Integrated Gradients

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

• If two inputs differ in exactly one feature and produce different predictions, that feature receives non-zero attribution.
• Guarantees that the explanation accounts for the total output change.
• A constant function always receives zero attributions for all features.

Two functionally equivalent networks—models that produce identical outputs for all possible inputs—will always receive identical attributions. This property is critical for model-agnostic auditing.

• The explanation depends solely on the mathematical function, not its architecture.
• Prevents explanation manipulation through functionally neutral architectural changes.
• Distinguishes Integrated Gradients from gradient-only methods like vanilla saliency maps.

If a model is a linear combination of two sub-models, the attribution is the same linear combination of the individual attributions. This enables compositional debugging of ensemble systems.

• Supports attribution for ensembles and stacked architectures.
• Allows decomposing a complex model's explanation into interpretable sub-components.
• Preserves the mathematical structure of the model in the explanation space.

Symmetric variables—features that play identical roles in the model and are swapped without changing the output—receive identical attributions. This ensures fair treatment of functionally equivalent inputs.

• Two pixels with identical color values in a symmetric image region get equal attribution.
• Prevents explanation bias toward arbitrary input ordering.
• Essential for explaining models with permutation-invariant architectures like Deep Sets.

Attributions are computed by accumulating gradients along a straight-line path from a baseline (representing absence of signal) to the actual input. This captures how each feature incrementally contributes to the prediction.

• The baseline is typically a zero embedding vector or black image.
• The integral is approximated via Riemann sum with 20–300 steps.
• Choosing a meaningful baseline is critical: a black image baseline explains why an object is present, not why it is dark.

The completeness axiom mathematically guarantees that Integrated Gradients is the unique path method satisfying both sensitivity and implementation invariance. This foundational result from cooperative game theory provides theoretical grounding.

• Formally: Σᵢ IGᵢ(x) = F(x) − F(x′), where x′ is the baseline.
• Links directly to the Aumann-Shapley value in cost-sharing theory.
• Provides a rigorous alternative to heuristic attribution methods lacking axiomatic guarantees.

A game-theoretic attribution framework that distributes the prediction difference between the actual input and a baseline among features. Unlike Integrated Gradients, SHAP satisfies the efficiency axiom, guaranteeing that feature contributions sum exactly to the model output. SHAP computes Shapley values by averaging marginal contributions across all possible feature coalitions, making it computationally heavier but theoretically grounded in cooperative game theory.

A noise-reduction technique that sharpens gradient-based saliency maps by averaging gradients over multiple noisy copies of the input. Where Integrated Gradients addresses saturation through path integration, SmoothGrad addresses visual noise through stochastic smoothing. The two methods are complementary and often combined: applying SmoothGrad to Integrated Gradients produces cleaner, more interpretable attributions.

The simplest gradient-based attribution method, computing the element-wise product of the input and its gradient. This approach partially addresses the sharpness of raw gradients but fails to solve the saturation problem where features with zero gradient still influence predictions. Integrated Gradients generalizes this by accumulating gradients along a continuous path, making Gradient × Input a degenerate case with a single-step baseline.

A backpropagation-based attribution method that explains the difference between a neuron's activation and a reference activation. DeepLIFT uses multipliers and a summation-to-delta property analogous to Integrated Gradients' completeness axiom. The key difference: DeepLIFT approximates gradients with finite differences using a linear composition rule, while Integrated Gradients computes true path integrals for theoretical rigor.

The choice of baseline critically shapes Integrated Gradients attributions. Common strategies include:

• Zero baseline: Black images or zero vectors, simple but may produce artifacts
• Blurred baseline: Gaussian-blurred versions of the input, preserving spatial structure
• Maximum entropy baseline: Uniform noise, representing maximum uncertainty
• Domain-specific baselines: Empty text embeddings or silence in audio Poor baseline selection can produce misleading attributions that highlight irrelevant features.

An extension of Integrated Gradients that replaces the single baseline with an expectation over multiple baselines drawn from a background distribution. This eliminates the need to choose a specific reference point and produces attributions that are more robust to baseline selection bias. Expected Gradients connects Integrated Gradients to the Aumann-Shapley cost-sharing framework from cooperative game theory.