Uniform Discretized Integrated Gradients: An effective attribution based method for explaining large language models

Ayan Kundu; Swarnava Sinha Roy

REVIEW 4 major objections 3 minor 1 cited by

Uniform Discretized Integrated Gradients: An effective attribution based method for explaining large language models

T0 review · 4 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper proposes UDIG, which computes word attributions by integrating model gradients along a nonlinear path of uniformly spaced, monotonic anchor words, and reports that it outperforms IG and DIG on sentiment classification and…

desk verdict A plausible attribution variant whose main claim is currently confounded by its baseline switch; worth refereeing with an ablation as the condition. read the letter →

arxiv 2412.03886 v1 pith:BFPARLCJ submitted 2024-12-05 cs.CL cs.AI

Swarnava Sinha Roy , Ayan Kundu This is my paper · ORCID

classification cs.CLcs.AI

keywords LargeLanguageModelsExplainableAIInputattributionIntegratedGradientsDiscretizedUniforminterpolationBERTQuestionAnswering

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Integrated Gradients explains a model by accumulating gradients along a straight line from a baseline to the input, but in embedding spaces those intermediate points are not real words, so the gradients can be unrepresentative. This paper proposes Uniform Discretized Integrated Gradients (UDIG), which samples evenly spaced points on the straight line and then replaces each point with a nearby real word, called an anchor, that is made monotonic with respect to the baseline and the target word. UDIG also uses the MASK token as the baseline instead of PAD, because the model output at MASK is closer to zero. On BERT and DistilBERT, across SST2, IMDb, Rotten Tomatoes, and SQuAD, UDIG improves log-odds, comprehensiveness, and sufficiency relative to IG and DIG in almost all comparisons. The paper interprets this as evidence that keeping the interpolation points uniform and bounded yields more accurate attribution scores.

What carries the argument

The central object is the interpolation path itself. Starting from a target word embedding $w$ and a baseline $w_0$ (the MASK token embedding), UDIG places $K$ uniformly spaced points on the line segment, searches each point's neighborhood for candidate words, and selects an anchor by either the Greedy rule (pick the word closest to its monotonic form) or the Max-Count rule (pick the word with the most monotonic dimensions). The selected anchors are then made monotonic dimension-wise, which lets the Riemann sum $\sum_{k} \frac{\partial F(x_{ij}^k)}{\partial x_{ij}} (x_{ij}^{k+1}-x_{ij}^k)$ approximate the path integral in equation (1). Because the attribution score is still a path integral, it satisfies sensitivity, implementation invariance, and completeness; the MASK baseline's near-zero model output is what makes the completeness identity numerically meaningful.

What would settle it

On a model and dataset where the vocabulary is sparse along the baseline-to-target direction, compute UDIG's median delta-percent error and compare attribution rankings with a dense numerical integral; if the delta stays near or above the reported roughly 25 percent even after up-sampling, or if rankings change when the neighborhood size $k$ changes, the claim that the perturbed anchors still track the intended integral would be falsified.

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Extended reading notes

Core claim

The paper's central claim is that attribution quality for language models improves when a path-integral method is both discretized onto real vocabulary words and kept geometrically disciplined. UDIG fixes $K$ evenly spaced points on the straight segment from a baseline embedding to the target word embedding, replaces each point with a nearby anchor word from the vocabulary, and perturbs non-monotonic dimensions so the selected anchors are monotonic with respect to the baseline and target. This keeps the path nonlinear enough to accumulate meaningful word-level gradients while staying close enough to the straight line to avoid the erratic paths that hurt DIG. The paper's empirical claim is that, on BERT and DistilBERT across SST2, IMDb, Rotten Tomatoes, and SQuAD, UDIG yields better log-odds, comprehensiveness, and sufficiency scores than IG and DIG in almost all comparisons, and that its median delta-percent error is consistently lower, supporting the view that uniform, bounded interpolation points make attribution computation more faithful.

Load-bearing premise

The method assumes that the anchor words, after their non-monotonic dimensions are perturbed to enforce monotonicity, stay close enough to the straight line from the baseline to the target that gradients at those anchors approximate the true integral, and the paper offers no worst-case bound on that distance, only median delta percentages.

Editorial extensions

If this is right

If UDIG's uniform, bounded path is the reason for the improvement, then up-sampling the path should reduce the delta-percent error further; the paper reports lower median delta than DIG on every tested model and dataset.
For models that contain a MASK token, the paper recommends using MASK rather than PAD as the attribution baseline because its model output is closer to zero and it behaves less like a learned class-specific token.
For models without a MASK token, such as GPT-style models, the paper says a zero embedding vector or the PAD token is a suitable fallback baseline.
At a fixed computation budget, UDIG needs fewer interpolation steps than IG or DIG to reach the same runtime, and on the Rotten Tomatoes set those fewer steps still improve log-odds and comprehensiveness.
The method is intended to apply to any predictive language model with token embeddings, not only to classification and question answering.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension, not tested in the paper, is to replace the Greedy and Max-Count anchor rules with an anchor choice that explicitly minimizes post-perturbation distance to the straight line; the residual delta error suggests such a rule could push completeness error below the reported medians.
The delta-percent error could serve as a practical screening test for any path-based attribution method, since it quantifies how far a discretized path is from satisfying the completeness identity.
UDIG's largest delta reductions occur on cases where DIG's delta is very high, such as the Rotten Tomatoes BERT setting, which suggests the method has the most to gain in embedding spaces where real words lie sparse and far from linear interpolations.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: UDIG is a real algorithmic variant of DIG, not a rehash. The idea—sample uniform points along the straight line first, then snap to nearby vocabulary anchors and enforce monotonicity—is clean, and the delta-error comparison is a nice diagnostic. But the paper's headline claim, that uniform/bounded interpolation is what drives the improvement, is not supported by the experiments as presented. Section 2.3 switches the baseline from PAD to MASK at the same time and states that MASK gave better metrics. Tables 2–5 therefore vary two things at once. Without an ablation (UDIG with PAD, DIG with MASK), the causal story is untested. The stress-test note is right. What the paper does well: the delta-error tables (6–9) consistently show UDIG below DIG, and these are computed on the completeness gap rather than on the downstream faithfulness metrics, so they are less contaminated by the baseline switch. The time-complexity appendix is candid, and the code is released. Soft spots besides the confound: no error bars or significance tests; the IMDB and SQuAD sampling is described with 'around' and no seed; the delta error is still 15–31% after up-sampling, so completeness is only approximate; and the claim that MASK is a better baseline is plausible but not isolated. The notation in Equation (1) is also a bit garbled, though that is minor. Net: this is a legitimate incremental contribution to token-attribution methods. It does not open a new line of research, but for practitioners comparing IG/DIG/UDIG on a budget it is informative. The central claim should be reworded as 'our combination of uniform anchoring and the MASK baseline gives better results,' pending an ablation. That ablation is straightforward with the released code. Who should read it: people doing explainability for transformer classifiers and QA models, especially those using IG-style path integrals in discrete embedding spaces. It is not for theorists; there is no worst-case guarantee and the delta analysis is empirical. Recommendation: send it to review. The method is plausible, the code exists, and the missing control is fixable. A good referee would ask for the ablation, variance, and detailed sampling. I would not cite it in its current form for the superiority claim, but the delta-error comparison is worth remembering.

Referee Report

4 major / 3 minor

Summary. The paper proposes Uniform Discretized Integrated Gradients (UDIG), a path-based attribution method that modifies Discretized Integrated Gradients (DIG) by placing interpolation points uniformly along the straight line between the target word embedding and a MASK-token baseline, then selecting nearest vocabulary words as anchors and enforcing monotonicity before applying a Riemann-sum approximation. The method is evaluated on binary sentiment classification (SST-2, IMDb, Rotten Tomatoes) using BERT and DistilBERT, and on extractive question answering using SQuAD, with log odds, comprehensiveness, and sufficiency as evaluation metrics. The paper reports that UDIG outperforms IG and DIG on almost all metric/task combinations, also shows lower median delta approximation error, and introduces the MASK token as a baseline recommendation.

Significance. If the reported gains are robust, UDIG is a simple, practical improvement over DIG that offers a useful trade-off between following the straight-line IG path and staying close to actual vocabulary words. The release of code on GitHub is a strength, as is the use of external faithfulness metrics (log odds, comprehensiveness, sufficiency) rather than self-similarity of attributions. The main limitations are the absence of an ablation separating the baseline change from the interpolation change, an unsupported implementation-invariance claim, and insufficient reporting of variance and sampling details. These issues currently prevent the reader from fully attributing the observed improvements to the proposed interpolation strategy.

major comments (4)

[§2.3, Tables 2–5, §5] The comparison between UDIG and DIG varies two factors simultaneously: the interpolation algorithm and the baseline token (MASK in §2.3 versus PAD for DIG). The paper itself notes in §2.3 that the MASK token 'resulted in better performance metrics and word attributions,' and the central conclusion in §5 that 'keeping the interpolation points uniform and bounded yields better results' is therefore not isolated by the experiments. An ablation is needed that runs UDIG with the PAD baseline and DIG with the MASK baseline; without it, the superiority in Tables 2–5 could be due entirely to the baseline change.
[§2.4] The claim that UDIG satisfies Implementation Invariance because it is a path-integral method is not valid as stated. The interpolation path in §2.2 is constructed through nearest-neighbor search in the model's word-embedding space and then by perturbing anchor embeddings to enforce monotonicity. Two functionally equivalent models with different embedding parameterizations, for instance after permuting embedding dimensions, would generally select different anchor words and hence different paths, leading to different attribution scores. The paper should either restrict the axiom claim to Sensitivity and Completeness, or define a class of models under which the path is invariant.
[§4.2, Tables 6–9] The paper claims in §2.4 that UDIG satisfies the Completeness axiom, but the approximation error reported in §4.2 is substantial: median delta error is 25.89% for BERT on SST-2 and 25.95% on Rotten Tomatoes, and several DIG rows exceed 100%. No threshold is given for what counts as acceptable delta error, even though §4.2 says the error 'must be within a threshold.' Since the Riemann-sum approximation in Eq. (2) is the quantity actually used to produce attributions, a high delta error weakens the direct link to the exact path integral and the completeness property. The authors should report the distribution of delta errors, justify why the approximation remains faithful at these error levels, or temper the completeness claim.
[§3.1, §3.2, Tables 2–5] The empirical support is under-specified. For IMDb, §3.1 states that 'around 650 examples' were sampled; for SQuAD, §3.2 says 950 question-answer pairs were sampled 'based on the length of the context,' but the exact counts, sampling procedure, and random seeds are not given. In addition, Tables 2–5 report single point estimates with no variance, confidence intervals, or significance tests. Without these details, it is impossible to assess whether the reported improvements are stable or within noise, and the experiments are not fully reproducible as described.

minor comments (3)

[Title and §2.2] The term 'bounded' in the title and conclusion is never formally defined. The description of selecting anchors 'in the vicinity' of each point lacks a definition of the neighborhood radius or how boundedness is quantified after the monotonicity perturbation.
[Table 1] Table 1 reports model outputs at a single baseline sequence for each length, with no standard errors or repetition. The claim that MASK is 'almost zero' and 'equally unbiased' would be stronger with multiple random sequences and measures of variability.
[Eq. (1)] The typesetting of Eq. (1) is unclear; the integral notation and the definitions of the variables are hard to parse, making it difficult to match Eq. (1) to the Riemann-sum expression in Eq. (2).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: UDIG's attribution formula is a path integral over an explicitly constructed interpolation, and its reported gains are measured against external faithfulness metrics rather than derived from its own inputs.

full rationale

The paper's derivation chain is self-contained. UDIG is defined in Eq. (1) as a path integral of gradients along an interpolated path, and Eq. (2) approximates it by a Riemann sum under an explicit monotonicity constraint. The interpolation algorithm in Section 2.2 constructs anchor words by nearest-neighbor search and monotonicity perturbation, and the baseline choice in Section 2.3 is an empirical model-output observation, not a quantity fitted to the evaluation metrics. The evaluation in Section 4.1 compares against external metrics (Log Odds, Comprehensiveness, Sufficiency) computed from model predictions before and after masking, which are not equivalent to any fitted parameter in UDIG. The up-sampling factor f=1 is selected based on the internal delta approximation error, not on the reported faithfulness metrics, so it is not a fitted input renamed as a prediction. No load-bearing self-citation appears: the cited DIG and IG methods are prior external work. The skeptic's concern that the MASK baseline change is confounded with the uniform-bounded interpolation change is a valid experimental-design criticism, but it is not circularity: the paper does not define or derive its success measure from its construction. Therefore the circularity burden is not met under the stated rules.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The attribution integral inherits the axioms of path-based methods, but the specific anchor selection heuristics and the MASK baseline choice are introduced ad hoc. No fitted parameters are used to construct the attribution formula itself; the evaluation metrics are external faithfulness measures. However, several implementation choices (K, k, similarity metric, f) are under-specified and may influence the comparison.

free parameters (5)

K (number of interpolation steps) = Not specified for each experiment; appendix shows K=10, 30, 60
The number of steps between baseline and input directly controls the Riemann sum approximation; the paper does not report the exact K used for each dataset in the main tables.
k (neighborhood size for nearest-word search)
The number of nearest words considered when selecting anchor words is not given, but it affects which anchor is chosen and therefore the attribution path.
Similarity metric for nearest-word search
The paper says 'a similarity score defined between word embeddings' but does not specify cosine or Euclidean, which can change the anchor selection.
f (up-sampling factor) = 1
The authors state 'we have suitably chosen f = 1 for a low delta error percentage', making it a hyperparameter tuned against the internal completeness error.
Baseline token = MASK
MASK is chosen over PAD because it yields near-zero model outputs at higher sequence lengths, but this is a design choice rather than a fitted constant.

assumptions (4)

standard math Riemann summation of the gradient path is a valid approximation when the interpolated points are monotonic (Section 2.1).
The completeness axiom and the integral approximation rely on monotonicity; this is a standard assumption for Riemann sums but must hold for every dimension of every anchor.
domain assumption The language model is differentiable with respect to all token embeddings along the chosen path (Section 2).
Gradient-based attribution requires well-defined gradients at the anchor points; subword tokenization and embedding lookup must be treated as continuous.
ad hoc to paper Nearest-word search with the chosen similarity metric identifies anchors that are faithful to the linear path (Section 2.2).
The paper does not justify a specific distance measure or neighborhood size, yet the quality of the anchor path depends on it.
ad hoc to paper The MASK token is an appropriate near-zero baseline because its model output is close to zero at higher sequence lengths (Section 2.3, Table 1).
Table 1 shows MASK outputs near 0.05-0.08 for lengths 25-100, but 1.36 for length 1, so the near-zero property is sequence-length-dependent.

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Cite this review

Pith. "Pith review of Uniform Discretized Integrated Gradients: An effective attribution based method for explaining large language models." pith.science (2026). https://pith.science/paper/BFPARLCJ

@misc{pith2026241203886,
  author       = {Pith},
  title        = {Pith review of: Uniform Discretized Integrated Gradients: An effective attribution based method for explaining large language models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BFPARLCJ}},
  note         = {Machine review of arXiv:2412.03886}
}

read the original abstract

Integrated Gradients is a well-known technique for explaining deep learning models. It calculates feature importance scores by employing a gradient based approach computing gradients of the model output with respect to input features and accumulating them along a linear path. While this works well for continuous features spaces, it may not be the most optimal way to deal with discrete spaces like word embeddings. For interpreting LLMs (Large Language Models), there exists a need for a non-linear path where intermediate points, whose gradients are to be computed, lie close to actual words in the embedding space. In this paper, we propose a method called Uniform Discretized Integrated Gradients (UDIG) based on a new interpolation strategy where we choose a favorable nonlinear path for computing attribution scores suitable for predictive language models. We evaluate our method on two types of NLP tasks- Sentiment Classification and Question Answering against three metrics viz Log odds, Comprehensiveness and Sufficiency. For sentiment classification, we have used the SST2, IMDb and Rotten Tomatoes datasets for benchmarking and for Question Answering, we have used the fine-tuned BERT model on SQuAD dataset. Our approach outperforms the existing methods in almost all the metrics.

Figures

Figures reproduced from arXiv: 2412.03886 by the authors.

**Figure 3.** Visualization of explanations by IG, DIG and UDIG for predicting the start and end position of the [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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Reference graph

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Reviewed August 11, 2026 · model on record in the stance chip above.

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