# On the optimization dynamics of RLVR: Gradient gap and step size thresholds

arXiv: https://arxiv.org/abs/2510.08539 · HTML: https://mlsanity.com/paper/2510.08539/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 49 references that are in the map.
- First posted on arXiv 9 Oct 2025.

## Builds on (most cited first, 29 of 29)

- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [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)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [Fine-Tuning Language Models from Human Preferences](https://mlsanity.com/paper/1909.08593/) (2019)
- [Understanding R1-Zero-Like Training: A Critical Perspective](https://mlsanity.com/paper/2503.20783/) (2025)
- [Kimi k1.5: Scaling Reinforcement Learning with LLMs](https://mlsanity.com/paper/2501.12599/) (2025)
- [Group Sequence Policy Optimization](https://mlsanity.com/paper/2507.18071/) (2025)
- [Process Reinforcement through Implicit Rewards](https://mlsanity.com/paper/2502.01456/) (2025)
- [Principled Reinforcement Learning with Human Feedback from Pairwise or K-wise Comparisons](https://mlsanity.com/paper/2301.11270/) (2023)
- [A Finite Time Analysis of Two Time-Scale Actor Critic Methods](https://mlsanity.com/paper/2005.01350/) (2020)
- [On the Convergence Rates of Policy Gradient Methods](https://mlsanity.com/paper/2201.07443/) (2022)
- [Self-Evolving Curriculum for LLM Reasoning](https://mlsanity.com/paper/2505.14970/) (2025)
- [Provable Benefits of Actor-Critic Methods for Offline Reinforcement Learning](https://mlsanity.com/paper/2108.08812/) (2021)
- [Reinforcement Learning with Verifiable Rewards: GRPO's Effective Loss, Dynamics, and Success Amplification](https://mlsanity.com/paper/2503.06639/) (2025)
- [Dueling RL: Reinforcement Learning with Trajectory Preferences](https://mlsanity.com/paper/2111.04850/) (2021)
- [Is RLHF More Difficult than Standard RL?](https://mlsanity.com/paper/2306.14111/) (2023)
- [Outcome-based Exploration for LLM Reasoning](https://mlsanity.com/paper/2509.06941/) (2025)
- [Human-in-the-loop: Provably Efficient Preference-based Reinforcement Learning with General Function Approximation](https://mlsanity.com/paper/2205.11140/) (2022)
- [Offline Reinforcement Learning for LLM Multi-Step Reasoning](https://mlsanity.com/paper/2412.16145/) (2024)
- [Is a Good Foundation Necessary for Efficient Reinforcement Learning? The Computational Role of the Base Model in Exploration](https://mlsanity.com/paper/2503.07453/) (2025)
- [SPEED-RL: Faster Training of Reasoning Models via Online Curriculum Learning](https://mlsanity.com/paper/2506.09016/) (2025)
- [Asymmetric REINFORCE for off-Policy Reinforcement Learning: Balancing positive and negative rewards](https://mlsanity.com/paper/2506.20520/) (2025)
- [Exploration-Driven Policy Optimization in RLHF: Theoretical Insights on Efficient Data Utilization](https://mlsanity.com/paper/2402.10342/) (2024)
- [Accelerating RL for LLM Reasoning with Optimal Advantage Regression](https://mlsanity.com/paper/2505.20686/) (2025)
- [PILAF: Optimal Human Preference Sampling for Reward Modeling](https://mlsanity.com/paper/2502.04270/) (2025)
- [Trajectory Bellman Residual Minimization: A Simple Value-Based Method for LLM Reasoning](https://mlsanity.com/paper/2505.15311/) (2025)
- [Taming "data-hungry" reinforcement learning? Stability in continuous state-action spaces](https://mlsanity.com/paper/2401.05233/) (2024)

## Built on by (most cited first, 4 of 4)

- [Don't Let Gains FADE: Breaking Down Policy Gradient Weights in RL](https://mlsanity.com/paper/2607.01490/) (2026)
- [Trust the Direction, Search the Step: Zero-and-First-Order Methods for LLM Fine-Tuning](https://mlsanity.com/paper/2610.02190/) (2026)
- [How RL Reshapes LLM Reasoning: Transferability, Coverage, and Scaling Laws](https://mlsanity.com/paper/2610.04158/) (2026)
- [Two Clocks: Why Larger Rollout Batches Learn Slower but Fail on Schedule in Reinforcement Learning with Verifiable Rewards](https://mlsanity.com/paper/2610.18853/) (2026)

## Abstract (arXiv)

Reinforcement Learning with Verifiable Rewards (RLVR), which uses simple binary feedback to post-train large language models, has found significant empirical success. However, a principled understanding of why it works is lacking. This paper builds a theoretical foundation for RLVR by analyzing its training process at both the full-response (trajectory) and token levels. Central to our analysis is a new quantity called the Gradient Gap, which formalizes the direction of improvement from low-reward to high-reward regions of the response space. We prove that convergence critically depends on aligning the update direction with this Gradient Gap. Moreover, we derive a sharp step-size threshold based on the magnitude of the Gradient Gap: below it, learning converges, whereas above it, performance collapses. Our theory further predicts how the critical step size must scale with response length and the success rate, thereby explaining why practical heuristics such as length normalization improve stability and showing that, with a fixed learning rate, the success rate can stagnate strictly below 100%. Importantly, our theory holds flexibly for any policy-gradient algorithm and so characterizes the dynamics of popular approaches such as REINFORCE and GRPO. We validate these predictions through controlled bandit simulations and language model experiments on post-training Qwen2.5-Math-7B with GRPO.