Shapley Value Attribution

In this paper, we provide a theoretically grounded and computationally efficient alternative to SHAP. To this end, we study feature attribution through the lens of cooperative game theory by formulating a class of XAI--TU games. Building on this formulation, we investigate equal-surplus-type and proportional-allocation-type attribution rules and propose a low-cost attribution rule, ESENSC_rev2, constructed by combining two polynomial-time closed-form rules while ensuring the null-player property in the XAI--TU domain. Extensive experiments on tabular prediction tasks demonstrate that ESENSC_rev2 closely approximates exact SHAP while substantially improving scalability as the number of features increases. These empirical results indicate that equal-surplus-type attribution rules can achieve favorable trade-offs between computational cost and approximation accuracy in high-dimensional explainability settings. To provide theoretical foundations for these findings, we establish an axiomatic characterization showing that ESENSC_rev2 is uniquely determined by efficiency, the null-player axiom, a restricted differential marginality principle, an intermediate inessential-game property, and axioms that reduce computational requirements. Our results suggest that axiomatically justified and computationally efficient attribution rules can serve as practical and theoretically principled substitutes for SHAP-based approximations in modern explainability pipelines.

In this article, we provide an axiomatic characterization of feature attribution for multi-output predictors within the Shapley framework. While SHAP explanations are routinely computed independently for each output coordinate, the theoretical necessity of this practice has remained unclear. By extending the classical Shapley axioms to vector-valued cooperative games, we establish a rigidity theorem showing that any attribution rule satisfying efficiency, symmetry, dummy player, and additivity must necessarily decompose component-wise across outputs. Consequently, any joint-output attribution rule must relax at least one of the classical Shapley axioms. This result identifies a previously unformalized structural constraint in Shapley-based interpretability, clarifying the precise scope of fairness-consistent explanations in multi-output learning. Numerical experiments on a biomedical benchmark illustrate that multi-output models can yield computational savings in training and deployment, while producing SHAP explanations that remain fully consistent with the component-wise structure imposed by the Shapley axioms.

Explaining the behavior of end-to-end audio language models via Shapley value attribution is intractable under native tokenization: a typical utterance yields over $150$ encoder frames, inflating the coalition space by roughly $10^{42}$ relative to text; individual audio frames lack standalone meaning; and token boundaries that bisect phonetic transitions introduce masking artifacts. We introduce Spectrogram-Guided Phonetic Alignment (SGPA), a four-stage pipeline that combines Connectionist Temporal Classification forced alignment with spectral boundary refinement to produce acoustically stable, word-aligned audio segments. Controlled diagnostics on LFM2-Audio-1.5B with VoiceBench show that SGPA yields a 43$\times$ reduction in model evaluations. Statistical testing confirms that SGPA significantly alters attribution concentration while preserving the global cumulative profile, establishing it as a feasibility-enabling layer for audio explainability.

Explainable artificial intelligence promises to yield insights into relevant features, thereby enabling humans to examine and scrutinize machine learning models or even facilitating scientific discovery. Considering the widespread technique of Shapley values, we find that purely data-driven operationalization of multivariate feature importance is unsuitable for such purposes. Even for simple problems with two features, spurious associations due to collider bias and suppression arise from considering one feature only in the observational context of the other, which can lead to misinterpretations. Causal knowledge about the data-generating process is required to identify and correct such misleading feature attributions. We propose cc-Shapley (causal context Shapley), an interventional modification of conventional observational Shapley values leveraging knowledge of the data's causal structure, thereby analyzing the relevance of a feature in the causal context of the remaining features. We show theoretically that this eradicates spurious association induced by collider bias. We compare the behavior of Shapley and cc-Shapley values on various, synthetic, and real-world datasets. We observe nullification or reversal of associations compared to univariate feature importance when moving from observational to cc-Shapley.

Shapley value-based methods have become foundational in explainable artificial intelligence (XAI), offering theoretically grounded feature attributions through cooperative game theory. However, in practice, particularly in vision tasks, the assumption of feature independence breaks down, as features (i.e., pixels) often exhibit strong spatial and semantic dependencies. To address this, modern SHAP implementations now include the Owen value, a hierarchical generalization of the Shapley value that supports group attributions. While the Owen value preserves the foundations of Shapley values, its effectiveness critically depends on how feature groups are defined. We show that commonly used segmentations (e.g., axis-aligned or SLIC) violate key consistency properties, and propose a new segmentation approach that satisfies the $T$-property to ensure semantic alignment across hierarchy levels. This hierarchy enables computational pruning while improving attribution accuracy and interpretability. Experiments on image and tabular datasets demonstrate that O-Shap outperforms baseline SHAP variants in attribution precision, semantic coherence, and runtime efficiency, especially when structure matters.

The SHAP (short for Shapley additive explanation) framework has become an essential tool for attributing importance to variables in predictive tasks. In model-agnostic settings, SHAP uses the concept of Shapley values from cooperative game theory to fairly allocate credit to the features in a vector $X$ based on their contribution to an outcome $Y$. While the explanations offered by SHAP are local by nature, learners often need global measures of feature importance in order to improve model explainability and perform feature selection. The most common approach for converting these local explanations into global ones is to compute either the mean absolute SHAP or mean squared SHAP. However, despite their ubiquity, there do not exist approaches for performing statistical inference on these quantities. In this paper, we take a semi-parametric approach for calibrating confidence in estimates of the $p$th powers of Shapley additive explanations. We show that, by treating the SHAP curve as a nuisance function that must be estimated from data, one can reliably construct asymptotically normal estimates of the $p$th powers of SHAP. When $p \geq 2$, we show a de-biased estimator that combines U-statistics with Neyman orthogonal scores for functionals of nested regressions is asymptotically normal. When $1 \leq p < 2$ (and the hence target parameter is not twice differentiable), we construct de-biased U-statistics for a smoothed alternative. In particular, we show how to carefully tune the temperature parameter of the smoothing function in order to obtain inference for the true, unsmoothed $p$th power. We complement these results by presenting a Neyman orthogonal loss that can be used to learn the SHAP curve via empirical risk minimization and discussing excess risk guarantees for commonly used function classes.

Data valuation is a natural framework for understanding which preference datasets matter most when aligning a Large Language Model (LLM) using multiple sources. The standard game-theoretic approach assigns each dataset a contribution score via the Shapley value. In practice, however, Shapley-based valuation is computationally prohibitive because it requires fine-tuning a separate model for every possible coalition of preference datasets, i.e., an exponential number of alignments. We address this challenge for a broad family of preference-optimization objectives, including DPO and IPO, that learn directly from log-policy ratios with respect to a reference policy. We introduce Sequential Preference Optimization, an offline procedure that applies existing preference optimization methods sequentially, source by source, updating the current policy after each dataset. Under exact optimization, this procedure yields an additive composition rule in reward space and an equivalent arithmetic composition rule in policy space. This observation enables an efficient approximation of the Shapley value: we train one model per preference dataset and reconstruct coalition policies at inference time from the singleton models, reducing the required alignments from exponential to linear in the number of sources. Leveraging this property, we compute Shapley values for several real-world preference datasets and reveal how each source drives model alignment.

With origins in game theory, probabilistic values like Shapley values, Banzhaf values, and semi-values have emerged as a central tool in explainable AI. They are used for feature attribution, data attribution, data valuation, and more. Since exact computation can require exponential time, research has focused on efficient approximation methods using two techniques: Monte Carlo sampling and linear regression formulations. In this work, we present a new way of combining both of these techniques. Our approach is more flexible than prior algorithms, allowing for linear regression to be replaced with any function family whose probabilistic values can be computed efficiently. This allows us to harness the accuracy of tree-based models like XGBoost; with cross-fitting, the method also produces unbiased estimates. From experiments across eight datasets, we find that our methods give state-of-the-art performance for estimating probabilistic values. For Shapley values, the error of our methods can be $11\times$ lower than Permutation SHAP (the most popular Monte Carlo method), $6.5\times$ lower than Kernel SHAP (the most popular linear regression method), and $3.4\times$ lower than Leverage SHAP (the prior state-of-the-art Shapley value estimator). For more general probabilistic values, we can obtain error $390\times$ lower than the best estimator from prior work.

Contribution attribution is a fundamental problem in federated learning (FL), yet it remains challenging in peer-to-peer FL (P2PFL), where participant influence propagates through repeated model exchanges. Recent work, DICE, addresses this issue by directly tracking multi-round propagation paths. However, its practical deployment is hindered by two challenges: (i) communication explosion caused by multi-hop propagation path tracking, and (ii) vulnerability to misreporting attacks. To overcome these challenges, we consider contribution attribution in P2PFL with a model-free coordinator used solely for auditing. Based on this setting, we propose TRIP-Shapley, a propagation-aware Shapley-inspired surrogate for contribution attribution. Instead of explicitly tracking multi-round paths, TRIP-Shapley computes round-wise contribution scores independently and propagates only scalar contribution values. This enables attribution of influence accumulated throughout the entire training process while maintaining practical communication costs. In addition, we introduce lightweight mechanisms to detect and mitigate misreporting attacks. Rather than theoretically approximating Shapley values, TRIP-Shapley is empirically evaluated against Shapley values as a reference. Experiments demonstrate communication savings of up to $45.8\times$ relative to the compared DICE configurations, with favorable cost--reference-alignment trade-offs overall, although alignment is weaker in some FMNIST and 25-client settings. The proposed defenses reduce contribution inflation under the two evaluated attacks.

Large Language Models (LLMs) increasingly power search engines and AI assistants that retrieve and summarize content from many sources. By serving answers directly, these systems obscure the original content creators' contributions, threatening the compensation that sustains a healthy content ecosystem. We frame this as a problem of fair document valuation and compensation, and propose a framework based on the Shapley value. Because exact Shapley computation is prohibitively expensive at scale, we develop Cluster Shapley, an approximation that groups semantically similar documents via LLM embeddings and computes Shapley values at the cluster level, with formal bounds on both the approximation error and the induced revenue-attribution error. On Amazon product review data, off-the-shelf approximations such as Monte Carlo sampling and Kernel SHAP perform suboptimally in LLM settings, whereas Cluster Shapley substantially improves the efficiency--accuracy frontier. Simple attribution heuristics (e.g., equal or relevance-based allocation), though computationally cheap, yield highly unfair outcomes. Our approach is agnostic to the exact LLM used, the summarization process used, and the evaluation procedure, which makes it broadly applicable to a variety of summarization settings.

Originally introduced in game theory, Shapley values have emerged as a central tool in explainable machine learning, where they are used to attribute model predictions to specific input features. However, computing Shapley values exactly is expensive: for a general model with $n$ features, $O(2^n)$ model evaluations are necessary. To address this issue, approximation algorithms are widely used. One of the most popular is the Kernel SHAP algorithm, which is model agnostic and remarkably effective in practice. At the time of our original submission, no strong non-asymptotic complexity guarantees were known for Kernel SHAP. We address this issue by introducing Leverage SHAP, a lightweight modification of Kernel SHAP that provides provably accurate Shapley value estimates with just $O(n\log n)$ model evaluations. Our approach takes advantage of a connection between Shapley value estimation and agnostic active learning by employing leverage score sampling, a powerful regression tool. Beyond theoretical guarantees, we find that Leverage SHAP achieves an approximately $50\%$ reduction in error compared to the highly optimized implementation of Kernel SHAP in the widely used SHAP library [Lundberg & Lee, 2017].

Originating in game theory, Shapley values are widely used for explaining a machine learning model's prediction by quantifying the contribution of each feature's value to the prediction. This requires a scalar prediction as in binary classification, whereas a multiclass probabilistic prediction is a discrete probability distribution, living on a multidimensional simplex. In such a multiclass setting the Shapley values are typically computed separately on each class in a one-vs-rest manner, ignoring the compositional nature of the output distribution. In this paper, we introduce Shapley compositions as a well-founded way to properly explain a multiclass probabilistic prediction, using the Aitchison geometry from compositional data analysis. We prove that the Shapley composition is the unique quantity satisfying linearity, symmetry and efficiency on the Aitchison simplex, extending the corresponding axiomatic properties of the standard Shapley value. We demonstrate this proper multiclass treatment in a range of scenarios.

Classical data valuation defines a data point's value through the finite marginal contribution $U(C\cup\{i\})-U(C)$, but estimating this quantity over coalitions requires repeated training and does not describe the contribution made along a stochastic training path. We ask whether marginal contributions of data points can be estimated from one coupled trajectory while retaining a verifiable relation to coalition-based values. To this end, we introduce Neural Dynamic Data Valuation (NDDV), which models each data point as a controlled stochastic state and computes a first-order marginal-contribution score via the adjoint equation of the Stochastic Maximum Principle (SMP). This raw sensitivity is then calibrated by a mass-preserving redistribution that increases one data point's participation while redistributing the same total weight over the remaining data points. We prove that the resulting backward adjoint recursion is the exact reverse-mode adjoint of the frozen-aggregate Euler system, bound its discrepancy from the mean-field sensitivity, and express each finite coalition marginal as an integral of local sample-weight sensitivities. These results yield pair-specific error bounds and sufficient conditions for ordering agreement with Shapley, Banzhaf, and leave-one-out values. Experiments on existing benchmarks evaluate marginal-contribution fidelity, score-release cost, corrupted-sample detection, ablations, and failure regimes. NDDV is a one-run, trajectory-conditioned estimator, not an unconditional replacement for cooperative-game values.