Integrated Influence Attribution

• Integrated Influence is a data attribution method that quantifies each training sample's contribution by integrating changes along a continuous baseline-to-data path.
• It addresses limitations of leave-one-out approaches by capturing collective effects and establishing an explicit, counterfactual baseline.
• The method uses Euler discretization and Hessian compression to efficiently approximate sample influences in complex machine-learning models.

Searching arXiv for the exact "Integrated Influence" paper and closely related uses of the term to ground the article. Integrated Influence most commonly denotes a data attribution method with baseline that measures how individual training samples contribute to a test loss by integrating their effect along a continuous transition from a baseline dataset to the observed training dataset, rather than by perturbing one sample at a time (Yang et al., 7 Aug 2025). In this sense, it is a response to two limitations of prevailing leave-one-out (LOO) attribution methods: local-based explanation, which can miss collective or contextual effects among samples, and the absence of an explicit baseline, which limits counterfactual explanation (Yang et al., 7 Aug 2025). The same phrase also appears in other literatures in more domain-specific senses, including network autoregression, cortical network control, human–robot interaction, and graph learning, but the most explicit formalization under the title “Integrated Influence” is the 2025 machine-learning attribution framework (Yang et al., 7 Aug 2025).

1. Motivation and conceptual scope

The central criticism underlying Integrated Influence is that many attribution methods estimate the effect of a training example by removing or perturbing one point at a time. The paper argues that this induces a local bias: a point may matter because it participates in a shared geometric or statistical structure with other points, even if its isolated LOO effect is negligible (Yang et al., 7 Aug 2025). The toy kernel-regression example in the paper is designed precisely to show that a point near a test point can have zero LOO influence even though it jointly supports the prediction through local structure.

The second motivation is the need for a baseline dataset. In the paper’s formulation, attribution should be relative to a reference state, so that explanations can answer counterfactual questions such as why a model behaves as it does under the observed training set rather than under a deliberately less informative baseline (Yang et al., 7 Aug 2025). This converts attribution from a purely local perturbation problem into what the paper describes as a data degeneration process from a baseline dataset to the current dataset.

A useful way to distinguish Integrated Influence from standard influence-function practice is therefore methodological rather than merely computational. Standard LOO-style attribution asks what happens when one sample is removed locally. Integrated Influence asks how the model and test loss change as the entire dataset is continuously restored from a baseline, while retaining a per-sample decomposition of that global change (Yang et al., 7 Aug 2025).

2. Formal construction

Let the training set be

$$D_{\text{train}}=\{z_i\}_{i=1}^N,\qquad z_i=(x_i,y_i),$$

with model $f(\cdot;\theta)$, and let $z=(x,y)$ be a test sample (Yang et al., 7 Aug 2025). The baseline dataset is defined as

$D'_{\text{train}}=\{(x_i,y_i')\}_{i=1}^N,$

so the features are preserved while the targets are replaced by baseline targets $y_i'$ (Yang et al., 7 Aug 2025).

For each sample $i$, the paper defines a target path

$p_i(0)=y_i',\qquad p_i(1)=y_i,$

and then a dataset path

$$T(t)=\{(x_i,p_i(t))\}_{i=1}^N,\qquad T(0)=D'_{\text{train}},\quad T(1)=D_{\text{train}}.$$

This path is the backbone of the method: attribution is accumulated continuously along $t\in[0,1]$ rather than read off from a single endpoint perturbation (Yang et al., 7 Aug 2025).

The total change in test loss is

$$\Delta = L_{\text{test}}(z;\theta^*) - L_{\text{test}}(z;\theta'),$$

where $f(\cdot;\theta)$0 is trained on $f(\cdot;\theta)$1 and $f(\cdot;\theta)$2 on $f(\cdot;\theta)$3 (Yang et al., 7 Aug 2025). If

$f(\cdot;\theta)$4

then the paper writes

$f(\cdot;\theta)$5

and, after differentiating the optimality condition,

$f(\cdot;\theta)$6

with $f(\cdot;\theta)$7 the Hessian of the training loss (Yang et al., 7 Aug 2025).

Because only targets vary along the path, the derivative of the training gradient decomposes samplewise. The resulting per-sample attribution is the Integrated Influence

$f(\cdot;\theta)$8

where

$f(\cdot;\theta)$9

The paper’s key identity is

$z=(x,y)$0

so the overall test-loss change between baseline and observed training data is additively decomposed into samplewise path integrals (Yang et al., 7 Aug 2025).

The interpretation given in the paper is explicit. $z=(x,y)$1 measures sensitivity of the test loss to model parameters, $z=(x,y)$2 maps training-gradient changes to changes in the optimum, and $z=(x,y)$3 measures how changing sample $z=(x,y)$4’s target perturbs the training gradient (Yang et al., 7 Aug 2025).

3. Baseline construction, path choice, and numerical approximation

The paper emphasizes that the baseline is not arbitrary. Its main construction uses unlearning-based baseline construction: $z=(x,y)$5 followed by

$z=(x,y)$6

This is intended to produce a baseline that is non-informative for the test sample while remaining close to the original training distribution (Yang et al., 7 Aug 2025).

For most experiments, the paper uses the linear path

$z=(x,y)$7

and for classification it applies a sparsity trick to avoid dense target vectors causing unstable gradients (Yang et al., 7 Aug 2025). A plausible implication is that the path is a modeling choice rather than a uniquely determined object; the paper explicitly lists path dependence as a limitation.

Exact evaluation of the integral is intractable, so the method uses an Euler-style discretization over $z=(x,y)$8 intermediate datasets: $z=(x,y)$9 The algorithm summarized in the paper is: construct the baseline by unlearning, initialize $D'_{\text{train}}=\{(x_i,y_i')\}_{i=1}^N,$0, step through discretized datasets $D'_{\text{train}}=\{(x_i,y_i')\}_{i=1}^N,$1, approximate $D'_{\text{train}}=\{(x_i,y_i')\}_{i=1}^N,$2, compute $D'_{\text{train}}=\{(x_i,y_i')\}_{i=1}^N,$3 and $D'_{\text{train}}=\{(x_i,y_i')\}_{i=1}^N,$4, and accumulate the discrete contribution (Yang et al., 7 Aug 2025).

Since Hessian inversion is the main computational bottleneck, the paper uses Hessian compression

$D'_{\text{train}}=\{(x_i,y_i')\}_{i=1}^N,$5

with a low-dimensional projection matrix $D'_{\text{train}}=\{(x_i,y_i')\}_{i=1}^N,$6, and also notes that inverse Hessian-vector products can be used directly for further savings (Yang et al., 7 Aug 2025).

4. Relation to influence functions and adjacent attribution methods

A major theoretical claim is that the classical Influence Function (IF) is a special case of Integrated Influence. The paper’s Lemma 2.5 states that if $D'_{\text{train}}=\{(x_i,y_i')\}_{i=1}^N,$7 and $D'_{\text{train}}=\{(x_i,y_i')\}_{i=1}^N,$8 are treated as constant along the path and the baseline target is chosen so that its gradient contribution vanishes, then the Integrated Influence formula reduces to the influence-function form (Yang et al., 7 Aug 2025). For losses whose gradient is linear in the target, including MSE and cross-entropy-like settings, the paper states that IF corresponds to the case $D'_{\text{train}}=\{(x_i,y_i')\}_{i=1}^N,$9 with an appropriate baseline.

For MSE, Corollary 2.6 gives the baseline target

$y_i'$0

which leads to the interpretation that IF implicitly replaces each label by the model’s own prediction and reads off attribution from a one-step path (Yang et al., 7 Aug 2025). This is the basis of the paper’s broader claim that IF is often too local and can miss structured contributions even when it is mathematically valid.

The experimental comparison set includes IF, TracIn, TRAK, and Integrated Influence (Yang et al., 7 Aug 2025). The article’s central contrast is not that the baselines are gradient-based while Integrated Influence is not; all are gradient-based. The distinction is that Integrated Influence introduces a baseline-relative, path-integrated attribution with explicit counterfactual semantics.

A separate but related development is Delta-Influence, which does not use a baseline dataset but instead measures a difference-of-influences signal under test-time transformations to expose poisoned training points through influence collapse (Li et al., 2024). Delta-Influence asks which training points lose influence when a poisoned test point is transformed in a way that severs poison-specific associations, whereas Integrated Influence asks how per-sample contribution accumulates along a baseline-to-data path (Li et al., 2024). This suggests two distinct post-LOO directions in influence analysis: path-integrated attribution and transformation-induced differential attribution.

5. Empirical evaluation, capabilities, and limitations

The paper evaluates Integrated Influence on two tasks: data attribution and mislabeled example identification (Yang et al., 7 Aug 2025). For attribution quality it uses Linear Datamodeling Score (LDS), defined as the Spearman rank correlation between the true loss of models retrained on random subsets and the predicted loss obtained by summing sample importance scores over those subsets. For mislabeled-example identification it uses AUC, with negative self-influence as the score (Yang et al., 7 Aug 2025).

In synthetic linear regression, the setup uses 100 training samples, 10 features, Gaussian noise in training and test labels, and MSE loss. The reported trend is that when training noise is low, IIF and IF are similar, but as noise increases, IIF degrades less than IF and TracIn; the paper also reports that IIF remains strongest under mismatched noise distributions such as Gaussian versus Laplacian (Yang et al., 7 Aug 2025).

On MNIST data attribution, the paper uses 5000 training samples, 1000 test samples, a 4-layer MLP, and cross-entropy loss. The reported result is that IIF achieves the highest LDS, outperforming the second-best method by about 18% (Yang et al., 7 Aug 2025). The paper also presents a proponents/opponents analysis for a test image that looks like a “4” but resembles a “9”, showing that different baselines can be used to ask different counterfactual questions; proponents are samples that reduce loss toward the “4” interpretation, while opponents are samples that resist that change (Yang et al., 7 Aug 2025).

For mislabeled example identification, the datasets are MNIST and CIFAR-10, with 1000 training samples and 10\% labels randomly flipped. The models are a 4-layer MLP for MNIST and ResNet-9 for CIFAR-10. The paper reports that all methods perform similarly on MNIST, that IIF achieves the best AUC overall, and that on CIFAR-10 IF runs out of memory, whereas IIF remains feasible and attains the best result among the scalable methods (Yang et al., 7 Aug 2025).

The paper’s limitations are explicit. Baseline quality is critical; a poor baseline can yield poor explanations. The method carries computational overhead because it evaluates multiple intermediate datasets and approximate Hessians. Attribution is path dependent, and the linear path is chosen for convenience rather than uniqueness. Finally, the unlearning used to construct the baseline is approximate in practice (Yang et al., 7 Aug 2025). These caveats are substantive rather than peripheral, because the framework’s explanatory semantics depend directly on the baseline and path choices.

6. Broader meanings of “integrated influence” across research areas

The phrase integrated influence is not unique to data attribution. Several papers use closely related language to denote a joint or multi-component treatment of influence, though the mathematical objects differ substantially.

Domain	Meaning of integrated influence	Representative paper
Data attribution	Baseline-relative path integral of per-sample influence	(Yang et al., 7 Aug 2025)
Relational time series	Joint sender-side and receiver-side influence in a low-rank network autoregression	(Minhas et al., 2017)
Cognitive neuroscience	Subcortical influence aligned with cortical integration or segregation	(Nestor, 2024)
Human–robot interaction	Intentional regulation of robot influence over humans across short-term and long-term adaptation	(Sagheb et al., 18 Mar 2025)
Item-level social influence	Integration of social-network latent positions with item-response latent space	(Park et al., 2021)
Multi-label graph learning	Integrated label influence from propagation and transformation in GNN training	(Sun et al., 1 Jul 2026)

In Social Influence Regression (SIR), influence is decomposed into sender-side and receiver-side matrices within a bilinear or vector-autoregressive relational model. If one adopts the phrase “integrated influence” in that context, the integration is that sender and receiver influence are estimated jointly and explained through exogenous covariates rather than treated as latent residual structure (Minhas et al., 2017).

In the fMRI study of cortical reconfiguration, the phrase refers to a hypothesis that basal ganglia and cerebellar systems differentially align with cortical integration and segregation. There, integration is measured by lower cortical modularity, and subcortical “influence” is proxied by eigenvector centrality; only basal ganglia influence is reported to precede cortical integration, while cerebellar influence is more correlative (Nestor, 2024).

In long-term human–robot interaction, the unified MOMDP framework models influence as purposeful behavior change maintained over repeated encounters by reasoning over latent short-term strategy $y_i'$1 and long-term adaptation parameters $y_i'$2. Here “integrated” refers to a framework that unifies prior Stackelberg-style and latent-state methods within a single partially observable control formalism (Sagheb et al., 18 Mar 2025).

In item-level social-influence modeling, the integrated latent-space approach links a friendship-network latent space to a latent-space item-response model so that overall social influence is measured by $y_i'$3 and $y_i'$4, while differential social influence is read from the interaction map across items (Park et al., 2021). In multi-label node classification, Label Influence Propagation constructs an integrated label influence matrix

$y_i'$5

combining propagation-induced and gradient-induced label effects before propagating them over a label graph to reweight losses (Sun et al., 1 Jul 2026).

Across these literatures, the shared intuition is that influence is rarely exhaustively characterized by a single local effect. What changes from field to field is the object being integrated: baseline-relative sample contributions in machine learning, sender and receiver processes in relational models, subcortical and cortical dynamics in neuroscience, or propagation and optimization effects in graph learning.

7. Significance

Integrated Influence, in its principal machine-learning sense, reframes data attribution as a counterfactual, baseline-relative, path-integrated decomposition of test-loss change (Yang et al., 7 Aug 2025). Its importance lies in three connected claims. First, attribution should not be restricted to one-point perturbations when training data exhibit collective structure. Second, explanations become more flexible and counterfactual when they are stated relative to a baseline dataset. Third, classical influence functions can be recovered as a limiting or simplified case rather than treated as the only natural formulation (Yang et al., 7 Aug 2025).

The broader literature shows that the phrase integrated influence has become a recurring label for methods that combine multiple influence channels into a single formal object, whether those channels are sender and receiver effects, short-term and long-term adaptation, or structural and optimization interactions (Minhas et al., 2017, Sagheb et al., 18 Mar 2025, Sun et al., 1 Jul 2026). In that broader sense, Integrated Influence names not a single universal theory but a recurring methodological move: replacing isolated local influence scores with models that integrate influence across paths, roles, timescales, or mechanisms.