Absolute Shapley Value for Data Attribution

• Absolute Shapley Value is a data attribution metric that measures the magnitude of each data point’s influence by summing absolute changes in model performance.
• It adapts the classical Shapley framework to capture both positive and negative contributions, making it ideal for identifying beneficial and harmful data points.
• Empirical studies, such as on the Iris dataset, demonstrate that Absolute Shapley Value more effectively discriminates high-impact data compared to traditional Shapley approaches.

The Absolute Shapley Value is a variant of the classical Shapley value from cooperative game theory, designed for attribution problems with potentially negative marginal contributions, such as data valuation in machine learning. While the original Shapley value quantifies average marginal utility, Absolute Shapley Value measures the magnitude of influence (regardless of sign) contributed by each participant, making it suitable in scenarios where strong negative effects are as significant as strong positive ones. It has been specifically advanced to address data contribution analysis in model training, where adding data points may either improve or degrade performance, contrary to standard cooperative game settings (Liu, 2020).

1. Formal Definitions

Let $N = \{1, 2, \dots, n\}$ denote the set of players or data tuples. The utility (value) function $v: 2^N \rightarrow \mathbb{R}$ assigns a real-valued score to each coalition $S \subseteq N$, commonly the prediction accuracy of a model trained on $S$.

The classical Shapley value for player $i \in N$ is:

$$\phi_i = \sum_{S \subseteq N \setminus \{i\}} \frac{|S|!\,(|N|-|S|-1)!}{|N|!} \bigl[ v(S \cup \{i\}) - v(S) \bigr]$$

Negative marginal contributions $v(S \cup \{i\}) - v(S) < 0$ frequently occur in data-driven machine learning settings.

To address this, two main variants are introduced:

• Zero Shapley Value: Only positive marginal contributions are considered:

$$\phi_i^0 = \sum_{S \subseteq N \setminus \{i\}} \frac{|S|!\,(|N|-|S|-1)!}{|N|!} \max\{0, v(S \cup \{i\}) - v(S)\}$$

• Absolute Shapley Value: All marginal contributions are replaced by their absolute values:

$$\phi_i^{abs} = \sum_{S \subseteq N \setminus \{i\}} \frac{|S|!\,(|N|-|S|-1)!}{|N|!} \left| v(S \cup \{i\}) - v(S) \right|$$

Alternatively, both can be defined by post-processing the standard Shapley value:

• $\phi_i^0 = \max\{\phi_i, 0\}$
• $v: 2^N \rightarrow \mathbb{R}$0

These variants share the intent that Zero Shapley captures only supportive contributions, whereas Absolute Shapley treats both large positive and large negative effects as highly important (Liu, 2020).

2. Theoretical Properties

The original Shapley value satisfies the following axioms:

• Group rationality (Efficiency): $v: 2^N \rightarrow \mathbb{R}$1.
• Fairness (Symmetry & Dummy):
  • Symmetry: If $v: 2^N \rightarrow \mathbb{R}$2 for all $v: 2^N \rightarrow \mathbb{R}$3, then $v: 2^N \rightarrow \mathbb{R}$4.
  • Dummy: If $v: 2^N \rightarrow \mathbb{R}$5 for all $v: 2^N \rightarrow \mathbb{R}$6, then $v: 2^N \rightarrow \mathbb{R}$7.
• Additivity: For value functions $v: 2^N \rightarrow \mathbb{R}$8, $v: 2^N \rightarrow \mathbb{R}$9.

When taking the absolute or zero modification:

• Group rationality and additivity do not hold. For Zero Shapley, total contributions satisfy $S \subseteq N$0; for Absolute Shapley, $S \subseteq N$1.
• Symmetry is preserved in both variants.
• Dummy property holds for Zero Shapley, but for Absolute Shapley, a dummy receives $S \subseteq N$2 if and only if $S \subseteq N$3.
• Interpretation mismatch: Zero Shapley may undervalue "harmful" data, while Absolute Shapley assigns high importance to datapoints that induce either large improvement or degradation (Liu, 2020).

The loss of classic axiomatic properties by Absolute (and Zero) Shapley values represents a deliberate trade-off in favor of quantifying the magnitude of influence.

3. Computation

Enumerating all $S \subseteq N$4 coalitions is generally infeasible, so the standard estimator uses Monte Carlo permutation sampling.

Pseudocode for estimating all three variants simultaneously is as follows:

$S$5

Each permutation entails $S \subseteq N$5 complexity, where $S \subseteq N$6 is single model training cost. As $S \subseteq N$7 increases, convergence rate is standard $S \subseteq N$8 (Liu, 2020).

4. Empirical Evaluation on the Iris Dataset

Experiments on the Iris dataset (150 samples, 4 features, 3 classes; 130 for training, 20 for testing) used both Logistic Regression (LR) and Support Vector Machine (SVM). The utility function $S \subseteq N$9 was test-set classification accuracy after training the model on $S$0. Marginal contributions were computed dynamically, with $S$1 random permutations.

Results after ranking tuples by each value variant and training on the top- or bottom-$S$2 ($S$3) tuples are:

Definition	LR (top-K)	LR (bottom-K)	SVM (top-K)	SVM (bottom-K)
ORI	100.00	100.00	93.33	93.33
ZERO	100.00	63.33	96.66	90.00
ABS	100.00	60.00	96.66	90.00

For the original Shapley value (ORI), both top- and bottom-ranked sets achieve similar accuracy, indicating bottom-K points have high negative impact. ZERO removes some harmful points but fails to fully separate influential data. ABS yields the clearest separation: lowest-ABS tuples lead to the lowest accuracy, demonstrating its efficacy in detecting influential (positive or negative) data. This result was consistent across both models, and the magnitude of the bottom-K accuracy drop (100% to 60–63% in LR) evidences the distinctiveness of Absolute Shapley Value (Liu, 2020).

5. Application Contexts and Practical Implications

Absolute Shapley Value is directly relevant in data marketplaces where compensation or scrutiny is based on any substantial influence—positive or negative—on model outputs. It is suitable for identifying both highly beneficial and highly detrimental data points, such as outliers, poisoning attempts, or essential support vectors.

In model training and data debugging, points with large $S$4 are those that robustly affect the decision boundary, making ABS a natural mechanism for surfacing key data. Its application is particularly pertinent when the definition of “importance” is detached from the sign of the effect and solely concerned with perturbation magnitude (Liu, 2020).

A plausible implication is that domains sensitive to data quality or robustness—such as medical diagnosis, anomaly detection, or adversarial defense—can leverage Absolute Shapley to triage and prioritize cases for further analysis.

6. Limitations and Directions for Future Research

The principal limitations stem from the loss of classic Shapley axioms. Both Absolute and Zero Shapley values lack efficiency and additivity, so the total “payout” need not match total system utility. Interpretation of individual scores as fair payments or attributions is thereby undermined. Monte Carlo estimation remains computationally demanding for large-scale or expensive models.

Future research directions include hybrid approaches that redistribute the efficiency budget (especially for Zero Shapley), renormalization schemes for Absolute Shapley to partially reinstate group rationality, and more scalable approximation techniques such as influence functions or incremental updates. Systematic validation on larger datasets and complex models such as deep neural networks is an open problem (Liu, 2020).

7. Summary

Absolute Shapley Value departs from the classical framework by prioritizing the magnitude of each participant’s effect, regardless of polarity, thus elevating the role of both helpful and harmful data. Empirical results on canonical datasets demonstrate its superior ability to discriminate influential from negligible contributors. This comes at the cost of forsaking some foundational fairness guarantees, but provides a rigorous method for applications valuing or monitoring high-impact data contributions (Liu, 2020).