SVARM-IQ: Efficient Shapley Interaction Estimation

• SVARM-IQ is a sampling-based framework for efficiently approximating any-order Shapley interaction indices in explainable AI using a novel stratified representation.
• It leverages stratified decomposition to reuse samples across multiple interaction orders, ensuring unbiased estimates and reducing computational cost.
• Empirical evaluations show that SVARM-IQ outperforms traditional permutation and kernel-based techniques across diverse domains such as language, vision, and synthetic cooperative games.

SVARM-IQ denotes a sampling-based framework for efficiently approximating any-order Shapley-based interaction indices in explainable artificial intelligence (XAI). It leverages a novel stratified representation to maximize sample reuse across interaction orders, providing theoretical guarantees regarding unbiasedness and estimation error, and attains state-of-the-art empirical performance compared to traditional permutation-based and kernel-based sampling techniques. SVARM-IQ is designed for broad applicability, handling interaction indices such as the Shapley Interaction Index, Shapley-Taylor, and Faithful Shapley Interaction Index across model architectures and domains, including language, vision, and synthetic cooperative games.

1. Foundational Principles and Motivation

SVARM-IQ addresses the computational infeasibility inherent in the exact calculation of Shapley values and their interaction extensions. The classical Shapley value evaluates individual feature contributions in a cooperative game setup by considering all possible feature coalitions. For interaction indices such as the Shapley Interaction Index (SII), the complexity further intensifies, requiring exponentially many coalition value evaluations ($2^n$ for $n$ features). Practical XAI scenarios typically involve large-scale models where brute-force evaluation is prohibitive, necessitating tractable approximation algorithms.

SVARM-IQ builds on cardinal interaction index (CII) theory, wherein the interaction $I_K$ for a feature subset $K \subseteq \mathcal{N}$ is formalized as a weighted aggregation of discrete derivative functionals over the value function $\nu$. Previous approaches employed permutation sampling, kernel-based estimation, or restricted-order interaction indices, but suffered from inefficient sample reuse and high estimator variance, particularly beyond low-order ($k=2$) settings.

2. Stratified Representation and Algorithmic Framework

The core methodological innovation of SVARM-IQ is the stratified decomposition of CIIs. For player set $\mathcal{N}$, value function $\nu$, and target subset $K$ of order $k$, the stratified formulation is:

$$I_K = \sum_{\ell=0}^{n-k} \left[ \binom{n-k}{\ell} \lambda_{k,\ell} \sum_{W \subseteq K} (-1)^{k-|W|} I_{K,\ell}^W \right]$$

where

$$I_{K,\ell}^W = \frac{1}{\binom{n-k}{\ell}} \sum_{S \subseteq \mathcal{N} \setminus K, |S| = \ell} \nu(S \cup W)$$

Algorithmically, SVARM-IQ samples coalitions $A \subseteq \mathcal{N}$, drawing both the coalition size and its membership according to a designed probability distribution. For each candidate $K$, the procedure computes intersection $W = A \cap K$ and $\ell = |A| - |W|$, and updates the corresponding stratum estimate $I_{K,\ell}^W$ using the observed coalition value $v = \nu(A)$. Each sample thus efficiently updates all interaction candidates $K$, resulting in maximal reuse of computed model outputs compared to permutation methods, which typically update only one or a few indices per evaluation.

To address variance and redundancy, the algorithm further partitions the coalition-size strata into "border sizes" (where the number of possible coalitions is small) and "implicit sizes" (large coalition families). Border sizes are fully enumerated up front, while implicit strata are approximated via random sampling, optimizing the allocation of the computational budget.

3. Theoretical Guarantees and Error Analysis

SVARM-IQ provides explicit non-asymptotic guarantees regarding estimator bias and variance. Each interaction estimate $\hat{I}_K$ is shown to be unbiased ($\mathbb{E}[\hat{I}_K] = I_K$). Variance and mean squared error (MSE) bounds are derived in terms of the remaining computational budget $\tilde{B}$ and the per-stratum variance $\sigma_{K,\ell,W}^2$, yielding the following generic error bound:

$$\operatorname{MSE} (\hat{I}_K) \leq \frac{\gamma_k}{\tilde{B}} \left / \sum_{W \subseteq K} \sum_{\ell \in \mathcal{L}_k^{|W|}} \binom{n-k}{\ell}^2 \lambda_{k,\ell}^2 \sigma_{K,\ell,W}^2 \right.$$

For pairwise interactions ($k=2$), tailored sampling probabilities (denoted $P_2$) are proposed to further minimize variance. Chebyshev-type results yield probabilistic bounds quantifying the likelihood that estimated values deviate from their true interactions by more than a prescribed $\epsilon$.

The stratification methodology guarantees that, as the computational budget increases—even modestly relative to full enumeration—SVARM-IQ converges more rapidly and stably to accurate estimates than competing approaches.

4. Empirical Evaluation and Benchmarking

SVARM-IQ is empirically validated on multiple XAI scenarios, comprising both deep learning and synthetic settings. In natural language tasks, SVARM-IQ analyzes token interactions in a fine-tuned DistilBERT model applied to IMDB sentiment classification. The estimated higher-order interaction indices disclose combinatorial feature synergies that are missed by additive attributions.

In computer vision, SVARM-IQ is deployed on Vision Transformer and ResNet18 architectures, quantifying interaction at the level of image patches. It successfully elucidates complementary interactions (e.g., contiguous facial regions) and redundancy (negative interaction for semantically overlapping image areas).

Synthetic testbeds, notably SOUM cooperative games, are utilized to rigorously compare MSE and precision at top-$k$ (Prec@10) metrics against permutation methods and SHAP-IQ. Across datasets and model types, SVARM-IQ consistently achieves lower MSE and higher precision with only 7–10% of the total coalition evaluations, substantiating its superior budget-efficiency.

Model/Domain	Interaction Order	Baseline MSE	SVARM-IQ MSE	Prec@10 Improvement
DistilBERT (IMDB)	k=2, 3	High	Low	Significant
ViT/ResNet18 (Vision)	k=2	Moderate	Low	Significant
SOUM Synthetic Game	k=2, 3, 4	High	Very Low	Significant

5. Comparison to Existing Approaches

SVARM-IQ is systematically compared with permutation-based sampling and recent methods such as SHAP-IQ. Permutation sampling typically updates a minimal subset of interaction indices per coalition evaluation and exhibits elevated estimator variance, especially for higher-order interactions. SVARM-IQ's stratified algorithm enables concurrent updates across all candidate indices. Empirical benchmarks substantiate that SVARM-IQ achieves faster error decay and higher top-$k$ precision over identical computational budgets, with optimal performance noted for pairwise and three-way interactions. The maximized sample reuse is the key driver of this efficiency.

6. Broader Implications for Explainable AI

SVARM-IQ extends the scope of XAI by enabling practitioners to interrogate not only individual feature attributions (classical SV) but also intricate feature group interactions, accommodating any interaction order. This capability is critical in applications where collective feature dynamics—such as gene groups in genomics, phrase structures in language, or pixel clusters in vision—are pivotal to model behavior. The method's model-agnostic character allows unified interpretability across architectures.

Moreover, SVARM-IQ’s suite of theoretical guarantees supports confidence in its adoption for high-stakes contexts. Its framework is robust to extension for approximating other indices (SII, STI, FSI), potentially guiding future developments in feature selection, model diagnostics, and fairness audits.

A plausible implication is that SVARM-IQ, by efficiently quantifying higher-order interactions, may become foundational in post-hoc explanation frameworks, rendering complex models more transparent and interpretable, and facilitating principled decision making in both research and deployment environments.

7. Relationship to Broader IQ Measurement and Collective Intelligence

While SVARM-IQ pertains strictly to feature interaction explanation in XAI, connections to broader IQ constructs and the measurement of intelligence in artificial or collective systems are evident. For instance, SVARM-IQ’s methodological rigor—unbiasedness, multi-dimensional stratification, quantitative error control—is analogous to the principles underlying AI IQ measurement frameworks, such as the standard intelligent system model (Liu et al., 2015). SVARM-IQ's capacity to interrogate multi-faceted feature synergies aligns with modern perspectives on intelligence as an emergent, multi-dimensional phenomenon, paralleling recent exploration into the amplification of group intelligence via conversational swarm architectures (Rosenberg et al., 25 Jan 2024).

This suggests SVARM-IQ's stratified, distributed estimation approach could inform or be integrated with collective intelligence systems, where quantification of synergy and redundancy among diverse agents or features is central to understanding group performance and emergent IQ.