# Scaling Influence Functions in LLMs through Eigenbasis-Corrected One-Bit Gradient Projection

arXiv: https://arxiv.org/abs/2609.37842 · HTML: https://mlsanity.com/paper/2609.37842/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 34 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Builds on (most cited first, 28 of 28)

- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [Understanding Black-box Predictions via Influence Functions](https://mlsanity.com/paper/1703.04730/) (2017)
- [Tulu 3: Pushing Frontiers in Open Language Model Post-Training](https://mlsanity.com/paper/2411.15124/) (2024)
- [XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks](https://mlsanity.com/paper/1603.05279/) (2016)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [Estimating Training Data Influence by Tracing Gradient Descent](https://mlsanity.com/paper/2002.08484/) (2020)
- [LESS: Selecting Influential Data for Targeted Instruction Tuning](https://mlsanity.com/paper/2402.04333/) (2024)
- [New insights and perspectives on the natural gradient method](https://mlsanity.com/paper/1412.1193/) (2014)
- [2 OLMo 2 Furious](https://mlsanity.com/paper/2501.00656/) (2024)
- [Studying Large Language Model Generalization with Influence Functions](https://mlsanity.com/paper/2308.03296/) (2023)
- [TRAK: Attributing Model Behavior at Scale](https://mlsanity.com/paper/2303.14186/) (2023)
- [Fast Approximate Natural Gradient Descent in a Kronecker-factored Eigenbasis](https://mlsanity.com/paper/1806.03884/) (2018)
- [If Influence Functions are the Answer, Then What is the Question?](https://mlsanity.com/paper/2209.05364/) (2022)
- [Scaling Up Influence Functions](https://mlsanity.com/paper/2112.03052/) (2021)
- [DataInf: Efficiently Estimating Data Influence in LoRA-tuned LLMs and Diffusion Models](https://mlsanity.com/paper/2310.00902/) (2023)
- [FastIF: Scalable Influence Functions for Efficient Model Interpretation and Debugging](https://mlsanity.com/paper/2012.15781/) (2020)
- [What is Your Data Worth to GPT? LLM-Scale Data Valuation with Influence Functions](https://mlsanity.com/paper/2405.13954/) (2024)
- [Scalable Influence and Fact Tracing for Large Language Model Pretraining](https://mlsanity.com/paper/2410.17413/) (2024)
- [Procedural Knowledge in Pretraining Drives Reasoning in Large Language Models](https://mlsanity.com/paper/2411.12580/) (2024)
- [Efficient Sketches for Training Data Attribution and Studying the Loss Landscape](https://mlsanity.com/paper/2402.03994/) (2024)
- [Influence-Preserving Proxies for Gradient-Based Data Selection in LLM Fine-tuning](https://mlsanity.com/paper/2602.17835/) (2026)
- [Correcting Large Language Model Behavior via Influence Function](https://mlsanity.com/paper/2412.16451/) (2024)
- [GraSS: Scalable Data Attribution with Gradient Sparsification and Sparse Projection](https://mlsanity.com/paper/2505.18976/) (2025)
- [Which Data Attributes Stimulate Math and Code Reasoning? An Investigation via Influence Functions](https://mlsanity.com/paper/2505.19949/) (2025)
- [IF-GUIDE: Influence Function-Guided Detoxification of LLMs](https://mlsanity.com/paper/2506.01790/) (2025)
- [LoRIF: Low-Rank Influence Functions for Scalable Training Data Attribution](https://mlsanity.com/paper/2601.21929/) (2026)
- [QLESS: A Quantized Approach for Data Valuation and Selection in Large Language Model Fine-Tuning](https://mlsanity.com/paper/2502.01703/) (2025)
- [Interaction-Aware Influence Functions for Group Attribution](https://mlsanity.com/paper/2605.15675/) (2026)

## Abstract (arXiv)

Influence functions estimate how individual training examples affect the behavior of large language models (LLMs). Analyzing how training data influence different behaviors of an LLM involves repeated influence computation. Reusing stored training gradients reduces the computational cost, but storing full gradients is prohibitively expensive at LLM scale. We study how to compress these gradients while preserving influence estimates for future queries that are unknown at storage time. Through a worst-case analysis, we characterize the optimal fixed-dimensional linear representation and propose eigenbasis-corrected one-bit gradient projection (EOGP) to approximate it at scale. Specifically, EOGP uses EK-FAC to reduce gradient dimensionality, then applies PCA within the retained subspace to learn compression directions from the training gradients. We then apply one-bit quantization to the resulting coordinates, allowing more coordinates to be retained within a fixed storage budget. On GPT-2, EOGP predicts retraining outcomes more accurately than the evaluated compression baselines while using one-sixteenth of their per-example storage. On OLMo 2 SFT models from 1B to 32B parameters, EOGP remains competitive with the baselines allocated over 100 times as much storage per example.