Integrated Gradients

Intuition

A plain gradient tells you the local slope, which can be zero for a saturated neuron even when the feature mattered. Integrated gradients (IG) accumulate gradients along the whole path from a baseline (e.g. a black image) to the input, so nothing that mattered along the way is missed.

Precise explanation (Sundararajan et al., 2017)

Completeness: $\sum_j \text{IG}_j = f(x) - f(x')$ (by the fundamental theorem of calculus along the path). Approximate the integral with a midpoint Riemann sum over m steps: $\alpha_k = (k + 0.5)/m$, $k = 0..m-1$, average the gradients and multiply by $(x - x')$. Error shrinks as m grows (compare MA-12). For linear models IG equals $w_j(x_j - x'_j)$ — again matching XA-01/XA-11.

Worked example

f(x) = x₀²·x₁, x = (1, 2), baseline 0: ∂f/∂x₀ = 2x₀x₁, ∂f/∂x₁ = x₀². On the path (α, 2α): ∫ 4α² dα = 4/3 → IG₀ = 4/3; ∫ α² dα = 1/3 → IG₁ = 2/3; sum = 2 = f(x) ✓.

Common mistakes

• Forgetting the (x − x′) factor.
• Endpoint Riemann sums with few steps (biased); use midpoints.
• Choosing a baseline that is not "neutral" for the task and misreading results.

Why this matters

IG is a standard attribution method for neural networks (images, text) and satisfies axioms that plain gradients violate.

References

• Axiomatic Attribution for Deep Networks — Mukund Sundararajan, Ankur Taly, Qiqi Yan (S027)
• Interpretable Machine Learning (3rd edition) — Christoph Molnar (S024)