Elucidating the Design Space of LM Data Attribution

Steven Yu; Sheng-Yu Wang; Kimia Hamidieh; Antonio Torralba; David Bau

cs.CLOct 29, 2026

Elucidating the Design Space of LM Data Attribution

Authors: Steven Yu, Sheng-Yu Wang, Kimia Hamidieh, Antonio Torralba, David Bau

Organizations: Northeastern University · MIT

Abstract

Data attribution methods for Language Models (LMs) are typically evaluated using downstream tasks like fact retrieval. However, existing evaluations vary multiple dataset properties simultaneously, making it impossible to isolate exactly why an attribution algorithm fails. To diagnose these failures, we explore the design space of LLM data attribution, with a primary focus on evaluation. We introduce the Controlled Attribution Benchmark (CATT) for fact tracing, in which we fine-tune LLMs on entirely fictitious knowledge, providing unambiguous ground truth for attribution. This synthetic setup allows us to systematically alter one dataset configuration at a time, such as document redundancy and lexical distractors. With this configurable evaluation, we demonstrate that redundancy and distractors consistently degrade attribution performance. Rather than proposing a new estimator, we examine two design choices that are underexplored: the loss used to compute gradients and how per-token attribution scores are aggregated into a document score. We find that the true causal evidence within a training document is highly localized, but standard methods dilute this evidence into noise by averaging attribution scores across the entire document. We find that replacing standard cross-entropy loss gradients with the negative log-odds gradients surfaces the critical tokens from other uninformative ones. We can then aggregate only the top token scores to improve ground truth recall by 35 to 72 percentage points across multiple baselines on our diagnostic benchmark. The negative log-odds correction also generalizes to FTRACE-TREX, improving counterfactual retraining performance for every method we test, by up to 4.9x.

Figures & tables

Figure 1: The right loss focuses attribution on relevant tokens. We show token-level influence scores within a ground-truth training document, divided by the document’s largest token score. Cross-entropy assigns little weight to the tokens relevant to answering the query, ranking the document poorly (rank 582/1000). Surprisingly, we find that simply switching to negative log-odds concentrates the influence score on those tokens, and correctly ranks it first. Result is from our Controlled Attribution Benchmark .

Figure 2: CATT: fact tracing tasks with controlled datasets. We fine-tune Qwen3.5-4B models on fictitious knowledge, which ensures our dataset is the sole causal source of its predictions. Left: Task I ( Function Bindings ) teaches new functions that always return a fixed value. Right: Task II ( Routing Rules ) teaches rules that sort objects based on their properties. Given a query phrased unlike any training document, we test whether attribution methods can correctly rank ground-truth documents. Unlike static benchmarks ( Akyürek et al., 2022 ; Jiao et al., 2026 ) , we diagnose performance by systematically controlling dataset properties: (A) introducing redundancy by teaching the same fact across

N

varied document styles, and (B) injecting lexical distractors, which are irrelevant documents that perfectly mimic the query format to expose methods that rely on superficial text overlap. Task II omits distractors because each fact uses a disjoint set of attribute words, making it difficult to construct clear lexical overlaps. Browse the datasets on our project page .

Figure 3: Existing attribution methods struggle with redundancy and distractors. We evaluate attribution methods on CATT. Recall@

N

of each method’s attribution scores as the number of training documents per fact,

N

, increases, at

|\mathcal{F}|=50

facts. On Function Bindings, only TRAK and EK-FAC among the gradient-based methods recover the ground-truth documents, and both lose recall as

N

grows; every other gradient-based method stays below

0.35

. Distractors drive TRAK to 0 at every

N

and lower EK-FAC. On Rule Routings, no gradient-based method exceeds

0.32

. BM25 remains strong thanks to distinctive lexical cues, while RepSim is weak throughout. Results are averaged over three seeds with error bars showing

\pm 2

standard errors (smaller bars are hidden by the markers).

Figure 4: Token selection and negative log-odds loss help isolate ground truth. Document scores for ground-truth (red) and other (gray) documents. When every token is scored under cross-entropy, the two distributions overlap (Recall@

N

0.01). Keeping only each document’s top 10 tokens (0.30) or switching to negative log-odds (0.35) partly separates them, and combining both separates them completely (1.00). This result is from Grad-Dot on a model trained on Function Bindings dataset with distractors

(N{=}50,|\mathcal{F}|=50)

. View these plots for each dataset, method, and model seed on our project page .

Figure 5: The loss determines what type of tokens occupy the top positions. (a) Per-token attributions for one query on one of its ground-truth Rule Routings documents, under cross-entropy (top) and negative log-odds (bottom). Boxes mark the ten highest-scoring positions. Negative log-odds moves the document from rank $1499$ to rank $12$ of $1500$ . (b) Each token type’s share of the top-10 positions divided by its share of all positions. Cross-entropy favors item names. Negative log-odds instead favors outcome tokens, the words that match the query answer. Appendix D.4 shows the same pattern on Function Bindings tasks. Explore per-token attributions for other queries and methods in our demo .

Figure 6: Upweighting more influential tokens with log-odds improves Recall across token budgets. Recall@

N

as the number of retained tokens

k

varies, under cross-entropy (top) and negative log-odds (bottom). Columns show the three CATT settings. The yellow regions mark full-document scoring (“all”), with the upper-right endpoint in each column corresponding to the prior-work loss and aggregation choices. Negative log-odds with small token budgets substantially improve several methods, especially with distractors. Gains persist across multiple

k

values; retaining more tokens often reduces recall. Results are averaged over all dataset configurations and model seeds at

|\mathcal{F}|=50

with error bars showing

\pm 2

standard errors.

Figure 7: Generalization to external FTRACE-TREX benchmarks. Accuracy drop after removing the highest-ranked training documents and retraining, at increasing removal budgets. Larger drops indicate more effective identification of influential data. In each panel, dashed: prior work; light: with negative log-odds; dark: negative log-odds over the top 10 tokens (TRAK’s dense projection has no per-token scores); dotted: BM25. Mean over eight retrain seeds (Appendix E ), error bars

\pm 2

standard errors.

Appendix figures & tables19 assets

Supplementary material from the paper’s appendix.

Appendix

Genre	Description passed to the prompt (“Write …”)
technical reference	a formal technical reference page: precise definition, type signature, domain and range, then a short properties list
practical guide	a practical developer guide that leads with working code, then explains when you would reach for this function
test suite	an annotated unit-test file: assertions with brief comments explaining what each one pins down, plus a short note on coverage
tutorial	a beginner tutorial that builds understanding step by step, with a worked example and a short recap
mathematical	a mathematical treatment using set notation and universal quantification, framing the function as a constant map
REPL session	an annotated REPL transcript where someone explores the function interactively, with prose between the prompts

Appendix

Table 1: The seven genres that fill {style} in Listing 1 .

	Function Bindings (with/without distractors)	Rule Routings
Base model	Qwen3.5-4B + synthetic tokens	Qwen3.5-4B (unmodified)
LoRA rank / $\alpha$ / dropout	64 / 128 / 0.05	same
LoRA targets	all linear layers
LoRA initialization	PEFT default, fixed seed 2000	same
Fully trained modules	embed_tokens , lm_head ; gradient zeroed on the 248,077 original rows	none
Optimizer	AdamW, $\beta=(0.9,0.95)$ , $\epsilon=10^{-8}$	same

Appendix

Table 2: Fine-tuning recipe. “Same” means the Rule Routings value equals the Function Bindings value.

	Function Bindings		+ distractors		Rule Routings
$N$	epochs	steps	epochs	steps	epochs	steps
1	200	1,000	200	2,000	200	1,000
3	–	–	–	–	67	1,005
5	40	1,000	40	2,000	40	1,000
10	20	1,000	20	2,000	20	1,000
20	10	1,000	10	2,000	10	1,000

Appendix

Table 3: Epochs and optimizer steps per cell at

|\mathcal{F}|=50

. Adding distractors doubles the corpus at fixed epochs.

Task	$N$	$\|\mathcal{D}\|$	tokens / fact doc.	training tokens	queries	chance R@ $N$
Function Bindings	1	50	267 [242, 291]	13,362	200	0.02
	5	250	266 [236, 295]	66,592		0.02
	10	500	258 [207, 295]	129,216		0.02
	30	1,500	242 [200, 295]	362,834		0.02
	50	2,500	239 [197, 295]	596,836		0.02
Function Bindings	1	100	267 [242, 291]	13,959	200	0.01

Appendix

Table 4: Training sets at

|\mathcal{F}|=50

|\mathcal{D}|

is the candidate pool an attribution method ranks: fact documents, plus distractors where present. Token counts use the Qwen3.5 tokenizer extended with the added tokens. “Tokens / fact doc.” gives the mean and [min, max] over fact documents. Distractors are about 12 tokens each. The last column is Recall@

N

for a random ranking,

N/|\mathcal{D}|

Figure 8: Fine-tuned models at

|\mathcal{F}|=50

. (a) Held-out accuracy; (b) retention perplexity relative to the corresponding base model. Lines show the mean over seeds 1001–1003 and shading shows the range. The hollow Rule Routings point at

N=1

lies outside the attribution grid. (c–e) Training loss of seed 1002. ∗ The published runs with distractors left no loss log, so (d) shows the checkpointed retrain used by the trajectory methods.

Model	Accuracy	$\Delta$ vs. base
Qwen3.5-4B (base)	49.5 $\pm$ 0.5	–
Function Bindings	31.5 $\pm$ 0.4	$-18.0$
Function Bindings + distractors	19.7 $\pm$ 0.4	$-29.8$
Rule Routings	33.9 $\pm$ 0.4	$-15.5$
Qwen2.5-3B-Instruct	37.2 $\pm$ 0.4	$-12.3$
Qwen2.5-1.5B-Instruct	30.1 $\pm$ 0.4	$-19.4$

Appendix

Table 6: MMLU-Pro accuracy (%,

\pm

binomial standard error) on the full test set, 5-shot answer-letter log-likelihood. Fine-tuned models are at

|\mathcal{F}|=50

N=10

, seed 1001. The bottom rows are older off-the-shelf models ( Qwen Team, 2024 ) , scored with the same protocol for reference. Chance is 11.1%.

Method	Representation	Curvature / kernel	Normalization	Token terms	Job
Grad-Dot	full gradient ( $p{=}1.3{\times}10^{8}$ )	none	none	yes	EK-FAC
Grad-Cos	full gradient	none	global $\ell_{2}$	no	EK-FAC
EK-FAC	full gradient	EK-FAC, CE empirical Fisher; $\delta{=}10^{-12}$	none	yes	EK-FAC
LoGra-Dot	layerwise projection ( $126{,}976$ )	none	none	yes	Layerwise-Proj.
LoGra	layerwise projection	per-module CE Fisher; ridge $10^{-3}$ rel.	none	yes	Layerwise-Proj.
Layerwise TRAK	layerwise projection	training Gram (dual); ridge $10^{-6}$ rel.; $Q$	none	yes	Layerwise-Proj.

Appendix

Table 7: Attribution methods as run. Methods in the same job share a pass over the model, so Section C.2 reports their cost jointly.

		A100-80GB hours per model, by $M$					Sweep as recorded
Job (methods it produces)	Per-token terms	50	500	2,500	5,000	A100 jobs	jobs	GPU-h
EK-FAC, Grad-Dot, Grad-Cos	no	0.43	2.85	(14.9)	(29.6)	23	125	820
	yes	(1.34)	(4.79)	21.7	38.9	6
LoGra, LoGra-Dot, layerwise TRAK, TrackStar	no	0.09	0.38	(1.67)	(3.27)	24	144	266
	yes	0.15	1.02	4.22	6.58	31
TRAK (dense projection, $k{=}4{,}000$ and $126{,}976$ )	–	0.14	0.33	1.61	2.65	45	135	100

Appendix

Table 8: Wall-clock cost per trained model, by job. Each job produces every method listed for it, on both training variants. Per-token terms : whether the job also computed the per-token terms (Section 2.3 ); the published results use the “yes” configuration, and “–” marks jobs without terms. The middle columns give the median over

|\mathcal{F}|=50

jobs with

M

candidate documents that ran on one A100-80GB, pooled over tasks and seeds. Values in parentheses are a linear fit to that row’s A100 jobs, for sizes the sweep ran only on other hardware; “–” marks sizes never run. A100 jobs : the number of jobs behind the row. Sweep as recorded : every job of the sweep, over

|\mathcal{F}|\in\{25,50,100\}

, and their total GPU-hours, on a mix of A100, H100 and H200 GPUs. BM25 runs on CPU.

Figure 9: Compute cost of the attribution methods. (a) Wall-clock hours per trained model against the number of candidate documents

M

, at

|\mathcal{F}|=50

and

|\mathcal{Q}|=200

, for jobs run on one A100-80GB. Points are individual jobs and lines join medians. Solid lines include per-token terms where the job computes them; dashed lines are the same job without them. (b) Total recorded GPU-hours over the whole sweep, stacked by

|\mathcal{F}|

and pooled over tasks, on a mix of A100, H100 and H200 GPUs. Each job produces every method in its group (Table 7 ).

		Minutes
Job	Stage	$M{=}500$	$M{=}2{,}500$	$M{=}2{,}500$
		no terms	no terms	with terms
EK-FAC	Kronecker factors: covariance pass	22	(108)	94
	Eigendecomposition (float64)	5	(5)	5
	Eigenvalue-correction pass	18	(89)	74
	Scoring, CE variant	87	(435)	556

Appendix

Table 9: Per-stage wall-clock of sweep jobs on one A100-80GB, Function Bindings,

|\mathcal{F}|=50

|\mathcal{Q}|=200

. Parenthesized entries are linear estimates from measured no-terms jobs at

M\in\{50,250,500\}

; all other entries are medians over the seeds run on this host. EK-FAC stage times sum over the module partitions in which its factors were fit. Model loading and metric scoring are not shown.

Figure 10: Recall@

N

with log-odds top-10 scoring. The counterpart of Figure 3 . Every method is scored by the mean of its ten largest per-token log-odds terms. Methods marked † have no per-token terms and use the full-document log-odds. Both TRAK variants are drawn without

Q

, so that no method’s score carries a document weight. With

Q

, TRAK falls to about 0 on the distractor task at every

N

: the decoys are fit poorly (

Q\approx 0.17

–

0.20

) and the ground-truth documents are memorized (

Q\approx 0.007

–

0.009

), so

Q

ranks a decoy first for every query. The panel titles and method names follow the main-text figures.

Method	$N{=}1$	$N{=}5$	$N{=}10$	$N{=}30$	$N{=}50$	Mean
Document-level score (variants of Figure 3 )
Grad-Dot	0.01 $\pm$ .01	0.03 $\pm$ .02	0.12 $\pm$ .02	0.09 $\pm$ .01	0.11 $\pm$ .01	0.07
Grad-Cos	0.01 $\pm$ .01	0.03 $\pm$ .02	0.12 $\pm$ .03	0.12 $\pm$ .02	0.13 $\pm$ .01	0.08
EK-FAC	0.70 $\pm$ .02	0.74 $\pm$ .01	0.67 $\pm$ .02	0.37 $\pm$ .02	0.45 $\pm$ .02	0.59
LoGra	0.02 $\pm$ .02	0.05 $\pm$ .01	0.09 $\pm$ .01	0.10 $\pm$ .01	0.12 $\pm$ .01	0.08
LoGra-Dot	0.01 $\pm$ .02	0.03 $\pm$ .01	0.07 $\pm$ .01	0.04 $\pm$ .01	0.04 $\pm$ .01	0.04

Appendix

Table 10: Recall@

N

on Function Bindings , mean

\pm

standard deviation over seeds, at

|\mathcal{F}|=50

. Top block : document-level scores, as in Figure 3 . Bottom block : log-odds top-10 scores, as in Figure 10 ; † : no per-token terms, full-document log-odds. Mean : over

N

. A missing

\pm

means a single seed. “–”: not run.

Method	$N{=}1$	$N{=}5$	$N{=}10$	$N{=}30$	$N{=}50$	Mean
Document-level score (variants of Figure 3 )
Grad-Dot	0.00 $\pm$ .00	0.00 $\pm$ .00	0.00 $\pm$ .00	0.00 $\pm$ .00	0.01 $\pm$ .00	0.00
Grad-Cos	0.01 $\pm$ .01	0.01 $\pm$ .01	0.04 $\pm$ .03	0.07 $\pm$ .01	0.08 $\pm$ .01	0.04
EK-FAC	0.07 $\pm$ .06	0.02 $\pm$ .02	0.01 $\pm$ .01	0.03 $\pm$ .02	0.18 $\pm$ .04	0.06
LoGra	0.00 $\pm$ .00	0.00 $\pm$ .00	0.00 $\pm$ .00	0.00 $\pm$ .00	0.02 $\pm$ .01	0.00
LoGra-Dot	0.00 $\pm$ .00	0.00 $\pm$ .00	0.00 $\pm$ .00	0.00 $\pm$ .00	0.00 $\pm$ .00	0.00

Appendix

Table 11: Recall@

N

on Function Bindings with Distractors . Layout as in Table 10 . The corpus has

2|\mathcal{F}|N

documents, one decoy per ground-truth document.

Method	$N{=}1$	$N{=}3$	$N{=}5$	$N{=}10$	$N{=}30$	$N{=}50$	Mean
Document-level score (variants of Figure 3 )
Grad-Dot	0.02 $\pm$ .00	0.02 $\pm$ .00	0.02 $\pm$ .01	0.02 $\pm$ .01	0.07 $\pm$ .01	0.06 $\pm$ .01	0.04
Grad-Cos	0.02 $\pm$ .00	0.04 $\pm$ .01	0.03 $\pm$ .01	0.03 $\pm$ .00	0.08 $\pm$ .01	0.06 $\pm$ .01	0.05
EK-FAC	0.07 $\pm$ .07	0.26 $\pm$ .04	0.28 $\pm$ .01	0.32 $\pm$ .01	0.05 $\pm$ .02	0.12 $\pm$ .01	0.19
LoGra	0.01 $\pm$ .01	0.05 $\pm$ .01	0.04 $\pm$ .01	0.03 $\pm$ .00	0.07 $\pm$ .01	0.11 $\pm$ .01	0.06
LoGra-Dot	0.01 $\pm$ .00	0.02 $\pm$ .00	0.02 $\pm$ .01	0.02 $\pm$ .00	0.04 $\pm$ .00	0.06 $\pm$ .00	0.03

Appendix

Table 12: Recall@

N

on Rule Routings . Layout as in Table 10 . The

N=1

column lies outside the Rule Routings grid (Appendix B.2 ) and is shown for completeness; the mean is over

N\in\{5,10,30,50\}

	Function Bindings		w/ Distractors		Rule Routings		w/ Distractors, decoy AUC
Method	Doc.	Top-10	Doc.	Top-10	Doc.	Top-10	Doc.	Top-10
Grad-Dot	0.46	1.00	0.39	1.00	0.50	0.85	0.44	1.00
Grad-Cos †	0.46	1.00	0.31	0.91	0.50	0.71	0.30	0.63
EK-FAC	0.69	1.00	0.53	1.00	0.52	0.95	0.45	1.00
LoGra	0.52	0.99	0.50	0.98	0.51	0.79	0.50	1.00
LoGra-Dot	0.48	1.00	0.44	1.00	0.50	0.78	0.48	1.00

Appendix

Table 13: AUC averaged over

N

|\mathcal{F}|=50

, for the document-level score (Doc.) and the log-odds top-10 score (Top-10). The last two columns are decoy AUC on Function Bindings with Distractors: each ground-truth document ranked against the decoys of its own query. † : no per-token terms; the Top-10 column uses the full-document log-odds. TRAK’s Doc. columns include

Q

and its Top-10 columns do not. The retrieval references have one configuration.

		Cross-entropy terms				Neg. log-odds terms
Task	Method	mean	sum	top-3	top-10	mean	sum	mean ${}\times Q$	top-3	top-10
Function Bindings	Grad-Dot	0.07	0.07	0.14	0.12	0.99	0.99	0.98	1.00	1.00
	EK-FAC	0.59	0.59	0.87	0.89	1.00	1.00	1.00	1.00	1.00
	LoGra	0.08	0.08	0.12	0.09	0.89	0.89	0.86	0.93	0.92
	LoGra-Dot	0.04	0.04	0.05	0.06	0.93	0.93	0.90	0.97	0.98
	Layerwise TRAK (no $Q$ )	0.04	0.04	0.03	0.04	0.65	0.64	0.62	0.84	0.87

Appendix

Table 14: Recall@

N

of every scoring variant, averaged over

N

|\mathcal{F}|=50

, for the methods with per-token terms. Top-

k

averages the

k

largest terms under either loss (Section 2.3 ). “mean

{}\times Q

” multiplies the full-document log-odds by

1-\bar{p}

. For Layerwise TRAK, every column except that one is unweighted. TrackStar has no sum columns, because its normalized score is the same under the sum and the mean. Bold: best in the row, within 0.005.

Figure 11: Recall@

N

|\mathcal{F}|=25

. The counterpart of Figure 3 , which is at

|\mathcal{F}|=50

, with the same methods and document-level scores. The panel titles and method names follow the main-text figures.

		$\|\mathcal{F}\|{=}25$		$\|\mathcal{F}\|{=}50$		$\|\mathcal{F}\|{=}100$
Task	Method	Doc.	Top-10	Doc.	Top-10	Doc.	Top-10
Function Bindings	Grad-Dot	0.13	1.00	0.12	1.00	0.12	0.99
	EK-FAC	0.62	1.00	0.67	1.00	0.22	1.00
	LoGra	0.08	0.94	0.09	0.82	0.11	0.78
	TRAK †	0.32	0.32	0.24	0.25	0.16	0.18
	Layerwise TRAK	0.56	0.77	0.61	0.80	0.61	–

Appendix

Table 15: Recall@

N

N=10

for

|\mathcal{F}|\in\{25,50,100\}

, mean over seeds. Doc.: document-level score; Top-10: log-odds top-10 ( † : no per-token terms, full-document log-odds; TRAK’s Top-10 column without

Q

). “–”: not run.

			Drop on the 50 facts			At $B=11{,}522$
Method	Score	Pooling	$B=2{,}000$	$5{,}000$	$11{,}522$	Prop. removed	Same-rel. drop
TRAK	Published	All	$9\pm 3$	$10\pm 2$	$8\pm 2$	13%	4.2
	Neg. log-odds	All	$13\pm 3$	$28\pm 2$	$40\pm 4$	24%	11.7
EK-FAC	Published	All	$13\pm 1$	$25\pm 2$	$30\pm 2$	17%	10.9
	Published	Top 10	$14\pm 2$	$19\pm 3$	$34\pm 2$	17%	11.9
	Neg. log-odds	All	$18\pm 2$	$28\pm 2$	$44\pm 2$	27%	15.3

Appendix

Table 16: FTRACE-TREX remove-and-retrain: drop in top-1 accuracy (percentage points) on the 50 facts at each removal budget, mean over eight retraining seeds

\pm

one standard error. Score : the training-document score as published (TRAK’s log-odds with

Q_{d}

, cross-entropy for EK-FAC and Grad-Dot) or the negative log-odds; Pooling : summed over all positions or the mean of the top 10 per-position terms. The last two columns are at

B=11{,}522

. Prop. removed : mean fraction of each fact’s proponents in the removal set. Same-rel. drop : the accuracy drop on the other 5,797 novel facts that share a relation with one of the 50. On the 2,590 novel facts in other relations, every set moves accuracy by at most 4 points. BM25 and the ground truth use two seeds at

B=2{,}000

and

5{,}000

and three at

11{,}522

. † One seed.

Training Data Attribution

Training data attribution identifies which examples, sources, or groups shape a model’s predictions, capabilities, and failures.

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Data Attribution in Large Language Models via Bidirectional Gradient Optimization

Frédéric Berdoz, Luca A. Lanzendörfer, Kaan Bayraktar, Roger Wattenhofer

ETH Zurich

Gradient-Based Attribution LLM Interpretability Training Data Attribution

Open paper AskPDF arXiv

Large Language Models (LLMs) are increasingly deployed across diverse applications, raising critical questions for governance, accountability, and data provenance. Understanding which training data most influenced a model's output remains a fundamental open problem. We address this challenge through training data attribution (TDA) for auto-regressive LLMs by expanding upon the inverse formulation: How would training data be affected if the model had seen the generated output during training? Our method perturbs the base model using bidirectional gradient optimization (gradient ascent and descent) on a generated text sample and measures the resulting change in loss across training samples. Our framework supports attribution at arbitrary data granularity, enabling both factual and stylistic attribution. We evaluate our method against baselines on pretrained models with known datasets, and show that it outperforms previous work on influence metrics, thereby enhancing model interpretability, an essential requirement for accountable AI systems.

Abstract

Training Data Attribution

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