Gradient SHAP

Gradient SHAP is built upon the Expected Gradients framework, which computes feature importance by averaging the model's gradients over a background distribution of samples. This unifies the axiomatic guarantees of Integrated Gradients with the game-theoretic fairness of SHAP. Instead of selecting a single arbitrary baseline, it integrates out the reference by sampling from a dataset, making attributions more robust and representative of the model's global behavior.

To approximate the Shapley value computation efficiently, Gradient SHAP adds Gaussian noise to the input features. This noise acts as a smoothing mechanism that estimates the model's behavior under partial feature occlusion. By repeatedly sampling from the background distribution and adding noise, the method avoids the exponential computational complexity of exact Shapley value calculations while maintaining the completeness axiom, ensuring the sum of attributions equals the output difference from the average background prediction.

Gradient SHAP satisfies critical axioms from both the SHAP and Integrated Gradients frameworks:

• Completeness: The sum of all feature attributions equals the model output minus the expected baseline output.
• Sensitivity-n: A feature that does not influence the output receives zero attribution.
• Implementation Invariance: Two functionally identical networks produce identical attributions regardless of architecture. This dual compliance makes it admissible for both regulatory auditing and model debugging.

Exact SHAP values require evaluating the model on all 2^N feature coalitions, which is intractable for high-dimensional inputs like images. Gradient SHAP reduces this to a linear number of forward and backward passes by leveraging gradient information. The number of samples drawn from the background distribution and the noise scale are tunable hyperparameters that control the bias-variance trade-off of the approximation. More samples yield higher fidelity to true SHAP values at increased compute cost.

While Integrated Gradients integrates gradients along a straight-line path from a single baseline to the target input, Gradient SHAP integrates over a distribution of baselines. This distinction is crucial: a single baseline (e.g., a black image) can introduce artifacts if it lies far from the training manifold. Gradient SHAP's distributional approach produces attributions that are more aligned with the model's training data density, reducing gradient saturation artifacts where important features receive near-zero attribution.

Gradient SHAP is implemented in the open-source SHAP Python library as shap.GradientExplainer. It expects a differentiable model (e.g., TensorFlow or PyTorch) and a background dataset. The explainer computes attributions by:

• Sampling background instances and adding Gaussian noise.
• Computing gradients of the output with respect to the input for each perturbed sample.
• Averaging the gradient × input products across all samples. This integrates seamlessly with existing deep learning pipelines for post-hoc explainability.