Integrated Gradients

Integrated Gradients is the only path-based attribution method known to satisfy both the Sensitivity and Implementation Invariance axioms simultaneously.

• Sensitivity (a): If an input differs from the baseline in exactly one feature and produces a different prediction, that feature must receive non-zero attribution.
• Sensitivity (b): If the model's output is mathematically independent of a feature, that feature must receive zero attribution.
• Implementation Invariance: Two functionally equivalent networks—regardless of architecture—must produce identical attributions. This eliminates the gradient saturation problem that plagues simple gradient-based saliency maps.

The method computes attribution by integrating gradients along a straight-line path from a baseline input x' to the actual input x.

The core formula:

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IG_i(x) = (x_i - x'_i) × ∫[α=0 to 1] ∂F(x' + α(x - x'))/∂x_i dα

Key components:

• Baseline (x'): A neutral reference point representing the absence of signal—typically a black image, zero embedding vector, or average transaction profile.
• Interpolation steps: The integral is approximated using Riemann summation across m steps (typically 20–300).
• Gradient accumulation: Gradients are computed at each interpolated point and averaged, capturing how the model's output changes as features scale from absent to present.

Integrated Gradients satisfies the Completeness axiom: the sum of all feature attributions equals the difference between the model's output at the input and the baseline.

Why this matters for fraud detection:

• The total attribution sum precisely accounts for the deviation from a 'normal' baseline transaction to the flagged anomalous transaction.
• No attribution is lost or created—every unit of the anomaly score is distributed among the input features.
• This enables auditable explanations: a compliance officer can verify that the sum of individual feature contributions exactly reconstructs the model's fraud score.

This property is a direct consequence of the fundamental theorem of calculus applied along the integration path.

The choice of baseline critically shapes the explanation. The baseline must represent 'absence of information' in the model's input space.

Common baselines for fraud models:

• Zero vector: For normalized numerical features, representing a null transaction.
• Global average: The mean transaction vector across the training distribution.
• Gaussian noise baseline: Multiple baselines drawn from a noise distribution, with attributions averaged to reduce baseline arbitrariness.
• Nearest-neighbor baseline: The closest non-fraudulent transaction in embedding space.

Best practice: Use multiple baselines and average the resulting attributions to mitigate the sensitivity of explanations to any single baseline choice. This is known as Expected Gradients.

While originally developed for image classifiers, Integrated Gradients adapts effectively to the tabular, high-dimensional feature spaces typical in financial fraud detection.

Implementation considerations:

• Feature scaling: All features must be normalized to a common range before integration to ensure the path is geometrically meaningful.
• Categorical features: Require embedding layers; the baseline is set to the embedding of a designated 'unknown' or 'absent' category.
• Temporal features: Time-based features (hour of day, days since last transaction) use a baseline representing a 'neutral' temporal state.
• Interaction effects: Integrated Gradients captures feature interactions implicitly through the gradient path, unlike additive methods that assume independence.

Example: For a flagged wire transfer, IG can reveal that the combination of high amount AND unusual beneficiary country—not either alone—drove the anomaly score.

Integrated Gradients occupies a distinct position in the explainability landscape, with specific trade-offs relative to model-agnostic alternatives.

IG vs. SHAP:

• IG is model-specific (requires gradient access); SHAP is model-agnostic.
• IG satisfies Implementation Invariance; Kernel SHAP does not.
• SHAP satisfies the Shapley axioms from game theory; IG satisfies path-based axioms.
• IG is computationally cheaper for deep networks than Kernel SHAP.

IG vs. LIME:

• IG provides global path consistency; LIME explanations are local linear approximations that may be unstable.
• IG requires no sampling of perturbed instances; LIME requires careful perturbation design.
• IG is deterministic given a fixed baseline; LIME involves randomness in sampling.