arxiv: 2604.14338 · v1 · submitted 2026-04-15 · 💻 cs.LG · stat.ML

Recognition: unknown

Path-Sampled Integrated Gradients

Firuz Kamalov , Fadi Thabtah , R. Sivaraj , Neda Abdelhamid

Authors on Pith no claims yet

Pith reviewed 2026-05-10 13:27 UTC · model grok-4.3

classification 💻 cs.LG stat.ML

keywords integrated gradientsfeature attributionpath samplingvariance reductionconvergence rateRiemann sumexplainable AI

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The pith

Path-sampled integrated gradients equal a weighted integral form when the weights match the sampling density's cumulative distribution function, turning a stochastic method into a faster deterministic one.

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

The paper presents path-sampled integrated gradients as a way to compute feature attributions by averaging gradients over points sampled along the straight line connecting a baseline input to the actual input. It shows that this averaging equals the path-weighted integrated gradients integral precisely when the weighting function is the cumulative distribution function of the sampling density. That match lets the random average be replaced by an evenly spaced sum, which cuts the approximation error from order one over square root of m to order one over m for smooth models. The same construction also lowers the variance of the resulting attribution scores by a factor of one third under uniform sampling. The approach keeps the standard properties of linearity and implementation invariance that make attributions consistent across models.

Core claim

Path-sampled integrated gradients computes attributions as the expected value of the gradient along the interpolation path times the input difference, with the expectation taken over baselines drawn from a chosen density. When the weighting function is set to the cumulative distribution function of that density, the expectation is identical to the path-weighted integrated gradients. The equivalence converts the Monte Carlo estimate into a deterministic Riemann sum whose error converges at rate O(m^{-1}) for differentiable models. Under uniform sampling the construction reduces attribution variance by exactly one third relative to standard integrated gradients while preserving linearity and a

What carries the argument

The equivalence between the path-sampled expectation and path-weighted integrated gradients, realized by setting the weight function equal to the cumulative distribution function of the sampling density.

If this is right

Attribution scores can be obtained from a fixed grid of points along the path instead of random samples, giving quadratic improvement in convergence speed.
The variance of the attribution vector is strictly lower by a factor of one third when uniform sampling is used.
Linearity and implementation invariance continue to hold, so the new scores remain consistent with standard axioms.
The method applies directly to any differentiable model without requiring changes to the underlying gradient computation.

Where Pith is reading between the lines

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

The same weighting-to-density match might be applied to other integral-based attribution methods to obtain similar convergence gains.
In settings with noisy gradients the built-in variance reduction could produce more stable explanations without additional smoothing steps.
Because the approach is deterministic once the grid is chosen, it may simplify reproducibility checks across different implementations.
The technique could be tested on non-uniform sampling densities to see whether the variance reduction factor changes in predictable ways.

Load-bearing premise

The weighting function must be set exactly to the cumulative distribution function of the sampling density and the model must be differentiable at every point along the straight-line path.

What would settle it

For a smooth model such as a linear classifier, compute the attribution error using m evenly spaced points and check whether the error shrinks proportionally to 1/m rather than 1 over square root of m as m grows.

Figures

Figures reproduced from arXiv: 2604.14338 by Fadi Thabtah, Firuz Kamalov, Neda Abdelhamid, R. Sivaraj.

read the original abstract

We introduce path-sampled integrated gradients (PS-IG), a framework that generalizes feature attribution by computing the expected value over baselines sampled along the linear interpolation path. We prove that PS-IG is mathematically equivalent to path-weighted integrated gradients, provided the weighting function matches the cumulative distribution function of the sampling density. This equivalence allows the stochastic expectation to be evaluated via a deterministic Riemann sum, improving the error convergence rate from $O(m^{-1/2})$ to $O(m^{-1})$ for smooth models. Furthermore, we demonstrate analytically that PS-IG functions as a variance-reducing filter against gradient noise - strictly lowering attribution variance by a factor of 1/3 under uniform sampling - while preserving key axiomatic properties such as linearity and implementation invariance.

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

Path sampling reformulates IG as an expectation that converts to a Riemann sum with O(1/m) convergence and a claimed 1/3 variance cut under uniform sampling when weights match the CDF.

read the letter

The main thing to know about this paper is that it introduces path-sampled integrated gradients by taking an expectation over baselines sampled on the interpolation path, then proves equivalence to a weighted IG when the weight function is the CDF of the sampling density. This lets them use a deterministic Riemann sum for faster convergence and claims an analytical variance reduction by a factor of one third under uniform sampling. The error rate improves from the typical Monte Carlo O(m^{-1/2}) to O(m^{-1}) for smooth models. They do a good job deriving everything from standard properties of expectations and integrals without any circular steps. The preservation of the axiomatic properties is checked explicitly, which is solid. The soft spots are the assumptions that the model must be smooth along the path and that the weighting must match the CDF precisely; deviations would invalidate the convergence and variance results. Without any experiments in the abstract, it's unclear how these theoretical gains translate to real models with noisy or non-smooth gradients. The variance reduction is also tied specifically to uniform sampling. This paper is for specialists in machine learning interpretability who are interested in refining attribution techniques. A reader working on similar methods would get some value from the technical details and proofs. It deserves serious peer review because the claims are mathematically grounded and the improvements are quantifiable. I would recommend sending it to referees.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces path-sampled integrated gradients (PS-IG), a generalization of feature attribution that computes the expected value of attributions over baselines sampled along the linear interpolation path from a baseline to the input. It proves that PS-IG is mathematically equivalent to path-weighted integrated gradients provided the weighting function equals the cumulative distribution function of the sampling density. This equivalence is used to replace stochastic Monte Carlo estimation with a deterministic Riemann sum, yielding an improved error convergence rate of O(m^{-1}) versus O(m^{-1/2}) for smooth models. The paper further derives an analytical result that PS-IG acts as a variance-reducing filter, lowering attribution variance by a factor of exactly 1/3 under uniform sampling, while preserving linearity and implementation invariance.

Significance. If the stated equivalence and variance-reduction derivations hold, the work supplies a theoretically grounded mechanism for improving both the statistical efficiency and convergence properties of integrated-gradients attributions. The deterministic Riemann-sum reformulation and the explicit 1/3 variance factor under uniform sampling are potentially useful for practitioners working with noisy gradients in high-dimensional models. The explicit conditioning on model smoothness and exact CDF matching is clearly stated, which aids reproducibility.

major comments (2)

[§3] The central equivalence (abstract and §3) is derived from standard properties of expectation and Riemann sums; however, the manuscript should explicitly verify that the path-weighted formulation remains well-defined when the sampling density is non-uniform, because any mismatch between the weighting function and the CDF immediately invalidates both the O(m^{-1}) convergence claim and the variance-reduction factor.
[§4] The analytical variance calculation yielding a strict factor of 1/3 (abstract) assumes a specific noise model on the gradients along the path; the derivation should be expanded to show the precise noise assumptions (e.g., uncorrelated additive noise) and to confirm that the factor remains 1/3 only under uniform sampling, as stated in the weakest assumption.

minor comments (2)

Clarify the minimal differentiability requirement (C^1 versus C^2) needed for the O(m^{-1}) convergence rate to hold uniformly along the entire interpolation path.
The abstract claims preservation of 'key axiomatic properties'; a short table or paragraph explicitly listing which axioms (linearity, implementation invariance, etc.) are retained and which are not would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address each major comment point by point below, with revisions incorporated where appropriate to strengthen the presentation.

read point-by-point responses

Referee: [§3] The central equivalence (abstract and §3) is derived from standard properties of expectation and Riemann sums; however, the manuscript should explicitly verify that the path-weighted formulation remains well-defined when the sampling density is non-uniform, because any mismatch between the weighting function and the CDF immediately invalidates both the O(m^{-1}) convergence claim and the variance-reduction factor.

Authors: We agree that an explicit verification strengthens the section. In the revised manuscript, we have added a dedicated paragraph in §3 confirming that the path-weighted integrated gradients formulation is well-defined for arbitrary (including non-uniform) sampling densities, as long as the weighting function is exactly the CDF of that density. This is shown by direct substitution into the expectation operator, preserving the equivalence, the O(m^{-1}) Riemann-sum convergence for smooth models, and the validity of the variance claims under the matching condition. We also briefly note the consequences of a mismatch to highlight the necessity of CDF alignment. revision: yes
Referee: [§4] The analytical variance calculation yielding a strict factor of 1/3 (abstract) assumes a specific noise model on the gradients along the path; the derivation should be expanded to show the precise noise assumptions (e.g., uncorrelated additive noise) and to confirm that the factor remains 1/3 only under uniform sampling, as stated in the weakest assumption.

Authors: We appreciate the request for greater precision. The derivation in §4 assumes uncorrelated additive noise on the path gradients (i.e., the noise terms at distinct path points are independent with zero cross-covariance). Under this model, the variance reduction factor is exactly 1/3 only for uniform sampling; for non-uniform densities the factor is a different function of the sampling distribution. In the revision we have expanded the derivation to state these assumptions explicitly, included the general expression for the variance factor under arbitrary sampling, and reiterated that the 1/3 result holds specifically for the uniform case as claimed in the abstract. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from first principles

full rationale

The paper defines PS-IG as the expected attribution over path-sampled baselines and proves its equivalence to path-weighted IG precisely when the weighting function equals the CDF of the sampling density. This equivalence is obtained directly from the definition of expectation and the fundamental theorem of calculus, permitting replacement of the stochastic integral by a deterministic Riemann sum whose error rate improves from O(m^{-1/2}) to O(m^{-1}) under smoothness. The subsequent variance-reduction claim (factor of 1/3 under uniform sampling) follows analytically from the same noise model and linearity of expectation without any fitted parameters or self-referential definitions. No load-bearing self-citations, imported uniqueness theorems, or ansatzes appear in the derivation chain; all steps are conditioned on explicitly stated assumptions (differentiability along the path and exact CDF matching) that are independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard mathematical assumptions about differentiability and the ability to match a weighting function to a sampling density; no free parameters are introduced and no new entities are postulated.

axioms (2)

domain assumption The model output is differentiable along the linear interpolation path from baseline to input
Required for the gradient to be defined at every point on the path so that integration and sampling are valid.
domain assumption A sampling density exists whose cumulative distribution function can be exactly matched by a weighting function
This matching is the key condition stated for the equivalence between the stochastic expectation and the deterministic Riemann sum.

pith-pipeline@v0.9.0 · 5428 in / 1549 out tokens · 79341 ms · 2026-05-10T13:27:14.562068+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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