# Overcoming Prior Barriers: Supervised Fine-Tuning under Long-Tail Distribution

arXiv: https://arxiv.org/abs/2610.12345 · HTML: https://mlsanity.com/paper/2610.12345/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 32 references that are in the map.
- First posted on arXiv 8 Oct 2026.

## Builds on (most cited first, 11 of 11)

- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [Mistral 7B](https://mlsanity.com/paper/2310.06825/) (2023)
- [MAmmoTH: Building Math Generalist Models through Hybrid Instruction Tuning](https://mlsanity.com/paper/2309.05653/) (2023)
- [What Makes Good Data for Alignment? A Comprehensive Study of Automatic Data Selection in Instruction Tuning](https://mlsanity.com/paper/2312.15685/) (2023)
- [Skill-it! A Data-Driven Skills Framework for Understanding and Training Language Models](https://mlsanity.com/paper/2307.14430/) (2023)
- [Large-Scale Data Selection for Instruction Tuning](https://mlsanity.com/paper/2503.01807/) (2025)
- [Improving Influence-based Instruction Tuning Data Selection for Balanced Learning of Diverse Capabilities](https://mlsanity.com/paper/2501.12147/) (2025)
- [AlpaGasus: Training A Better Alpaca with Fewer Data](https://mlsanity.com/paper/2307.08701/) (2023)

## Abstract (arXiv)

Supervised fine-tuning (SFT) adapts pretrained large language models (LLMs) to downstream tasks, but the required concepts can receive substantially different levels of pretrained support. Frequent concepts are more likely to be well learned, whereas rare concepts may remain weakly represented. We introduce a novel notion named prior barrier to quantify how strongly the pretrained model supports competing concepts over the target concept. We observe that prior barriers follow a long-tail distribution, placing head and tail concepts at different starting points for SFT: head concepts face lower prior barriers, whereas tail concepts require additional instructions to overcome their higher prior barriers. Our theoretical analysis further derives a predictive risk bound for SFT under long-tail prior barriers, explicitly characterizing how the prior barrier and accumulated SFT evidence jointly determine predictive performance. Motivated by this prior barrier-dependent demand, we propose PASS, an adaptive SFT instruction selection method that constructs reference-derived concepts and estimates the distinguishing evidence provided by each instruction, and adaptively allocates the selection budget toward concepts that remain insufficiently covered under the current selection. In this way, PASS jointly considers which instructions can provide useful evidence and where additional supervision is needed under a limited budget. Experiments show that our method consistently outperforms seven state-of-the-art instruction selection methods on four backbone-budget settings. An ablation study further shows that PASS's adaptive allocation consistently improves over uniform allocation.