# The Unlearnability Phenomenon in RLVR for Language Models

arXiv: https://arxiv.org/abs/2605.16787 · HTML: https://mlsanity.com/paper/2605.16787/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 22 of its 30 references that are in the map.
- First posted on arXiv 16 May 2026.

## Placed in

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

## Builds on (most cited first, 22 of 22)

- [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)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [Understanding R1-Zero-Like Training: A Critical Perspective](https://mlsanity.com/paper/2503.20783/) (2025)
- [Does Reinforcement Learning Really Incentivize Reasoning Capacity in LLMs Beyond the Base Model?](https://mlsanity.com/paper/2504.13837/) (2025)
- [SimpleRL-Zoo: Investigating and Taming Zero Reinforcement Learning for Open Base Models in the Wild](https://mlsanity.com/paper/2503.18892/) (2025)
- [The Entropy Mechanism of Reinforcement Learning for Reasoning Language Models](https://mlsanity.com/paper/2505.22617/) (2025)
- [DeepResearcher: Scaling Deep Research via Reinforcement Learning in Real-world Environments](https://mlsanity.com/paper/2504.03160/) (2025)
- [Reasoning with Exploration: An Entropy Perspective](https://mlsanity.com/paper/2506.14758/) (2025)
- [Reasoning Models Know When They're Right: Probing Hidden States for Self-Verification](https://mlsanity.com/paper/2504.05419/) (2025)
- [Arithmetic Without Algorithms: Language Models Solve Math With a Bag of Heuristics](https://mlsanity.com/paper/2410.21272/) (2024)
- [The Invisible Leash: Why RLVR May or May Not Escape Its Origin](https://mlsanity.com/paper/2507.14843/) (2025)
- [Efficient Reinforcement Finetuning via Adaptive Curriculum Learning](https://mlsanity.com/paper/2504.05520/) (2025)
- [Not All Rollouts are Useful: Down-Sampling Rollouts in LLM Reinforcement Learning](https://mlsanity.com/paper/2504.13818/) (2025)
- [On the Interplay of Pre-Training, Mid-Training, and RL on Reasoning Language Models](https://mlsanity.com/paper/2512.07783/) (2025)
- [Improving Data Efficiency for LLM Reinforcement Fine-tuning Through Difficulty-targeted Online Data Selection and Rollout Replay](https://mlsanity.com/paper/2506.05316/) (2025)
- [Prompt Curriculum Learning for Efficient LLM Post-Training](https://mlsanity.com/paper/2510.01135/) (2025)
- [RLEP: Reinforcement Learning with Experience Replay for LLM Reasoning](https://mlsanity.com/paper/2507.07451/) (2025)
- [Improving Sampling Efficiency in RLVR through Adaptive Rollout and Response Reuse](https://mlsanity.com/paper/2509.25808/) (2025)
- [Revisiting Entropy in Reinforcement Learning for Large Reasoning Models](https://mlsanity.com/paper/2511.05993/) (2025)
- [The Reasoning Boundary Paradox: How Reinforcement Learning Constrains Language Models](https://mlsanity.com/paper/2510.02230/) (2025)
- [How Does RL Post-training Induce Skill Composition? A Case Study on Countdown](https://mlsanity.com/paper/2512.01775/) (2025)

## Built on by (most cited first, 3 of 3)

- [DRIFT: Difficulty Routing Self-DIstillation with Rhythm-Gated Exploration and Success BuFfer Training](https://mlsanity.com/paper/2606.30345/) (2026)
- [Unlearnable, or Unmeasured? On the Reliability of Difficulty Labels in RLVR](https://mlsanity.com/paper/2609.40115/) (2026)
- [What Will Post-Training Fix? Per-Problem Gains Are Shared Across Independent RL Runs, and Existing Checkpoints Predict Them Better Than A Priori Signals](https://mlsanity.com/paper/2610.04978/) (2026)

## Abstract (arXiv)

Reinforcement Learning with Verifiable Reward (RLVR) has proven effective in improving Large Language Model's (LLM) reasoning ability. However, the learning dynamics of RLVR remain underexplored. In this paper, we reveal a counterintuitive phenomenon: among hard examples that the model initially struggles with, a substantial subset remains unlearnable even when correct rollouts are present. To understand the phenomenon, we first demonstrate that existing optimization and sampling techniques fail to resolve unlearnability. With cross-example gradient analysis, we show that unlearnable examples have fundamental representation issue, characterized by low gradient similarity with the rest of the examples and ungeneralizable reasoning patterns. We further show that representation flaws are difficult to mitigate in RL, as data augmentation does not improve gradient similarity. Our study provides the first systematic characterization of unlearnable data in RLVR training and reveals fundamental limitations in current RL approaches for reasoning tasks. Code and data are available at urlhttps://github.com/yulinchen99/unlearnability-rlvr.