Title: Regression-adjusted Monte Carlo Estimators for Shapley Values and Probabilistic Values
Venue: NeurIPS 2025
Accepted: yes
Keywords: Shapley values

Regression-adjustment

Kernel SHAP

Maximum sample reuse

Permutation SHAP
Abstract: 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 all of these values require exponential time to compute exactly, 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, while still producing 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 is up to $6\times$ lower than Permutation SHAP (the most popular Monte Carlo method), $2.75\times$ lower than Kernel SHAP (the most popular linear regression method), and $1.75\times$ lower than Leverage SHAP (the prior state-of-the-art Shapley value estimator). For more general probabilistic values, we can obtain error up to $60\times$ lower than prior work.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a technique that combines Monte Carlo sampling with a regression-based formulation. This approach is claimed to reduce the variance of approximation, thereby improving the estimation of probabilistic values. The authors also introduce treeprob, a method for exactly computing all probabilistic values when interventional utility is used. Experimental results support the potential effectiveness of the proposed technique.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n- The proposed approach appears promising for developing variance-reduced Monte Carlo methods for approximating probabilistic values.\n\n**Weaknesses**\n#### Major\n- Basically, the proposed procedure can be summarized as follows: (1) learning a function $f$ to approximate the given utility function $v$, where $\\phi(f)$ is cheap to compute exactly; and then (2) approximate $\\phi(v)$ as $\\phi(f) + \\hat{\\phi}(v-f)$. In particular, $\\hat{\\phi}(v-f)$ is an approximation of $\\phi(v-f)$ computed using th Monte Carlo method introduced in Eq. (2). That being said,  $\\hat{\\phi}(v-f)$ could also be computed using any existing Monte Carlo methods. This suggests that all the considered Monte Carlo baselines could potentially benefit from the same learned function $f$. \n\n\n- On the other hand, while the proposed Monte Carlo method appears novel, it is unclear whether it achieves faster convergence rate either empirically or theoretically compared to the existing baselines.  \n\nAll in all, the paper lacks experiments demonstrating that the proposed Monte Carlo method is the most effective when paired with the learned $f$. \n\n\n#### Minor\n- I suggest that the authors include more details on the implementation of treeMSR. After examining the provided code, I found that the features used are binary; the baseline is the zero vector, and the explicand is the one vector. This important information is missing in the paper.\n\n- In the proof of theorem 2.1, the randomness involved in learning $f$ is not accounted. As a result, the stated result applies only to the specific Monte Carlo method defined in Eq. (3). The current statement of theorem 2.1 may therefore be misleading.\n\n\n**Typos**\n- According to Eq. (7), the first equality in Eq. (8) should be changed to $\\leq$. \n- In line 26 of Algorithm 2, it should be $\\phi[\\mathrm{node.feat}] \\gets \\phi[\\mathrm{node.feat}] + (\\mathrm{pos}\\_{e} + \\mathrm{neg}\\_{b})$.\n\nQUESTIONS:\n- Why not combine $\\mathcal{S}\\^{\\mathrm{train}}$ and $\\mathcal{S}\\^{\\mathrm{test}}$ for both learning and approximating, given that they are drawn from the same distribution?\n- Could the authors provide results of substituting other Monte Carlo baselines in both linearMSR and treeMSR?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-23"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes using the control variates (CV) method or regression sampling (RS) method for computing the Shapley value and its variants to address the well-known computational challenges. While CV and RS are well-established techniques in Monte Carlo estimation, to the best of my knowledge, this is the first proposal to apply these methods to Shapley value calculation, which is a noteworthy contribution.\n\nAs demonstrated in general Monte Carlo contexts, both CV and RS are undoubtedly effective. For Shapley value computation, the baseline function can be chosen as a linear function, since in that case the exponential sum required for the Shapley value can be computed exactly.\n\nThe idea is straightforward. Since existing sampling approaches such as LeverageSHAP are already highly optimized, direct performance comparison may be challenging. Nevertheless, the authors’ proposal is meaningful as an additional acceleration method for SHAP, complementing LeverageSHAP. I support acceptance of this paper.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\n- The first introduction of the CV or RS method to Shapley values. \n- Demonstrated variance reduction.\n\nWeakness:\n- Depending on how the performance is measured, improvement may not be drastic as compared to highly optimized alternatives.\n\nQUESTIONS:\nI found Fig. 1(b) confusing. The Correlated and Community data points do not appear to satisfy the unbiasedness property. Why do they seem so biased? In theory, the estimated values should be distributed both above and below the true value, so that the deviations cancel out on average. Why is this not observed in the figure?\n\nLIMITATIONS:\nCV and RS is well-known. Contribution can be seen incremental from a general Monte Carlo study perspective. However, application to Shapley value computation is novel enough. This would should be accepted.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposed a novel estimator for general probabilistic values such as Shapley values and Banzhaf values. The authors combined Monte Carlo sampling with regression-based methods to construct an unbiased estimator. Specifically, the proposed method used regression as a variance reduction technique for the Maximum Sample Reuse method, thereby integrating the strengths of both approaches.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n\n1.\tThe proposed estimator has several desirable properties: (1) it is unbiased, (2) it is applicable to any probabilistic value (e.g., Shapley value and Banzhaf value), (3) it allows the use of any learned function $f$  to approximate the value function $v$, and (4) the variance of the estimator depends on the quality of the surrogate function $f$. As such, when $f$ is accurate, the variance can be significantly reduced.\n\n2.\tThe authors analyze the error bound of the estimator under an arbitrary sampling distribution $D$ , which improves the theoretical soundness and generality of the method.\n\n3.\tThe authors implement the estimator using both linear and tree-based models for the surrogate function $f$, with the tree-based models exhibiting lower approximation error in practice.\n\n**Weaknesses:**\n\n1.\tIt would be helpful if the authors could provide a more detailed derivation of Equation (5) in the appendix, including a formal proof of its unbiasedness and an analysis of its variance.\n\n\n\n2.\tSince the variance of the proposed estimator depends on the surrogate function $f$, it would be beneficial to provide a more detailed discussion of how different choices of $𝑓$ affect the variance and estimation accuracy. For example, to what extent does the use of linear versus non-linear models influence the quality of the estimated probabilistic values? Could shallow neural networks serve as effective approximators in this context? A more detailed analysis of how the choice of $f$ impacts estimation error would strengthen the paper.\n\n\n \n3.\tIt is unclear how the computational complexity of the proposed method compares to existing approaches. In particular, it would be helpful to clarify whether the estimator has similar time complexity to the Maximum Sample Reuse method. A discussion or empirical comparison of the time complexity (or runtime) would improve the clarity of the contribution.\n\n4.  The authors could add a summary table in the appendix comparing the proposed method with existing baselines (e.g., in terms of unbiasedness, variance, computational cost, use of surrogate models, and other relevant properties), which would help readers better understand the advantages and limitations of each method.\n\nQUESTIONS:\nSee weaknesses\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe article addresses the estimation of probabilistic values such as Shapley values and Banzhaf values without the need to perform $2^{n}$ computations and accesses to the payoff function.\nThey propose using a simpler function, $f(S): 2^{[n]} \\to \\mathcal{R}$, to approximate $v(S)$. This function is then used to exploit the linearity property of probabilistic values to find a formulation that can estimate the contributions consistently, which is proven theoretically.\nThe proposed algorithm assumes that $\\phi(f)$ can be computed fast, i.e. $f$ is either linear or tree-based.\nThe article also provides intensive numerical experiments analyzing the sample complexity of the proposed algorithm and comparing it to the many published methods.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1) The article addresses an important problem in XAI which is the efficient computation of Shapley values\n2) It provides sound theoretical guarantees.\n3) The experimental results are thorough and convincingly demonstrate the method’s effectiveness. \n\nWeaknesses:\n1) The presentation lacks clarity in key areas, including both the preliminary literature overview, the discussion of closely related methods, and even the theoretical contribution\n2) The manuscript omits discussion of several relevant works in the literature on fast Shapley value estimation. While the literature is admittedly large, a more comprehensive review would strengthen the article.\n\nQUESTIONS:\n1) Equation (3) seems central to the proposed algorithm, but no intuition is provided. It’s also unclear which cited work this equation comes from. Can you clarify clarify its origin and provide some explanation?\n\n2) Theorem 2.1 is the main contribution of the article, I think it can be stated better, although equation (6) does entail consistency results, I believe the main contribution of the article can be more readily accessible if consistency ($E[\\hat{\\phi}_{i}] = \\phi_{i}$) and variance results are mentioned explicitly in the main body of the article.\n\n3) is the division between Monte Carlo and regression-based methods strictly correct? For instance, doesn’t KernelSHAP also involve Monte Carlo sampling, not just regression?\n\n4) Some explanations are hard to follow. For example, in line 35, the first sentence appears without context, and the topic shifts abruptly right after. This is also seen in sections 1.1\n\n5) Key related works like Mitchell et al., 2022 (\"Sampling Permutations for Shapley Value Estimation\") and Jethani et al., 2022 (\"FastSHAP\") are not mentioned. Even though FastSHAP is not based on Monte Carlo, it is relevant. Also, Mitchell et al. show that Owen's halves can be effective, yet this article only discusses Owen’s method in the context of KernelSHAP. I understand that the relevant literature is vast but it does draw attention that the method is not compared to such prevalent works\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}