Expected Gradients

Instead of relying on a single, often uninformative baseline (like a black image or zero vector), Expected Gradients averages attributions over multiple background samples drawn from the training data.

• Removes the subjective choice of a reference point
• Each background sample provides a distinct path integral
• The final attribution is the weighted average of these path integrals
• Background samples anchor explanations in the data manifold

Expected Gradients inherits the key axioms from Integrated Gradients while improving practical robustness:

• Completeness: The sum of attributions equals the difference between the model output and the average prediction over the background distribution
• Sensitivity: If a feature changes and alters the prediction, it receives non-zero attribution
• Implementation Invariance: Two functionally equivalent networks yield identical attributions
• Linearity: Attributions compose linearly across model ensembles

The algorithm computes attributions through a three-step process:

1. Sample background: Draw a set of reference inputs from the training distribution
2. Path integration: For each background sample, interpolate linearly to the target input and accumulate gradients along the path
3. Expectation: Average the resulting attributions across all background samples

This approximates the Aumann-Shapley value with respect to the data-generating distribution rather than an arbitrary origin.

Expected Gradients addresses the primary weakness of Integrated Gradients—baseline selection:

• Integrated Gradients: Attribution is relative to a single, user-chosen baseline that may lie far outside the data manifold
• Expected Gradients: Attribution is relative to the training distribution, providing contextually meaningful importance
• When the baseline is uninformative, Integrated Gradients can produce noisy or misleading attributions
• Expected Gradients naturally handles features where a zero baseline is nonsensical (e.g., categorical embeddings)

In practice, Expected Gradients is implemented by sampling from a background dataset:

• SHAP integration: Expected Gradients is the theoretical foundation of the GradientExplainer in the SHAP library
• Batch processing: Multiple background samples can be processed in parallel on GPU hardware
• Subsampling: For large datasets, a representative subset of background samples suffices
• Convergence: Attribution estimates stabilize with a modest number of background samples (often 100-500)
• The method is model-agnostic for any differentiable architecture

Expected Gradients excels in scenarios where baseline choice is ambiguous:

• Tabular data: Explaining credit risk or medical diagnosis models where a zero baseline has no real-world meaning
• NLP models: Attributing predictions to input tokens using background embeddings from the training corpus
• Image classification: Using a distribution of blurred or random images as the reference set
• Regulatory compliance: Providing auditable explanations grounded in the actual data distribution rather than an arbitrary reference point

A theoretical framework defining the mathematical properties a valid attribution method must satisfy. Key axioms include:

• Completeness: Attributions must sum to the output difference from the baseline.
• Sensitivity: Zero-attribution features must not affect the output.
• Implementation Invariance: Different networks computing the same function must yield identical attributions. Expected Gradients inherits these guarantees while removing the single-baseline dependency.

Quantitative evaluation criteria measuring how accurately an attribution map reflects true feature importance. Core protocols include:

• Deletion Metric: Remove features from most to least important; a sharp drop in probability indicates high faithfulness.
• Insertion Metric: Add features into a blurred baseline; a steep probability increase signals quality.
• ROAR: Retrain the model after removing top-k features; performance degradation validates the attribution method.

A model-agnostic implementation of Shapley additive explanations that estimates SHAP values using a weighted linear regression over sampled coalitions. While both SHAP and Expected Gradients satisfy completeness, KernelSHAP operates on discrete feature coalitions rather than continuous path integrals, making it more suitable for tabular data but computationally heavier for high-dimensional inputs.

A backpropagation-based attribution method that compares neuron activations to a reference activation and distributes contribution scores according to differences from that reference. DeepLIFT uses a linear composition rule to satisfy completeness. Expected Gradients can be seen as a rigorous, gradient-based generalization that avoids DeepLIFT's heuristic reference selection.

A sensitivity map sharpening technique that reduces visual noise in gradient-based attributions by averaging gradients from multiple noisy copies of the same input. This noise-averaging principle parallels Expected Gradients' core insight: averaging over multiple baselines removes the variance introduced by a single arbitrary reference point, yielding more robust and visually coherent attributions.