Counterfactual Proximity

Measures proximity as the sum of absolute differences across all features. This metric naturally promotes sparsity because deviations are penalized linearly, encouraging solutions where most features remain unchanged and only a few are altered.

• Formula: Σ |x_i - x'_i|
• Key Benefit: Generates highly interpretable, sparse counterfactuals.
• Use Case: Ideal for tabular data where changing a small number of categorical or binary features is preferred.

Calculates proximity as the straight-line distance between the original and counterfactual points in feature space. It penalizes large deviations more heavily due to squaring, distributing changes across many features in small increments.

• Formula: √(Σ (x_i - x'_i)²)
• Key Benefit: Finds the geometrically closest point across the decision boundary.
• Trade-off: Often results in dense changes to many features, which can reduce actionability for a human end-user.

A proximity metric that accounts for the data distribution by incorporating the inverse covariance matrix. It scales distances based on feature variance and correlation, ensuring the counterfactual lies within high-density regions.

• Plausibility: Prevents generating unrealistic outliers that technically cross the boundary but are not representative of real data.
• Mechanism: A deviation along a high-variance feature is penalized less than the same deviation along a low-variance feature.

Applies a custom weight vector to the distance calculation, reflecting the relative cost of changing each feature. This directly encodes domain knowledge about actionability.

• Immutable Features: Assigned infinite weight to prevent any change.
• Actionable Features: Lower weights for easily modifiable attributes like 'savings amount' vs. high weights for 'years of education'.
• Implementation: Often combined with L1 or L2 norms as a weighted sum: Σ w_i * |x_i - x'_i|.

While not a distance metric itself, the L0 norm (counting the number of changed features) is a critical property of proximity. A counterfactual is considered more interpretable if it alters only a handful of features.

• Human Constraint: Users can only process a limited number of recommended changes.
• Optimization: Direct L0 minimization is NP-hard, so L1 regularization is used as a convex relaxation to induce sparsity.

The closest counterfactual in raw Euclidean space is often an adversarial example—a point outside the training data manifold. Effective proximity must be balanced with a plausibility constraint.

• Density Estimation: Penalizes points in low-density regions.
• Autoencoder Reconstruction Error: Measures how well a counterfactual can be reconstructed by a model trained only on real data.
• Result: A slightly more distant but realistic recommendation is always preferred over a nonsensical, proximal one.

A distance metric that accounts for feature correlations and variances in the training data distribution. Unlike Euclidean (L2) distance, it measures distance in terms of standard deviations from the mean, ensuring the counterfactual lies within high-density regions. This prevents generating unrealistic instances that are technically close in raw feature space but are statistical outliers, directly enforcing plausibility.

A counterfactual instance that alters the prediction by changing only a small number of features. Sparsity is often enforced by adding an L1 regularization term (Lasso) to the objective function during generation. This is critical for human interpretability, as a user can only process a limited number of changes. The trade-off between proximity and sparsity is a central optimization challenge in counterfactual generation.

A counterfactual instance that lies within the high-density region of the training data distribution. Proximity alone is insufficient; a counterfactual can be close to the query point but still be an adversarial artifact that would never occur in reality. Plausibility metrics often use the log-likelihood of the counterfactual under the data distribution or the distance to the k-nearest neighbors in the training set.

A model-agnostic counterfactual generation algorithm that searches for the closest counterfactual by expanding a hypersphere around the query instance. It samples points on the surface of an expanding sphere until a point is found that crosses the decision boundary. This method directly optimizes for proximity without requiring gradient access, making it applicable to any black-box classifier.

A hard rule encoded into a counterfactual generation algorithm that prevents the modification of immutable features or enforces causal monotonicity. For example, a constraint might prevent decreasing a user's age or increasing their education level without also increasing their years of experience. These constraints define the boundary of the action set and ensure the generated counterfactual represents a realistic path to recourse.

A binary evaluation metric that verifies whether a generated counterfactual instance successfully flips the model's prediction to the target outcome. It is the most fundamental requirement: if validity is 0, the counterfactual is useless. However, optimizing solely for validity often produces distant or unrealistic instances, necessitating a multi-objective optimization that balances validity with proximity, sparsity, and plausibility.