# Reinforcement Learning with Verifiable Rewards: GRPO's Effective Loss, Dynamics, and Success Amplification

arXiv: https://arxiv.org/abs/2503.06639 · HTML: https://mlsanity.com/paper/2503.06639/

## Key facts

- Cited by 110 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 12 of its 26 references that are in the map.
- First posted on arXiv 9 Mar 2025.

## Placed in

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

## Builds on (most cited first, 12 of 12)

- [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)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Tulu 3: Pushing Frontiers in Open Language Model Post-Training](https://mlsanity.com/paper/2411.15124/) (2024)
- [DeepSeek-V3 Technical Report](https://mlsanity.com/paper/2412.19437/) (2024)
- [Deliberative Alignment: Reasoning Enables Safer Language Models](https://mlsanity.com/paper/2412.16339/) (2024)
- [Mirror Descent Policy Optimization](https://mlsanity.com/paper/2005.09814/) (2020)
- [Apple Intelligence Foundation Language Models](https://mlsanity.com/paper/2407.21075/) (2024)
- [Information Theoretic Guarantees For Policy Alignment In Large Language Models](https://mlsanity.com/paper/2406.05883/) (2024)
- [What is the Alignment Objective of GRPO?](https://mlsanity.com/paper/2502.18548/) (2025)
- [KL-Regularized RLHF with Multiple Reference Models: Exact Solutions and Sample Complexity](https://mlsanity.com/paper/2502.01203/) (2025)

## Built on by (most cited first, 30 of 110)

- [The Surprising Effectiveness of Negative Reinforcement in LLM Reasoning](https://mlsanity.com/paper/2506.01347/) (2025)
- [Online Difficulty Filtering for Reasoning Oriented Reinforcement Learning](https://mlsanity.com/paper/2504.03380/) (2025)
- [wd1: Weighted Policy Optimization for Reasoning in Diffusion Language Models](https://mlsanity.com/paper/2507.08838/) (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)
- [Reinforcement Learning with Verifiable yet Noisy Rewards under Imperfect Verifiers](https://mlsanity.com/paper/2510.00915/) (2025)
- [VCRL: Variance-based Curriculum Reinforcement Learning for Large Language Models](https://mlsanity.com/paper/2509.19803/) (2025)
- [Search and Refine During Think: Facilitating Knowledge Refinement for Improved Retrieval-Augmented Reasoning](https://mlsanity.com/paper/2505.11277/) (2025)
- [Revisiting Group Relative Policy Optimization: Insights into On-Policy and Off-Policy Training](https://mlsanity.com/paper/2505.22257/) (2025)
- [MoDoMoDo: Multi-Domain Data Mixtures for Multimodal LLM Reinforcement Learning](https://mlsanity.com/paper/2505.24871/) (2025)
- [APRIL: Active Partial Rollouts in Reinforcement Learning to Tame Long-tail Generation](https://mlsanity.com/paper/2509.18521/) (2025)
- [A Comprehensive Survey on Trustworthiness in Reasoning with Large Language Models](https://mlsanity.com/paper/2509.03871/) (2025)
- [What about gravity in video generation? Post-Training Newton's Laws with Verifiable Rewards](https://mlsanity.com/paper/2512.00425/) (2025)
- [Improving Discrete Diffusion Unmasking Policies Beyond Explicit Reference Policies](https://mlsanity.com/paper/2510.05725/) (2025)
- [Shop-R1: Rewarding LLMs to Simulate Human Behavior in Online Shopping via Reinforcement Learning](https://mlsanity.com/paper/2507.17842/) (2025)
- [VeriReason: Reinforcement Learning with Testbench Feedback for Reasoning-Enhanced Verilog Generation](https://mlsanity.com/paper/2505.11849/) (2025)
- [DisCO: Reinforcing Large Reasoning Models with Discriminative Constrained Optimization](https://mlsanity.com/paper/2505.12366/) (2025)
- [Skywork-R1V3 Technical Report](https://mlsanity.com/paper/2507.06167/) (2025)
- [EFRame: Deeper Reasoning via Exploration-Filter-Replay Reinforcement Learning Framework](https://mlsanity.com/paper/2506.22200/) (2025)
- [CTRLS: Chain-of-Thought Reasoning via Latent State-Transition](https://mlsanity.com/paper/2507.08182/) (2025)
- [Loong: Synthesize Long Chain-of-Thoughts at Scale through Verifiers](https://mlsanity.com/paper/2509.03059/) (2025)
- [Student-Centered Distillation Narrows the Agentic Gap Between Small and Large LLMs](https://mlsanity.com/paper/2509.14257/) (2025)
- [Noise-corrected GRPO: From Noisy Rewards to Unbiased Gradients](https://mlsanity.com/paper/2510.18924/) (2025)
- [Falcon-H1R: Pushing the Reasoning Frontiers with a Hybrid Model for Efficient Test-Time Scaling](https://mlsanity.com/paper/2601.02346/) (2026)
- [LegalDelta: Enhancing Legal Reasoning in LLMs via Reinforcement Learning with Chain-of-Thought Guided Information Gain](https://mlsanity.com/paper/2508.12281/) (2025)
- [WS-GRPO: Weakly-Supervised Group-Relative Policy Optimization for Rollout-Efficient Reasoning](https://mlsanity.com/paper/2602.17025/) (2026)
- [Learning to Hint for Reinforcement Learning](https://mlsanity.com/paper/2604.00698/) (2026)
- [AJ-Bench: Benchmarking Agent-as-a-Judge for Environment-Aware Evaluation](https://mlsanity.com/paper/2604.18240/) (2026)
- [RLMR: Reinforcement Learning with Mixed Rewards for Creative Writing](https://mlsanity.com/paper/2508.18642/) (2025)
- [Quantile Advantage Estimation: Stabilizing RLVR for LLM Reasoning](https://mlsanity.com/paper/2509.22611/) (2025)
- [Revoking Amnesia: RL-based Trajectory Optimization to Resurrect Erased Concepts in Diffusion Models](https://mlsanity.com/paper/2510.03302/) (2025)

## Abstract (arXiv)

Group Relative Policy Optimization (GRPO) was introduced and used recently for promoting reasoning in LLMs under verifiable (binary) rewards. We show that the mean + variance calibration of these rewards induces a weighted contrastive loss in which the contrastive samples are synthetic data drawn from the previous policy. While GRPO was originally paired with clipping to keep updates near the old policy, we analyze variants that differ in reward normalization (mean-only vs mean + variance) and in how they regularize updates using KL divergence: either penalizing divergence from the previous model (mirror), penalizing divergence from a fixed reference model πref, or combining both forms of regularization. For each, the optimal policy πn admits an explicit form in terms of the binary reward and the first and second order statistics of the reward under πn-1, as well as the policies πn-1 and πref. Iterating results in a sequence πn whose probability of success (PoS) obeys a simple recurrence that converges to a fixed point determined by the reference PoS and the regularization strength. We further show that this fixed point exceeds the reference, demonstrating that GRPO amplifies the policy's probability of success.