Expected Gradients

Expected Gradients bridges two dominant attribution paradigms. It computes feature importance by averaging Integrated Gradients over a set of background samples, which mathematically recovers SHAP values as a special case. This unification means the method inherits the completeness axiom from Integrated Gradients and the game-theoretic fairness properties of Shapley values, providing a single, theoretically grounded explanation framework.

A critical weakness of standard Integrated Gradients is its sensitivity to the choice of a single baseline input (e.g., a black image or zero embedding). Expected Gradients eliminates this arbitrary choice by integrating over an entire background distribution of samples. This makes the attribution robust and removes the need for a practitioner to guess a 'neutral' reference point, which is often impossible for structured tabular data or text.

The method strictly adheres to the completeness axiom, also known as the efficiency property in Shapley values. This guarantees that the sum of all feature attributions for a given input equals the difference between the model's output for that input and the average prediction over the background distribution. No importance is created or destroyed; the prediction is fully decomposed and accounted for.

Computing the exact expectation over a large background dataset is computationally prohibitive. In practice, Expected Gradients is implemented by sampling a small number of reference inputs from the background and averaging their path integrals. This Monte Carlo approximation provides a computationally tractable method that maintains high fidelity, allowing it to scale to deep networks and high-dimensional inputs.

While both Expected Gradients and Gradient SHAP approximate SHAP values using gradients, they differ fundamentally in their path of integration. Expected Gradients follows a linear path from each background sample to the input, aligning with the Aumann-Shapley cost-sharing framework. Gradient SHAP instead uses a Gaussian noise-based approach centered on the input, which does not satisfy the same axiomatic guarantees.

Unlike some gradient-based methods that assume continuous input spaces, Expected Gradients naturally extends to discrete features like categorical variables or tokenized text. By defining an appropriate interpolation between the background sample and the input—such as replacing discrete tokens with their embedding vectors—the path integral can be computed, making the method applicable to NLP and tabular data models.