# ThinkPrior: Zero-Rollout Difficulty Priors for Cold-Start Prompt Selection in RLVR

arXiv: https://arxiv.org/abs/2609.09075 · HTML: https://mlsanity.com/paper/2609.09075/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 24 of its 35 references that are in the map.
- First posted on arXiv 8 Sep 2026.

## Placed in

- LLM reasoning / [Reinforcement learning for reasoning](https://mlsanity.com/lr/reinforcement-learning-for-reasoning/)

## Builds on (most cited first, 24 of 24)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [Solving Quantitative Reasoning Problems with Language Models](https://mlsanity.com/paper/2206.14858/) (2022)
- [Kimi k1.5: Scaling Reinforcement Learning with LLMs](https://mlsanity.com/paper/2501.12599/) (2025)
- [OlympiadBench: A Challenging Benchmark for Promoting AGI with Olympiad-Level Bilingual Multimodal Scientific Problems](https://mlsanity.com/paper/2402.14008/) (2024)
- [Prioritized Experience Replay](https://mlsanity.com/paper/1511.05952/) (2015)
- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [Tulu 3: Pushing Frontiers in Open Language Model Post-Training](https://mlsanity.com/paper/2411.15124/) (2024)
- [Data Selection for Language Models via Importance Resampling](https://mlsanity.com/paper/2302.03169/) (2023)
- [tinyBenchmarks: evaluating LLMs with fewer examples](https://mlsanity.com/paper/2402.14992/) (2024)
- [Automated Curriculum Learning for Neural Networks](https://mlsanity.com/paper/1704.03003/) (2017)
- [Prioritized Training on Points that are Learnable, Worth Learning, and Not Yet Learnt](https://mlsanity.com/paper/2206.07137/) (2022)
- [Superfiltering: Weak-to-Strong Data Filtering for Fast Instruction-Tuning](https://mlsanity.com/paper/2402.00530/) (2024)
- [Act Only When It Pays: Efficient Reinforcement Learning for LLM Reasoning via Selective Rollouts](https://mlsanity.com/paper/2506.02177/) (2025)
- [Online Difficulty Filtering for Reasoning Oriented Reinforcement Learning](https://mlsanity.com/paper/2504.03380/) (2025)
- [Can Prompt Difficulty be Online Predicted for Accelerating RL Finetuning of Reasoning Models?](https://mlsanity.com/paper/2507.04632/) (2025)
- [Prompt Curriculum Learning for Efficient LLM Post-Training](https://mlsanity.com/paper/2510.01135/) (2025)
- [Learning Latent Parameters without Human Response Patterns: Item Response Theory with Artificial Crowds](https://mlsanity.com/paper/1908.11421/) (2019)
- [Small Generalizable Prompt Predictive Models Can Steer Efficient RL Post-Training of Large Reasoning Models](https://mlsanity.com/paper/2602.01970/) (2026)
- [sGPO: Trading Inference FLOPs for Training Efficiency in RLVR](https://mlsanity.com/paper/2606.08854/) (2026)

## Built on by (most cited first, 2 of 2)

- [Prompts Live on an Arc: Gaussian Curricula in Fisher--Rao Coordinates for Rollout-Efficient GRPO](https://mlsanity.com/paper/2609.38018/) (2026)
- [Tev: Scaling Long-Horizon Agentic Environments with Dense Progress Rewards](https://mlsanity.com/paper/2610.18868/) (2026)

## Abstract (arXiv)

In reinforcement learning with verifiable rewards (RLVR) trained with group relative policy optimization (GRPO), the KL-free reward-advantage term studied here depends on within-group reward variation. If all rollouts in a group are correct or all are wrong, their group-relative advantages are identically zero; these zero-advantage silent groups provide no reward-advantage gradient, yet uniform sampling spends 39% of a run's rollouts on them. History-based prompt selection must first spend target-policy rollouts to estimate difficulty, creating a cold start with rollout waste; ThinkPrior instead uses an external anchor in one offline pass to construct a zero-rollout difficulty prior before the first target-policy rollout. The verifier-scored anchor pass rate supplies an external-anchor initialization for a Beta posterior; ThinkPrior selects by expected learnability and then updates from training outcomes, changing neither the loss nor the optimizer. On Qwen2.5-Math-7B across sixteen seeds, ThinkPrior more than halves early silent groups and cuts wasted rollouts through step 30 by nearly a fifth, while we detect no difference in final accuracy. On this 250-prompt pool the fixed-budget result is a reallocation rather than a net saving. The measured ThinkPrior+DAPO composition reduces generated rollouts by 10.6% while both arms retain the same 3840-rollout update budget. The prior requires no target-policy rollout before the first selection, but the posterior thereafter uses target-policy outcomes.