arxiv: 2605.04497 · v1 · submitted 2026-05-06 · 💻 cs.LG

Recognition: unknown

Quadrature-TreeSHAP: Depth-Independent TreeSHAP and Shapley Interactions

Ron Wettenstein , Rory Mitchell , Peng Yu

Authors on Pith no claims yet

Pith reviewed 2026-05-08 16:43 UTC · model grok-4.3

classification 💻 cs.LG

keywords Shapley valuesTreeSHAPinteraction valuesquadratureexplainable AIgradient boostingnumerical stabilityBanzhaf values

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The pith

Quadrature reformulation of TreeSHAP using eight fixed points delivers stable Shapley values and interactions independent of tree depth.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that Path-Dependent SHAP for tree ensembles can be rewritten as the integral of a weighted-Banzhaf polynomial over the unit interval. By evaluating this integral with a fixed eight-point Gauss-Legendre quadrature rule, Shapley values and arbitrary-order interaction values are recovered. The resulting algorithm is numerically more stable than the original recursive TreeSHAP, runs faster on both CPU and GPU, and scales independently of tree depth. These properties make higher-order interaction explanations feasible for large models. The approach is implemented in XGBoost and demonstrated on twelve benchmarks with reported speedups ranging from 1.06x to over 1000x depending on the order of interactions.

Core claim

By expressing the Banzhaf interaction values as expectations under a participation probability p and recovering the corresponding Shapley values through quadrature integration of the resulting polynomial, one obtains a depth-independent computation of Shapley values and any-order interactions that attains machine precision with only eight fixed quadrature nodes and improves numerical stability over direct recursion.

What carries the argument

The weighted-Banzhaf interaction polynomial, which encodes the expectations needed for Banzhaf values under a variable participation probability and permits recovery of Shapley values by a single integral over that probability.

If this is right

Shapley values for tree models can be computed to high precision without recursion over tree depth.
Pairwise and higher-order Shapley interactions become computationally practical for gradient-boosted trees.
CPU and GPU implementations benefit from fixed-point arithmetic without depth-dependent branching.
The method integrates into existing libraries such as XGBoost without changing model training.
Speedups increase with interaction order, reaching three orders of magnitude for high-order terms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the eight-point rule holds across all realistic trees, then production explainers can drop adaptive quadrature entirely.
The polynomial view may allow analytic solutions for trees with limited depth or specific split structures.
Similar quadrature reformulations could be applied to other attribution methods that currently rely on path enumeration.
Faster access to multi-way interactions supports routine auditing of feature dependencies in deployed models.

Load-bearing premise

The weighted-Banzhaf polynomial together with a fixed eight-point quadrature exactly reproduces the expectations required by the path-dependent Shapley definition for any tree structure and any feature distribution.

What would settle it

Run Quadrature-TreeSHAP with eight points and exact TreeSHAP on a deep tree with non-uniform feature values and observe whether the two sets of Shapley values differ by more than floating-point roundoff error.

Figures

Figures reproduced from arXiv: 2605.04497 by Peng Yu, Ron Wettenstein, Rory Mitchell.

**Figure 1.** Figure 1: First-order numerical error on FashionMNIST-sparse vs. max_depth view at source ↗

read the original abstract

Shapley values are a standard tool for explaining predictions of tree ensembles, with Path-Dependent SHAP being the most widely used variant. Despite substantial progress, existing methods still exhibit trade-offs between depth-dependent runtime, numerical stability, and support for higher-order interactions. To address these challenges, we introduce Quadrature-TreeSHAP, a quadrature-based reformulation of Path-Dependent TreeSHAP that is numerically stable, naturally extends to any-order Shapley interaction values and is practically insensitive to tree depth. Our implementation supports both CPU and GPU and is integrated into XGBoost. Our method is based on a weighted-Banzhaf interaction polynomial, which expresses Banzhaf interaction values as expectations under a feature participation probability $p$. Shapley values and any-order interaction values are then recovered by integrating these polynomials over $p$ from 0 to 1. We evaluate these integrals using Gauss-Legendre quadrature, and show that, in practice, only 8 fixed quadrature points are sufficient to reach machine precision. In fact, Quadrature-TreeSHAP with 8 fixed points achieves greater numerical stability than TreeSHAP. This fixed-point formulation removes depth dependence from the inner computation and enables efficient SIMD execution. We confirm these advantages empirically. On 12 XGBoost benchmarks, Quadrature-TreeSHAP computes Shapley values 1.06x-10.59x faster than TreeSHAP on CPU and 1.84x-6.95x faster than GPUTreeSHAP on GPU. Shapley pairwise interactions are 3.80x-58.11x faster on CPU, with higher-order interactions achieving speedups of up to 1200x compared to TreeSHAP-IQ.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Quadrature-TreeSHAP gives a fixed 8-point quadrature on the Banzhaf polynomial that speeds up interaction values and removes explicit depth loops, but the machine-precision claim for arbitrary tree depths is only shown empirically.

read the letter

The main thing to know is that this paper turns Path-Dependent TreeSHAP into an integral over a participation probability p and approximates it with 8 fixed Gauss-Legendre points. That move removes the per-path depth dependence from the inner loop and lets the same code handle any-order interactions without extra machinery. On the 12 XGBoost benchmarks they report, single Shapley values run 1-10x faster on CPU than standard TreeSHAP, pairwise interactions 4-58x faster, and higher-order terms up to 1200x faster than TreeSHAP-IQ, with better numerical stability in their tests. The XGBoost integration and CPU/GPU support are practical bonuses that make the method immediately usable.

Referee Report

1 major / 0 minor

Summary. The manuscript introduces Quadrature-TreeSHAP, a quadrature-based reformulation of Path-Dependent TreeSHAP. It defines a weighted-Banzhaf interaction polynomial that expresses Banzhaf values as expectations under a participation probability p, recovers Shapley values and arbitrary-order interactions by integrating the polynomial over p ∈ [0,1] via fixed 8-point Gauss-Legendre quadrature, and claims that this yields machine-precision results, removes depth dependence from the inner loop, improves numerical stability over TreeSHAP, and delivers speedups of 1.06×–10.59× for Shapley values and up to 1200× for higher-order interactions on 12 XGBoost benchmarks, with both CPU and GPU implementations integrated into XGBoost.

Significance. If the quadrature approximation is shown to be accurate to machine precision for the tree depths and feature distributions encountered in practice, the work would provide a practically useful advance in post-hoc explanation of tree ensembles: a depth-independent algorithm that is faster, more stable, and naturally supports higher-order Shapley interactions. The reported empirical speed and stability numbers on 12 benchmarks constitute concrete evidence of practical benefit.

major comments (1)

[Abstract and quadrature construction] The central claim that 8 fixed Gauss-Legendre points achieve machine precision rests on the integrand being a polynomial of degree ≤15. For any root-to-leaf path of length d the contribution to the integrand contains a term of exact degree d. Gauss-Legendre quadrature with 8 nodes is algebraically exact only up to degree 15; beyond that the error is nonzero. The manuscript asserts machine precision “in practice” on 12 benchmarks but supplies neither an a-priori error bound, an analysis of the leading coefficients, nor the maximum depths of the trees used in those benchmarks. This assumption underpins both the numerical-stability claim and the assertion of depth independence.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful and constructive review. The major comment raises an important point about the theoretical basis for the quadrature precision claim. We address it directly below and will revise the manuscript to incorporate additional supporting information.

read point-by-point responses

Referee: The central claim that 8 fixed Gauss-Legendre points achieve machine precision rests on the integrand being a polynomial of degree ≤15. For any root-to-leaf path of length d the contribution to the integrand contains a term of exact degree d. Gauss-Legendre quadrature with 8 nodes is algebraically exact only up to degree 15; beyond that the error is nonzero. The manuscript asserts machine precision “in practice” on 12 benchmarks but supplies neither an a-priori error bound, an analysis of the leading coefficients, nor the maximum depths of the trees used in those benchmarks. This assumption underpins both the numerical-stability claim and the assertion of depth independence.

Authors: We agree that 8-point Gauss-Legendre quadrature is algebraically exact only for polynomials of degree ≤15 and that paths with d>15 introduce higher-degree terms for which the error is nonzero in principle. Our claim of machine precision is empirical, based on observed relative errors at the level of 1e-16 across the 12 XGBoost benchmarks. In the revised manuscript we will report the maximum tree depths present in those benchmarks and add a short analysis of the quadrature remainder. The leading coefficients of the weighted-Banzhaf polynomial are bounded (products of split probabilities ≤1), which keeps the 16th-derivative term small on the unit interval and explains why the observed error remains at machine epsilon even when d exceeds 15. The depth-independence claim does not rely on algebraic exactness; it follows from replacing the depth-dependent recursion with a fixed loop over the 8 quadrature nodes, each of which can be evaluated in constant time with respect to d. Numerical stability is likewise demonstrated empirically rather than asserted from exactness. We have revised the abstract, the quadrature-construction section, and the experimental section to include the tree-depth statistics and the brief error discussion. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper derives Quadrature-TreeSHAP by expressing Banzhaf interaction values via a weighted polynomial in participation probability p, then recovering Shapley and higher-order values through integration over p using standard Gauss-Legendre quadrature. This rests on the known equivalence between Banzhaf and Shapley values plus fixed-point numerical integration, with no self-definitional loops, no fitted parameters renamed as predictions, and no load-bearing self-citations or uniqueness theorems invoked from prior author work. The 8-point rule and machine-precision claim are presented as empirical observations on benchmarks rather than tautological reductions, leaving the central construction independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach relies on standard mathematical quadrature rules and the algebraic relation between Banzhaf and Shapley values; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)

standard math Gauss-Legendre quadrature with a fixed number of nodes accurately approximates the integral of the Banzhaf polynomial over p in [0,1]
Invoked when the paper states that 8 fixed points reach machine precision.

pith-pipeline@v0.9.0 · 5622 in / 1261 out tokens · 21501 ms · 2026-05-08T16:43:17.613461+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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By the distributive property:(T∪L)∩M(v) = (T∩M(v))∪(L∩M(v))
[62]

SinceT, Sare disjoint:T∩M(v) =T∩(M(v)\S)
[63]

SinceT, Sare disjoint andL⊆S, it follows thatTandLare disjoint
[64]

fv(T∪L) =R v ∅ Y j∈(T∪L)∩M(v) qv j =R v ∅   Y j∈T∩(M(v)\S) qv j     Y j∈L∩M(v) qv j  

SinceT, Lare disjoint,(T∩(M(v)\S))and(L∩M(v))are also disjoint. fv(T∪L) =R v ∅ Y j∈(T∪L)∩M(v) qv j =R v ∅   Y j∈T∩(M(v)\S) qv j     Y j∈L∩M(v) qv j   . Thus, (the setT∩(M(v)\S)is independent ofL): ∆Sfv(T) =R v ∅   Y j∈T∩(M(v)\S) qv j  X L⊆S (−1)|S|−|L| Y j∈L∩M(v) qv j . By Lemma 4 (see below): X L⊆S (−1)|S|−|L| Y j∈L∩M(v) qv j =    Y j∈S ...