Integrated Gradients

The sum of all feature attributions must equal the difference between the model's output at the input and the baseline. This ensures no attribution is created or destroyed.

• Mathematical form: Σᵢ IGᵢ(x) = F(x) - F(x')
• Practical implication: The total importance assigned to all pixels or features perfectly accounts for the entire prediction score.
• Failure mode: Methods like simple gradients or Grad-CAM do not satisfy this, meaning they can miss critical evidence or overstate irrelevant features.

If an input differs from the baseline in exactly one feature and that difference causes a change in the model's prediction, that feature must receive a non-zero attribution.

• Formal definition: If F(x) ≠ F(x') and x differs from x' only in feature i, then IGᵢ(x) ≠ 0.
• Contrast with LRP: Layer-wise Relevance Propagation can violate this by assigning zero relevance to a feature that demonstrably changes the output.
• Debugging utility: This property guarantees that any feature which mathematically influences the output will be highlighted in the explanation.

Two functionally equivalent networks—models that produce identical outputs for all possible inputs—must receive identical attributions, regardless of their internal architecture.

• Why it matters: A ResNet and a VGG network trained to the same function should produce the same explanation for a given input.
• Violation example: Guided Backpropagation and DeepLIFT violate this axiom because they depend on the specific activation patterns of the chosen architecture.
• Audit assurance: This property ensures that explanations reflect the learned function, not arbitrary implementation details of the model.

If a model is a linear combination of two sub-models (F = a·F₁ + b·F₂), the attribution for the combined model is the same linear combination of the attributions from the sub-models.

• Ensemble transparency: Attributions for an ensemble model are simply the weighted average of the attributions of its constituent models.
• Path independence: This axiom ensures that the attribution method behaves predictably under model composition.
• Practical use: Enables decomposition of explanations when using model stacking or mixture-of-experts architectures.

Two features that play exactly the same functional role in the model and have identical values must receive identical attributions.

• Formal condition: If swapping the values of features i and j leaves the model output unchanged for all inputs, then IGᵢ(x) = IGⱼ(x).
• Fairness implication: This prevents the explanation method from arbitrarily favoring one input dimension over another when the model treats them identically.
• Example: In a symmetric image classification task, two identical pixels on opposite sides of the image must get the same importance score.

Integrated Gradients is the unique path method that satisfies both Sensitivity and Implementation Invariance when using a straight-line path from baseline to input.

• Theoretical guarantee: No other path integral method can claim this combination of axioms.
• Baseline dependence: While the choice of baseline is a hyperparameter, the straight-line path is the canonical choice that preserves symmetry.
• Contrast with Expected Gradients: Expected Gradients sacrifices this uniqueness by averaging over multiple baselines, trading axiomatic purity for reduced baseline sensitivity.