# Data Difficulty and the Generalization--Extrapolation Tradeoff in LLM Fine-Tuning

arXiv: https://arxiv.org/abs/2605.12906 · HTML: https://mlsanity.com/paper/2605.12906/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 30 references that are in the map.
- First posted on arXiv 13 May 2026.

## Builds on (most cited first, 26 of 26)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [LlamaFactory: Unified Efficient Fine-Tuning of 100+ Language Models](https://mlsanity.com/paper/2403.13372/) (2024)
- [Solving Quantitative Reasoning Problems with Language Models](https://mlsanity.com/paper/2206.14858/) (2022)
- [s1: Simple test-time scaling](https://mlsanity.com/paper/2501.19393/) (2025)
- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [LESS: Selecting Influential Data for Targeted Instruction Tuning](https://mlsanity.com/paper/2402.04333/) (2024)
- [LIMO: Less is More for Reasoning](https://mlsanity.com/paper/2502.03387/) (2025)
- [User-friendly introduction to PAC-Bayes bounds](https://mlsanity.com/paper/2110.11216/) (2021)
- [Task2Vec: Task Embedding for Meta-Learning](https://mlsanity.com/paper/1902.03545/) (2019)
- [AIMO-2 Winning Solution: Building State-of-the-Art Mathematical Reasoning Models with OpenMathReasoning dataset](https://mlsanity.com/paper/2504.16891/) (2025)
- [On the Generalization of SFT: A Reinforcement Learning Perspective with Reward Rectification](https://mlsanity.com/paper/2508.05629/) (2025)
- [Physics of Language Models: Part 2.1, Grade-School Math and the Hidden Reasoning Process](https://mlsanity.com/paper/2407.20311/) (2024)
- [The Best Instruction-Tuning Data are Those That Fit](https://mlsanity.com/paper/2502.04194/) (2025)
- [Long-Short Chain-of-Thought Mixture Supervised Fine-Tuning Eliciting Efficient Reasoning in Large Language Models](https://mlsanity.com/paper/2505.03469/) (2025)
- [The Information Complexity of Learning Tasks, their Structure and their Distance](https://mlsanity.com/paper/1904.03292/) (2019)
- [Improving Generalization by Controlling Label-Noise Information in Neural Network Weights](https://mlsanity.com/paper/2002.07933/) (2020)
- [KL Guided Domain Adaptation](https://mlsanity.com/paper/2106.07780/) (2021)
- [BERTIN: Efficient Pre-Training of a Spanish Language Model using Perplexity Sampling](https://mlsanity.com/paper/2207.06814/) (2022)
- [SFT Doesn't Always Hurt General Capabilities: Revisiting Domain-Specific Fine-Tuning in LLMs](https://mlsanity.com/paper/2509.20758/) (2025)
- [Rethinking Data Selection at Scale: Random Selection is Almost All You Need](https://mlsanity.com/paper/2410.09335/) (2024)
- [Beyond Log Likelihood: Probability-Based Objectives for Supervised Fine-Tuning across the Model Capability Continuum](https://mlsanity.com/paper/2510.00526/) (2025)
- [I Learn Better If You Speak My Language: Understanding the Superior Performance of Fine-Tuning Large Language Models with LLM-Generated Responses](https://mlsanity.com/paper/2402.11192/) (2024)
- [Difficulty-Based Preference Data Selection by DPO Implicit Reward Gap](https://mlsanity.com/paper/2508.04149/) (2025)
- [AdaptiveLLM: A Framework for Selecting Optimal Cost-Efficient LLM for Code-Generation Based on CoT Length](https://mlsanity.com/paper/2506.10525/) (2025)

## Abstract (arXiv)

Data selection during supervised fine-tuning (SFT) can critically change the behavior of large language models (LLMs). Although existing work has studied the effect of selecting data based on heuristics such as perplexity, difficulty, or length, the reported findings are often inconsistent or context-dependent. In this work, we systematically study the role of data difficulty in fine-tuning from both empirical and theoretical perspectives, and find that there is no universally optimal difficulty level; rather, its effectiveness depends on the dataset size. We show that for a fixed data budget, there exists an optimal data difficulty for SFT, and that this optimal difficulty shifts toward harder data as the data budget increases. To explain this phenomenon, we conduct controlled synthetic experiments that reveal a simple underlying mechanism: the interplay between the (in-distribution) generalization gap and the extrapolation gap. We further support this mechanism through a theoretical analysis using PAC-Bayesian generalization bounds. Overall, our results clarify how data size and difficulty jointly affect the trade-off between generalization and extrapolation in SFT, providing guidance for difficulty-based data selection under certain model and data conditions.