Paired-sampling PermutationSHAP

• The paper introduces reversed permutation pairing that averages marginal contributions, yielding an unbiased, strictly consistent, and asymptotically normal estimator.
• It guarantees exact recovery for second-order interactions and additive recovery for additively decomposed games, ensuring strong structural validity.
• The method employs antithetic sampling to reduce variance, though it requires more value-function evaluations compared to alternative SHAP estimators.

Searching arXiv for papers on paired-sampling PermutationSHAP and closely related SHAP sampling work. Search query: paired-sampling PermutationSHAP Shapley Values Paired-Sampling Approximations Paired-sampling PermutationSHAP is a permutation-based Monte Carlo estimator of Shapley values in which each sampled permutation $\pi$ is coupled with its reversed permutation $\rho(\pi)=(\pi_q,\ldots,\pi_1)$, and the two induced marginal-contribution vectors are averaged. In the standard cooperative-game formulation, this leaves the estimator within the permutation-SHAP family, but changes its sampling geometry through a symmetry over complementary predecessor coalitions. The resulting method is unbiased, strictly consistent, and asymptotically normal, and it has two structural guarantees that are unusually strong for a sampling-based SHAP estimator: exactness for value functions with interactions of maximal order two, and additive recovery for additively decomposed games (Mayer et al., 18 Aug 2025). It is distinct both from paired complement sampling in KernelSHAP-style coalition regression and from “pairwise permutation” methods that jointly perturb pairs of correlated features in global feature-importance settings rather than pairing Monte Carlo permutations (Fumagalli et al., 26 Jan 2026, Maaslandand et al., 2021).

1. Cooperative-game setting and permutation SHAP

The underlying setting is the standard SHAP game with $q \ge 2$ players or features, indexed by

${\cal Q}=\{1,\ldots,q\},$

and a value function

$\nu: {\cal C} \mapsto \nu({\cal C})$

defined on coalitions ${\cal C}\subseteq {\cal Q}$. The paper assumes without loss of generality that $\nu(\emptyset)=0$. In this formulation, the Shapley value of feature $j$ is represented as an average over all feature orderings: $$\phi_j = \frac{1}{q!} \sum_{\pi}\nu\left({\cal C}_{\pi, j}\cup \{j\}\right) -\nu\left({\cal C}_{\pi, j}\right).$$ Here $\pi=(\pi_1,\ldots,\pi_q)$ is a permutation of $\rho(\pi)=(\pi_q,\ldots,\pi_1)$0, and $\rho(\pi)=(\pi_q,\ldots,\pi_1)$1 is the set of predecessors of $\rho(\pi)=(\pi_q,\ldots,\pi_1)$2 in $\rho(\pi)=(\pi_q,\ldots,\pi_1)$3. Thus $\rho(\pi)=(\pi_q,\ldots,\pi_1)$4 is the average marginal contribution of $\rho(\pi)=(\pi_q,\ldots,\pi_1)$5 across all permutations (Mayer et al., 18 Aug 2025).

Standard sampling PermutationSHAP approximates this expectation by drawing i.i.d. random permutations $\rho(\pi)=(\pi_q,\ldots,\pi_1)$6 uniformly from the $\rho(\pi)=(\pi_q,\ldots,\pi_1)$7 possible orderings and averaging the corresponding marginal contributions. A single sampled permutation naturally yields a full $\rho(\pi)=(\pi_q,\ldots,\pi_1)$8-vector of contributions, because every feature appears once along the same path. In the broader literature on permutation sampling for Shapley estimation, this estimator is the baseline against which structured permutation designs are compared, since exact evaluation is generally infeasible and naïve Monte Carlo can converge slowly (Mitchell et al., 2021).

2. Reverse-permutation pairing and the paired estimator

In paired-sampling PermutationSHAP, the pairing rule is explicit: for each sampled permutation $\rho(\pi)=(\pi_q,\ldots,\pi_1)$9, one also evaluates the reversed permutation

$q \ge 2$0

The paired estimator for feature $q \ge 2$1 is

$q \ge 2$2

The central combinatorial identity is that the predecessor coalition of $q \ge 2$3 in the reverse permutation is the complement of the original predecessor-plus-$q \ge 2$4 coalition: $q \ge 2$5 This complementarity is the reason the estimator is described as a symmetrized permutation average rather than merely a duplicated Monte Carlo sample (Mayer et al., 18 Aug 2025).

The variance rationale is antithetic. Averaging the contributions from $q \ge 2$6 and $q \ge 2$7 tends to cancel part of the permutation-induced variability, because if one ordering places a feature early, the reverse places it late. In covariance form,

$q \ge 2$8

so variance falls whenever the cross-covariance is sufficiently negative. This reverse-pairing construction is closely aligned with the “MC-antithetic” baseline studied in the permutation-sampling literature, where reverse-paired permutations were reported as the best overall among existing Shapley baselines, and where structured designs such as orthogonal spherical codes were interpreted as extending antithetic sampling from $q \ge 2$9 correlated samples to ${\cal Q}=\{1,\ldots,q\},$0 correlated samples (Mayer et al., 18 Aug 2025, Mitchell et al., 2021).

3. Statistical and structural guarantees

The paired estimator is unbiased because both ${\cal Q}=\{1,\ldots,q\},$1 and ${\cal Q}=\{1,\ldots,q\},$2 are uniformly distributed over permutations, so the expectation of the paired summand remains ${\cal Q}=\{1,\ldots,q\},$3. The paper further states that paired-sampling PermutationSHAP is strictly consistent, unbiased, and asymptotically normal: ${\cal Q}=\{1,\ldots,q\},$4 with

${\cal Q}=\{1,\ldots,q\},$5

This yields the usual ${\cal Q}=\{1,\ldots,q\},$6 Monte Carlo error rate and supports covariance-based uncertainty quantification through the paired summand rather than the unpaired marginal-contribution vector (Mayer et al., 18 Aug 2025).

Its more distinctive results are structural rather than asymptotic. If the value function has interactions of maximal order two, written as a bilinear or quadratic form on binary coalitions,

${\cal Q}=\{1,\ldots,q\},$7

then the exact Shapley values are

${\cal Q}=\{1,\ldots,q\},$8

and one single paired permutation already recovers them exactly: ${\cal Q}=\{1,\ldots,q\},$9 The exactness mechanism is that the original and reversed marginal contributions together activate all pairwise interactions involving $\nu: {\cal C} \mapsto \nu({\cal C})$0, independently of the permutation. Within this class, the Monte Carlo variance is therefore zero (Mayer et al., 18 Aug 2025).

The second major guarantee is the additive recovery property. Suppose the feature set is partitioned into nonempty groups $\nu: {\cal C} \mapsto \nu({\cal C})$1, and the value function decomposes additively across those groups: $\nu: {\cal C} \mapsto \nu({\cal C})$2 Then, for a single permutation,

$\nu: {\cal C} \mapsto \nu({\cal C})$3

Thus the blockwise attribution totals are already exact before any Monte Carlo averaging. The same paper emphasizes that paired KernelSHAP does not satisfy any similar property. It also proves a partial exactness result: if one subset of features interacts only internally and only up to order two, with no interactions with the remainder, then that subset’s Shapley values are exactly recovered from one paired permutation even if the rest of the value function is arbitrarily nonlinear (Mayer et al., 18 Aug 2025).

4. Distinctions from related “paired” and “permutation” constructions

Several neighboring literatures use similar language but refer to different mathematical objects. The distinctions are operationally important.

Construction	Pairing or permutation object	Role
Paired-sampling PermutationSHAP	A permutation $\nu: {\cal C} \mapsto \nu({\cal C})$4 and its reverse $\nu: {\cal C} \mapsto \nu({\cal C})$5	Monte Carlo estimation of Shapley values
Paired KernelSHAP / PolySHAP	A coalition $\nu: {\cal C} \mapsto \nu({\cal C})$6 and its complement $\nu: {\cal C} \mapsto \nu({\cal C})$7	Weighted regression over coalition evaluations
Pairwise Permutation Algorithm	A pair of features $\nu: {\cal C} \mapsto \nu({\cal C})$8 jointly permuted and relearned	Global feature importance under dependence

In the coalition-regression literature, paired sampling means complement-paired subsets. One paper proves that if subsets are sampled in pairs $\nu: {\cal C} \mapsto \nu({\cal C})$9 and ${\cal C}\subseteq {\cal Q}$0, then paired KernelSHAP outputs exactly the same Shapley value approximations as second-order PolySHAP, without ever fitting a degree ${\cal C}\subseteq {\cal Q}$1 polynomial. That result is specific to weighted least-squares coalition regression and is not a theorem about permutation-based SHAP estimators (Fumagalli et al., 26 Jan 2026).

A related regression-theoretic account shows that Shapley values depend only on the odd component of a set function, and that paired complement sampling orthogonalizes a regression objective into odd and even parts, thereby filtering out the irrelevant even component. Again, this is a formal result for paired coalition regression rather than a direct variance theorem for permutation Monte Carlo, although it provides a conceptual explanation for why complement symmetry is beneficial more generally (Fumagalli et al., 1 Feb 2026).

The Pairwise Permutation Algorithm is different in a more fundamental sense. Its “pairwise” mechanism means pairs of features, not paired Monte Carlo samples: it perturbs two correlated features at a time in the training data, retrains the model, and uses those pairwise relearned importances to correct single-feature global importance scores. It is not a Shapley estimator, does not average over permutations of feature orderings, and does not compute additive attributions whose sum equals a coalition-value difference (Maaslandand et al., 2021).

A further source of ambiguity is data permutation. Another line of work uses independent permutation of each data column to sample from a product-of-marginals distribution for safe aggregation of SHAP or KernelSHAP values. That construction concerns background-data preprocessing for global feature-discarding analysis, not feature-order permutation in PermutationSHAP (Bhattacharjee et al., 29 Mar 2025).

5. Empirical behavior and computational tradeoffs

The empirical picture in the paired-sampling PermutationSHAP paper is mixed in a way that reflects the method’s strong structure but nontrivial evaluation cost. In a nonlinear ${\cal C}\subseteq {\cal Q}$2 example with

${\cal C}\subseteq {\cal Q}$3

the CLT approximation based on ${\cal C}\subseteq {\cal Q}$4 matches empirical standard deviations well, and paired PermutationSHAP and paired KernelSHAP have approximation errors of similar order, with the permutation version somewhat better for one component. In a larger ${\cal C}\subseteq {\cal Q}$5 example with

${\cal C}\subseteq {\cal Q}$6

empirical bias is negligible, asymptotic standard errors from ${\cal C}\subseteq {\cal Q}$7 are accurate, and paired PermutationSHAP has significantly smaller standard deviations than paired KernelSHAP on the per-sample scale (Mayer et al., 18 Aug 2025).

The computational caveat is that the same paper counts each paired PermutationSHAP simulation as requiring ${\cal C}\subseteq {\cal Q}$8 value-function evaluations, whereas paired KernelSHAP requires ${\cal C}\subseteq {\cal Q}$9. Consequently, after adjusting for evaluation cost rather than raw per-sample variance, KernelSHAP may be preferable if the only concern is evaluation count. The paper’s practical preference for the permutation-based method instead rests on its structural guarantees, especially additive recovery and the interpretability of the covariance structure (Mayer et al., 18 Aug 2025).

The broader permutation-sampling literature places this reverse-pairing strategy within a larger design space. Reverse-paired Monte Carlo is already a strong baseline, but structured samplers can go further. Kernel herding and sequential Bayesian quadrature minimize discrepancy on permutation space, while orthogonal spherical codes and Sobol-based constructions generate more space-filling or negatively dependent batches of permutations. In tabular experiments, orthogonal spherical codes were the most consistently strong new method and were explicitly interpreted as a generalization of antithetic pairing beyond simple reverse-permutation pairs (Mitchell et al., 2021).

6. Scope, extensions, and neighboring formulations

The method’s clearest domain is any local explanation problem where the primary object is a fixed coalition value function $\nu(\emptyset)=0$0 and the objective is to estimate Shapley attributions by averaging marginal contributions along feature orderings. Within that scope, paired-sampling PermutationSHAP is best understood as a reverse-permutation symmetrization that is exact for quadratic interaction structure and blockwise exact in aggregate for additively decomposed games (Mayer et al., 18 Aug 2025).

Its transfer to more specialized games depends on how the coalition value function is defined. In preference modeling, for example, Pref-SHAP defines a pairwise comparison game over shared feature indices across both compared items: a coalition $\nu(\emptyset)=0$1 means that the features in $\nu(\emptyset)=0$2 are revealed for both items simultaneously, and the missing coordinates are integrated out through conditional expectations. The paper itself estimates SHAP values by sampled coalitions and weighted least squares rather than by permutation traversal, but its game semantics make clear that a permutation-based estimator for pairwise comparisons would need to add feature index $\nu(\emptyset)=0$3 by revealing $\nu(\emptyset)=0$4 and $\nu(\emptyset)=0$5 together, not by treating the two items as separate $\nu(\emptyset)=0$6 players (Hu et al., 2022).

The present literature therefore supports a precise delimitation. Paired-sampling PermutationSHAP has direct theory for reverse-permutation pairing, asymptotic inference, exactness for second-order interactions, and additive recovery (Mayer et al., 18 Aug 2025). Coalition-regression papers provide complementary but not interchangeable theory: complement-paired sampling can implicitly encode second-order information in KernelSHAP-style estimators (Fumagalli et al., 26 Jan 2026), and paired complements isolate the odd component relevant to Shapley values in regression formulations (Fumagalli et al., 1 Feb 2026). A plausible implication is that symmetry-based designs are valuable across SHAP estimators, but the strongest permutation-specific statements currently remain those proved for the reverse-paired permutation estimator itself.