Quantifying the Agreement Between Data-Influence and Data-Similarity to Understand LLM Behavior

Abstract: One way to understand LLM behavior is to trace its output back to the training data. Two types of measures are commonly used for output tracing: data-similarity and data-influence. The former is cheaper while the latter is believed to be more accurate. Even though many works have compared them for ground-truth tasks, no such comparisons exist for output tracing. Here, we fill this gap and precisely quantify the commonalities and differences between the two measures. We do this by first ranking the training documents according to each measure and then computing the overlap between the two rankings. Our main finding is that the two rankings agree significantly, but there is an asymmetry between them: The top documents of data-similarity are assigned more consistent ranks by data-influence than the other way around. This result is valid across a range of experiments involving OLMo2-1B, Qwen3-1.7B, LlaMa3.2-1B, Gemma3-1B, and GPT2. We exploit the asymmetry to obtain a favorable cost-accuracy trade-off by using the costly data-influence to refine the results of data-similarity.

• The paper demonstrates that data-similarity (BM25) and data-influence (gradient/Hessian) methods share roughly 11% overlap in top-100 rankings, highlighting a key divergence in tracing LLM outputs.
• It reveals an asymmetric relationship where similarity rankings are more consistently predicted by influence methods than the reverse, guiding computational resource prioritization.
• A hybrid strategy using BM25 pre-filtering followed by EK-FAC influence-based refinement achieves an AUC of 0.83, showcasing a scalable and cost-effective approach to LLM output tracing.

Quantifying the Agreement Between Data-Influence and Data-Similarity for LLM Output Tracing

Motivation and Framework

The paper addresses a foundational question in LLM interpretability: when tracing LLM outputs back to the training data, how closely do rankings obtained by data-similarity methods (e.g., BM25 n-gram matching) align with those obtained via data-influence (e.g., gradient or Hessian-based influence functions)? Data-similarity measures are resource-efficient but potentially less accurate; data-influence is computationally expensive but often considered the gold standard. Previous comparative studies focused on ground-truth attribution tasks, not on output tracing where absolute ground-truth is absent. This work fills that gap by formalizing, quantifying, and analyzing the overlap and divergences of these rankings, yielding empirical insights across OLMo2-1B, Qwen3-1.7B, LlaMa3.2-1B, Gemma3-1B, and GPT2 variants.

Methodology

The authors formalize two ranking paradigms:

• Data-Similarity: BM25-based ranking, utilizing token frequency-based feature representations without model outputs. No embedding-based methods considered due to computational expense.
• Data-Influence: Counterfactual influence as $I(i,j)=|f_i(\theta)-f_i(\theta_j)|$, with $\theta_j$ denoting LLM parameters after removal (or reweighting) of the $j$th training document. Practical computation leverages gradient-based (TracIn) and Hessian-based (EK-FAC) approximations.

100 diverse prompts are generated per model, each traced back to 100,000 training documents, yielding a pair of rankings for each prompt. Agreement metrics and rank-biased overlap [Webber et al., 2010] are used to quantify the intersection at various depths. Asymmetry is analyzed by measuring how well one ranking predicts the top entries of the other and via paired statistical tests.

Empirical Results

Agreement: Rank Overlap and Consistency

Across all models, the top-100 rankings from data-similarity and data-influence share approximately 11% overlap. The agreement at deeper ranks (top-1000, top-20,000) shows both measures capture related corpus regions but with significant divergence beyond the top tier. Agreement increases with improved influence estimators: gradient-only yields 1% overlap, Hessian-based EK-FAC achieves 11%.

Asymmetry: Directional Predictiveness

The asymmetry is a key empirical finding:

• The top documents from data-similarity are ranked more consistently by data-influence than vice versa.
• For OLMo2-1B, 34% of the data-similarity top-100 are within the data-influence top-1000; only 28% conversely.
• Across all considered models, data-influence rankings are more predictive of the data-similarity top selections than the reverse, statistically significant across depths except at symmetry points.

Cost-Accuracy Trade-Off

Proof-of-concept experiments reveal the practical utility of hybrid approaches. Pre-filtering candidates using BM25 data-similarity and then refining with EK-FAC data-influence yields an AUC of 0.83 in ROC evaluation. The inverse (influence first, similarity second) fares worse (AUC 0.63), reinforcing asymmetry and supporting the cost-effective application of influence refinement after similarity search.

Model and Estimator Robustness

Results are robust across LLMs and influence estimators. Larger models (up to 1.7B parameters) exhibit similar trends. Better influence approximations (full Hessian vs. diagonal vs. gradient) yield higher agreement and consistent asymmetry.

Implications and Outlook

Practical Implications

For LLM tracing in production systems such as OLMoTrace, the paper substantiates the use of data-similarity as a scalable first pass but recommends influence-based refinement for higher fidelity. The identified asymmetry provides a principled heuristic for prioritizing computational resources: expensive influence estimation is most impactful when deployed post similarity-filtering.

Theoretical Implications

The findings delineate the boundaries and interplay between statistical IR metrics (e.g., BM25) and parametric sensitivity metrics (influence functions) with respect to generative LLM behavior. The observed asymmetry reflects the fact that influence functions encode both data and parameterization, while similarity metrics are strictly corpus-based. The authors suggest this could inform dynamic routing of queries to tracing methods based on prompt topology.

Limitations and Future Directions

Current experiments are bottlenecked by computational resources, constraining prompt/document scale. Extending to 30B+ parameter models and billion-document corpora is feasible given modern accelerators but remains to be demonstrated. Topic-balanced prompt distributions and other similarity metrics (e.g., semantic embeddings, InfiniGram) are open avenues, as is automated routing between similarity and influence tracing.

Conclusion

This paper rigorously quantifies the complementary and asymmetric relationship between data-similarity and data-influence methods for tracing LLM outputs to training data (2606.23591). Empirical results establish significant but non-total agreement, with asymmetry favoring similarity-to-influence prediction. Hybrid tracing strategies leveraging similarity for candidate selection and influence for final ranking are supported both empirically and theoretically. The work encourages further study on scaling, topic conditioning, and metric diversification, enabling more interpretable and efficient LLM output attribution frameworks.