# Mechanistically Interpreting the Role of Sample Difficulty in RLVR for LLMs

arXiv: https://arxiv.org/abs/2605.28388 · HTML: https://mlsanity.com/paper/2605.28388/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 47 of its 57 references that are in the map.
- First posted on arXiv 27 May 2026.

## Placed in

- LLM reasoning / [Reinforcement learning for reasoning](https://mlsanity.com/lr/reinforcement-learning-for-reasoning/)
- Mechanistic interpretability / [Steering and interpreting LLM reasoning](https://mlsanity.com/mechinterp/steering-and-interpreting-llm-reasoning/)

## Builds on (most cited first, 30 of 47)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [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)
- [HybridFlow: A Flexible and Efficient RLHF Framework](https://mlsanity.com/paper/2409.19256/) (2024)
- [Sparse Autoencoders Find Highly Interpretable Features in Language Models](https://mlsanity.com/paper/2309.08600/) (2023)
- [Solving Quantitative Reasoning Problems with Language Models](https://mlsanity.com/paper/2206.14858/) (2022)
- [Understanding R1-Zero-Like Training: A Critical Perspective](https://mlsanity.com/paper/2503.20783/) (2025)
- [OlympiadBench: A Challenging Benchmark for Promoting AGI with Olympiad-Level Bilingual Multimodal Scientific Problems](https://mlsanity.com/paper/2402.14008/) (2024)
- [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)
- [Does Reinforcement Learning Really Incentivize Reasoning Capacity in LLMs Beyond the Base Model?](https://mlsanity.com/paper/2504.13837/) (2025)
- [Group Sequence Policy Optimization](https://mlsanity.com/paper/2507.18071/) (2025)
- [MetaMath: Bootstrap Your Own Mathematical Questions for Large Language Models](https://mlsanity.com/paper/2309.12284/) (2023)
- [Flow-GRPO: Training Flow Matching Models via Online RL](https://mlsanity.com/paper/2505.05470/) (2025)
- [Scaling and evaluating sparse autoencoders](https://mlsanity.com/paper/2406.04093/) (2024)
- [Process Reinforcement through Implicit Rewards](https://mlsanity.com/paper/2502.01456/) (2025)
- [Sparse Feature Circuits: Discovering and Editing Interpretable Causal Graphs in Language Models](https://mlsanity.com/paper/2403.19647/) (2024)
- [DanceGRPO: Unleashing GRPO on Visual Generation](https://mlsanity.com/paper/2505.07818/) (2025)
- [Gemma Scope: Open Sparse Autoencoders Everywhere All At Once on Gemma 2](https://mlsanity.com/paper/2408.05147/) (2024)
- [Absolute Zero: Reinforced Self-play Reasoning with Zero Data](https://mlsanity.com/paper/2505.03335/) (2025)
- [Reinforcement Learning for Reasoning in Large Language Models with One Training Example](https://mlsanity.com/paper/2504.20571/) (2025)
- [VAPO: Efficient and Reliable Reinforcement Learning for Advanced Reasoning Tasks](https://mlsanity.com/paper/2504.05118/) (2025)
- [Deep Think with Confidence](https://mlsanity.com/paper/2508.15260/) (2025)
- [ReMax: A Simple, Effective, and Efficient Reinforcement Learning Method for Aligning Large Language Models](https://mlsanity.com/paper/2310.10505/) (2023)
- [MedVLM-R1: Incentivizing Medical Reasoning Capability of Vision-Language Models (VLMs) via Reinforcement Learning](https://mlsanity.com/paper/2502.19634/) (2025)
- [Automatically Interpreting Millions of Features in Large Language Models](https://mlsanity.com/paper/2410.13928/) (2024)
- [Online Difficulty Filtering for Reasoning Oriented Reinforcement Learning](https://mlsanity.com/paper/2504.03380/) (2025)
- [Curriculum Reinforcement Learning from Easy to Hard Tasks Improves LLM Reasoning](https://mlsanity.com/paper/2506.06632/) (2025)

## Built on by (most cited first, 1 of 1)

- [DRIFT: Difficulty Routing Self-DIstillation with Rhythm-Gated Exploration and Success BuFfer Training](https://mlsanity.com/paper/2606.30345/) (2026)

## Abstract (arXiv)

Reinforcement Learning with Verifiable Reward (RLVR) is empirically shown to notably enhance the reasoning performance of large language models (LLMs), particularly in mathematics and programming. However, the mechanistic role of Sample Difficulty in RLVR remains poorly understood. In this paper, we investigate RLVR through the lens of difficulty-wise and one-sample analysis. We find that sample difficulty has a non-monotonic effect on RLVR: easy and medium-difficulty problems yield the strongest and most stable reasoning improvements, whereas overly hard problems often provide weak learning signals, induce degenerate behaviors such as answer repetition or skipping necessary computation, and can ultimately degrade the model's pre-existing capabilities. Beyond the obverse of response, we further analyze the model's internal feature dynamics using Temporal Sparse Autoencoders (T-SAE). Easy problems mainly reinforce direct-answer and basic-computation features while suppressing deliberative-reasoning features; hard problems activate reasoning-related features but become useful only when successful trajectories are sampled; medium-difficulty problems provide a more balanced signal, strengthening both computation and multi-step reasoning features. Motivated by these findings, we propose difficulty-adaptive strategies for hard-sample utilization, using backward-reasoning reformulation and T-SAE-guided training signals to improve reward density and credit assignment during RLVR. Overall, our results identify sample difficulty as a key factor governing both the optimization dynamics and representation evolution of RLVR.