., 25 Nov 2025): In large-scale email marketing, DANCE generated plausible, domain-compliant recourse for 3,248 out of 5,000 “bad” campaigns, with most changes reflecting actionable feature modifications such as sending hour or removal of disallowed symbols. Statistical tests validated the semantic alignment of recommendations.
• CARE Benchmarking (Rasouli et al., 2021): On Adult-Income, COMPAS, and Credit-Default datasets, CARE matched or outperformed other approaches across metrics (proximity: $2.97\pm1.9$; soundness: $0.98\pm0.1$; coherency error: $0.00$; actionability: $0.06\pm0.2$), driven by its knowledge-driven penalty structure.
• Comparative Evaluation (Bobek et al., 25 Nov 2025): In 140 OpenML datasets, DANCE achieved top-3 ranks for Probability, Sparsity, Plausibility, and Proximity, with statistical significance established via Friedman + Nemenyi tests.

Ablation studies show that removing plausibility constraints boosts diversity and sparsity at the expense of outcome probability and plausibility, highlighting inherent trade-offs (Bobek et al., 25 Nov 2025).

7. Limitations, Trade-offs, and Open Challenges

DANCE exposes design tensions central to counterfactual recourse:

• Diversity vs Plausibility: Stronger knowledge/causal constraints improve plausibility but contract the feasible space, lowering attainable diversity (Bobek et al., 25 Nov 2025, Mothilal et al., 2019).
• Complexity of Graph Learning: Structure estimation (e.g., DirectLiNGAM, NOTEARS) can be computationally expensive; the method’s scalability depends on pre-processing costs (Bobek et al., 25 Nov 2025).
• Hyperparameter Sensitivity: Tuning $(\lambda, \omega_p, \omega_s, \omega_r, \omega_d)$ is crucial for balancing objectives; misconfiguration risks implausible or trivial counterfactuals.
• Assumptions: Causal learning assumes acyclicity, no hidden confounding, and effective discretization for CPDs. Extensions to mixed types, time series, or richer generative models remain incomplete.
• Generalization and Comparative Diversity: While DANCE achieves leading scores in most factual metrics, explicit diversity (measured by DPP, Jaccard) can lag less-constrained methods (Bobek et al., 25 Nov 2025).

A plausible implication is that DANCE’s effectiveness relies not merely on constraint encoding, but on rigorous, context-attuned trade-off calibration.

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References:

(Mothilal et al., 2019) Explaining Machine Learning Classifiers through Diverse Counterfactual Explanations (Rasouli et al., 2021) CARE: Coherent Actionable Recourse based on Sound Counterfactual Explanations (Bobek et al., 25 Nov 2025) Actionable and diverse counterfactual explanations incorporating domain knowledge and causal constraints