# Beyond Pass@1: Self-Play with Variational Problem Synthesis Sustains RLVR

arXiv: https://arxiv.org/abs/2508.14029 · HTML: https://mlsanity.com/paper/2508.14029/

## Key facts

- Cited by 54 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 55 of its 68 references that are in the map.
- First posted on arXiv 19 Aug 2025.

## Placed in

- LLM reasoning / [Reinforcement learning for reasoning](https://mlsanity.com/lr/reinforcement-learning-for-reasoning/)
- Synthetic data and self-improvement for LLMs / [Self-play RL for reasoning](https://mlsanity.com/synth/self-play-rl-for-reasoning/)

## Builds on (most cited first, 30 of 55)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [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)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Efficient Memory Management for Large Language Model Serving with PagedAttention](https://mlsanity.com/paper/2309.06180/) (2023)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [HellaSwag: Can a Machine Really Finish Your Sentence?](https://mlsanity.com/paper/1905.07830/) (2019)
- [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)
- [MMLU-Pro: A More Robust and Challenging Multi-Task Language Understanding Benchmark](https://mlsanity.com/paper/2406.01574/) (2024)
- [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)
- [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)
- [SimpleRL-Zoo: Investigating and Taming Zero Reinforcement Learning for Open Base Models in the Wild](https://mlsanity.com/paper/2503.18892/) (2025)
- [From Generation to Judgment: Opportunities and Challenges of LLM-as-a-judge](https://mlsanity.com/paper/2411.16594/) (2024)
- [The Entropy Mechanism of Reinforcement Learning for Reasoning Language Models](https://mlsanity.com/paper/2505.22617/) (2025)
- [Open-Reasoner-Zero: An Open Source Approach to Scaling Up Reinforcement Learning on the Base Model](https://mlsanity.com/paper/2503.24290/) (2025)
- [Demystifying Long Chain-of-Thought Reasoning in LLMs](https://mlsanity.com/paper/2502.03373/) (2025)
- [From System 1 to System 2: A Survey of Reasoning Large Language Models](https://mlsanity.com/paper/2502.17419/) (2025)
- [Reinforcement Learning with Verifiable Rewards Implicitly Incentivizes Correct Reasoning in Base LLMs](https://mlsanity.com/paper/2506.14245/) (2025)
- [VAPO: Efficient and Reliable Reinforcement Learning for Advanced Reasoning Tasks](https://mlsanity.com/paper/2504.05118/) (2025)
- [Reasoning with Exploration: An Entropy Perspective](https://mlsanity.com/paper/2506.14758/) (2025)
- [DeepMath-103K: A Large-Scale, Challenging, Decontaminated, and Verifiable Mathematical Dataset for Advancing Reasoning](https://mlsanity.com/paper/2504.11456/) (2025)

## Built on by (most cited first, 30 of 54)

- [The Landscape of Agentic Reinforcement Learning for LLMs: A Survey](https://mlsanity.com/paper/2509.02547/) (2025)
- [A Survey of Reinforcement Learning for Large Reasoning Models](https://mlsanity.com/paper/2509.08827/) (2025)
- [SPICE: Self-Play In Corpus Environments Improves Reasoning](https://mlsanity.com/paper/2510.24684/) (2025)
- [Jointly Reinforcing Diversity and Quality in Language Model Generations](https://mlsanity.com/paper/2509.02534/) (2025)
- [SCOPE: Signal-Calibrated On-Policy Distillation Enhancement with Dual-Path Adaptive Weighting](https://mlsanity.com/paper/2604.10688/) (2026)
- [The Choice of Divergence: A Neglected Key to Mitigating Diversity Collapse in Reinforcement Learning with Verifiable Reward](https://mlsanity.com/paper/2509.07430/) (2025)
- [Guided Self-Evolving LLMs with Minimal Human Supervision](https://mlsanity.com/paper/2512.02472/) (2025)
- [RL Squeezes, SFT Expands: A Comparative Study of Reasoning LLMs](https://mlsanity.com/paper/2509.21128/) (2025)
- [Search Self-play: Pushing the Frontier of Agent Capability without Supervision](https://mlsanity.com/paper/2510.18821/) (2025)
- [Aegis: Automated Error Generation and Attribution for Multi-Agent Systems](https://mlsanity.com/paper/2509.14295/) (2025)
- [Composer 2 Technical Report](https://mlsanity.com/paper/2603.24477/) (2026)
- [Harder Is Better: Boosting Mathematical Reasoning via Difficulty-Aware GRPO and Multi-Aspect Question Reformulation](https://mlsanity.com/paper/2601.20614/) (2026)
- [Beyond the Sampled Token: Preserving Candidate Support in RLVR](https://mlsanity.com/paper/2510.14807/) (2025)
- [LoopTool: Closing the Data-Training Loop for Robust LLM Tool Calls](https://mlsanity.com/paper/2511.09148/) (2025)
- [The Best of N Worlds: Aligning Reinforcement Learning with Best-of-N Sampling via max@k Optimisation](https://mlsanity.com/paper/2510.23393/) (2025)
- [Find Your Optimal Teacher: Personalized Data Synthesis via Router-Guided Multi-Teacher Distillation](https://mlsanity.com/paper/2510.10925/) (2025)
- [EnvFactory: Scaling Tool-Use Agents via Executable Environments Synthesis and Robust RL](https://mlsanity.com/paper/2605.18703/) (2026)
- [CLPO: Curriculum Learning meets Policy Optimization for LLM Reasoning](https://mlsanity.com/paper/2509.25004/) (2025)
- [Position: The Hidden Costs and Measurement Gaps of Reinforcement Learning with Verifiable Rewards](https://mlsanity.com/paper/2509.21882/) (2025)
- [Dynamic Generation of Multi-LLM Agents Communication Topologies with Graph Diffusion Models](https://mlsanity.com/paper/2510.07799/) (2025)
- [Random Policy Valuation is Enough for LLM Reasoning with Verifiable Rewards](https://mlsanity.com/paper/2509.24981/) (2025)
- [Self-Harmony: Learning to Harmonize Self-Supervision and Self-Play in Test-Time Reinforcement Learning](https://mlsanity.com/paper/2511.01191/) (2025)
- [Best-of-Majority: Minimax-Optimal Strategy for Pass@k Inference Scaling](https://mlsanity.com/paper/2510.03199/) (2025)
- [Towards Understanding Self-play for LLM Reasoning](https://mlsanity.com/paper/2510.27072/) (2025)
- [Mnemis: Dual-Route Retrieval on Hierarchical Graphs for Long-Term LLM Memory](https://mlsanity.com/paper/2602.15313/) (2026)
- [When Silence Is Golden: Can LLMs Learn to Abstain in Temporal QA and Beyond?](https://mlsanity.com/paper/2602.04755/) (2026)
- [OmniOPD: Logit-Free On-Policy Distillation via Speculative Verification](https://mlsanity.com/paper/2606.01476/) (2026)
- [Detecting Data Contamination from Reinforcement Learning Post-training for Large Language Models](https://mlsanity.com/paper/2510.09259/) (2025)
- [Learning to Explore with Parameter-Space Noise: A Deep Dive into Parameter-Space Noise for Reinforcement Learning with Verifiable Rewards](https://mlsanity.com/paper/2602.02555/) (2026)
- [Efficient RLVR Training via Weighted Mutual Information Data Selection](https://mlsanity.com/paper/2603.01907/) (2026)

## Abstract (arXiv)

Reinforcement Learning with Verifiable Rewards (RLVR) has recently emerged as a key paradigm for post-training Large Language Models (LLMs), particularly for complex reasoning tasks. However, vanilla RLVR training has been shown to improve Pass@1 performance at the expense of policy entropy, leading to reduced generation diversity and limiting the Pass@k performance, which typically represents the upper bound of LLM reasoning capability. In this paper, we systematically analyze the policy's generation diversity from the perspective of training problems and find that augmenting and updating training problems helps mitigate entropy collapse during training. Based on these observations, we propose an online Self-play with Variational problem Synthesis (SvS) strategy for RLVR training, which uses the policy's correct solutions to synthesize variational problems while ensuring their reference answers remain identical to the originals. This self-improving strategy effectively maintains policy entropy during training and substantially improves Pass@k compared with standard RLVR, sustaining prolonged improvements and achieving absolute gains of 18.3% and 22.8% in Pass@32 performance on the competition-level AIME24 and AIME25 benchmarks, as well as on code generation tasks. Experiments on 12 reasoning benchmarks across varying model sizes from 3B to 32B consistently demonstrate the generalizability and robustness of SvS.