Integrated gradients: Definition & Examples

Integrated gradients (IG) assigns each input feature a contribution to a scalar model output’s change from a chosen baseline, by accumulating input gradients along the straight line between baseline and observation. For a quant model, it can help investigate why a predicted risk score differs from a reference market state. Its answer depends on that reference and on the output being explained.

A gradient saliency map measures sensitivity at or near the observation. IG instead follows the journey from reference to observation, including regions where the model’s response was stronger before flattening. This is the distinction behind the method introduced by Sundararajan, Taly and Yan.

Name the comparison before computing it

Let $f$ be the scalar output, $x$ the observed feature vector and $b$ the baseline. Standard IG is

\operatorname{IG}_i(x;b)=(x_i-b_i)\int_0^1 \frac{\partial f(b+\alpha(x-b))}{\partial x_i}\,d\alpha.

The multiplier $x_i-b_i$ matters: an average derivative alone has different units. IG has the units of the chosen output. A positive attribution adds to the output difference; a negative one offsets it. Neither sign establishes whether a feature is economically desirable.

For a credit model, explaining a default probability differs from explaining its pre-sigmoid logit. A nonlinear output transformation changes the gradients along the path; one cannot transform individual logit attributions into probability attributions afterward. Specify the target output or class, feature preprocessing and model evaluation settings before comparing explanations.

A zero baseline is a numerical convenience, not an automatic definition of “no information.” Zero in a standardised variable may represent a historical mean; zero in a raw price variable may be implausible. A reference market observation answers a different question from an all-zero vector. Sturmfels, Lundberg and Lee’s baseline study shows why such choices belong to the explanation itself.

One interaction, three allocations

Consider a synthetic score with dimensionless inputs,

f(x_1,x_2)=x_1x_2,\qquad b=(0,0),\qquad x=(2,3).

The score rises from $0$ to $6$ . On the straight path $(2\alpha,3\alpha)$ , the partial derivatives are $3\alpha$ and $2\alpha$ . Therefore

\operatorname{IG}_1=2\int_0^1 3\alpha\,d\alpha=3, \qquad \operatorname{IG}_2=3\int_0^1 2\alpha\,d\alpha=3.

At the endpoint, the raw gradient is $(3,2)$ ; multiplying it by the displacement gives $(6,6)$ , which sums to $12$ . That endpoint shortcut counts the product interaction twice. IG allocates the interaction across the journey.

Synthetic bilinear score from baseline (0,0) to target (2,3). Straight-line integrated gradients allocate 3 to each feature. Changing feature 1 first allocates 0 to feature 1 and 6 to feature 2; reversing the order allocates 6 and 0. Every path sums to 6.

Original synthetic example. The product interaction is split equally by the straight path; coordinate-by-coordinate paths allocate it to whichever feature changes second. Completeness holds for all three allocations.

The two bent paths in the figure are alternative path attribution methods, rather than standard IG. Change

x_1

from

0

2

while

x_2=0

, and the score stays zero. Changing

x_2

next creates the entire rise of

6

: the allocation is

(0,6)

. Reverse the order and it becomes

(6,0)

. These are exact allocations of the same model change, so adding up correctly cannot select the economically appropriate path.

Even retaining the straight path, changing the baseline to $(1,1)$ changes IG to $(2,3)$ . The total is now $5$ , because the reference score is $1$ . This is a different comparison, not a failed completeness check.

What completeness promises

Under conditions that justify the chain rule along the path and the fundamental theorem of calculus,

\sum_i\operatorname{IG}_i(x;b)=f(x)-f(b).

For example, these conditions hold when $f$ is continuously differentiable around the path. A more general sufficient formulation requires the output along the path to be absolutely continuous and its derivative to equal the gradient dotted with the path velocity almost everywhere. Ordinary continuous piecewise-smooth networks can meet this condition despite isolated activation kinks. Differentiability almost everywhere in the surrounding input space alone is insufficient to justify every exceptional path: inspect paths that run along nondifferentiable boundaries and the derivative conventions used there. Discrete decisions, hard thresholds and tree splits do not automatically inherit the smooth-path argument.

Completeness is an accounting identity for the fitted function. It does not establish that a feature caused a realised market outcome, that the baseline is neutral, or that the interpolated feature combinations are realistic. Correlated market inputs may travel through states absent from training data. In this example, the equal split is a path-based allocation of a pure interaction; it is not evidence that two real economic mechanisms contributed equally.

Check the integral, then the interpretation

Numerical IG replaces the integral with weighted gradient evaluations. Define the signed completeness residual as

r=\sum_i\widehat{\operatorname{IG}}_i-\bigl(f(x)-f(b)\bigr).

For the bilinear example, equally spaced right-endpoint sums give:

Integration rule	Points	Estimated contributions	Residual $r$
Right endpoints	20	$(3.15,3.15)$	$0.30$
Right endpoints	100	$(3.03,3.03)$	$0.06$
Midpoints	20	$(3,3)$	$0$

The midpoint rule is exact here because the path integrands are linear; that result does not extend to arbitrary networks. Captum’s API offers Riemann rules and Gauss–Legendre quadrature, a step-count setting, and a convergence delta. The standard formula here includes the displacement multiplier, corresponding to its multiply_by_inputs=True setting.

Use a finer grid or another integration rule to check individual attribution stability as well as the total. A small residual can conceal cancelling component errors; it cannot validate a baseline or supply causal evidence. Report the baseline, output scale, integration method and residual with the explanation, and examine whether conclusions survive defensible alternative baselines.

IG repeatedly evaluates ordinary gradients. DeepLIFT instead propagates contributions based on activation differences from a reference, as described by Shrikumar, Greenside and Kundaje. A shared reference-based interpretation does not make the methods interchangeable.

For the derivative machinery, Backpropagation as the Chain Rule explains how the backward pass reuses local computations. Its public preview is open; the full lesson requires membership. Attribution differentiates a selected output with respect to inputs, whereas training typically differentiates a loss with respect to parameters.

Sources

Sundararajan, Taly and Yan (2017), Axiomatic Attribution for Deep Networks, §§2–5: sensitivity, IG, completeness, paths and numerical approximation.
Sturmfels, Lundberg and Lee (2020), Visualizing the Impact of Feature Attribution Baselines, baseline dependence and missingness.
Captum, IntegratedGradients API, baseline, target, integration and convergence options; accessed 1 November 2026.
Shrikumar, Greenside and Kundaje (2017), Learning Important Features Through Propagating Activation Differences, reference activations and contribution propagation.

Name the comparison before computing it

One interaction, three allocations

What completeness promises

Check the integral, then the interpretation

Sources

Related terms

DeepLIFT and layer-wise relevance propagation

Gradient saliency and SmoothGrad