# KTAE: A Model-Free Algorithm to Key-Tokens Advantage Estimation in Mathematical Reasoning

arXiv: https://arxiv.org/abs/2505.16826 · HTML: https://mlsanity.com/paper/2505.16826/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 53 of its 61 references that are in the map.
- First posted on arXiv 22 May 2025.

## Placed in

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

## Builds on (most cited first, 30 of 53)

- [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)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Continuous control with deep reinforcement learning](https://mlsanity.com/paper/1509.02971/) (2015)
- [Playing Atari with Deep Reinforcement Learning](https://mlsanity.com/paper/1312.5602/) (2013)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [Addressing Function Approximation Error in Actor-Critic Methods](https://mlsanity.com/paper/1802.09477/) (2018)
- [Asynchronous Methods for Deep Reinforcement Learning](https://mlsanity.com/paper/1602.01783/) (2016)
- [HybridFlow: A Flexible and Efficient RLHF Framework](https://mlsanity.com/paper/2409.19256/) (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)
- [s1: Simple test-time scaling](https://mlsanity.com/paper/2501.19393/) (2025)
- [Kimi k1.5: Scaling Reinforcement Learning with LLMs](https://mlsanity.com/paper/2501.12599/) (2025)
- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [Large Language Monkeys: Scaling Inference Compute with Repeated Sampling](https://mlsanity.com/paper/2407.21787/) (2024)
- [Do NOT Think That Much for 2+3=? On the Overthinking of o1-Like LLMs](https://mlsanity.com/paper/2412.21187/) (2024)
- [SimpleRL-Zoo: Investigating and Taming Zero Reinforcement Learning for Open Base Models in the Wild](https://mlsanity.com/paper/2503.18892/) (2025)
- [Process Reinforcement through Implicit Rewards](https://mlsanity.com/paper/2502.01456/) (2025)
- [LIMO: Less is More for Reasoning](https://mlsanity.com/paper/2502.03387/) (2025)
- [Generative Verifiers: Reward Modeling as Next-Token Prediction](https://mlsanity.com/paper/2408.15240/) (2024)
- [Stop Overthinking: A Survey on Efficient Reasoning for Large Language Models](https://mlsanity.com/paper/2503.16419/) (2025)
- [Open-Reasoner-Zero: An Open Source Approach to Scaling Up Reinforcement Learning on the Base Model](https://mlsanity.com/paper/2503.24290/) (2025)
- [L1: Controlling How Long A Reasoning Model Thinks With Reinforcement Learning](https://mlsanity.com/paper/2503.04697/) (2025)
- [From System 1 to System 2: A Survey of Reasoning Large Language Models](https://mlsanity.com/paper/2502.17419/) (2025)

## Built on by (most cited first, 19 of 19)

- [Self-Distilled RLVR](https://mlsanity.com/paper/2604.03128/) (2026)
- [Towards a Unified View of Large Language Model Post-Training](https://mlsanity.com/paper/2509.04419/) (2025)
- [Unlocking Exploration in RLVR: Uncertainty-aware Advantage Shaping for Deeper Reasoning](https://mlsanity.com/paper/2510.10649/) (2025)
- [Your Group-Relative Advantage Is Biased](https://mlsanity.com/paper/2601.08521/) (2026)
- [Outcome-Grounded Advantage Reshaping for Fine-Grained Credit Assignment in Mathematical Reasoning](https://mlsanity.com/paper/2601.07408/) (2026)
- [Look Again, Think Slowly: Enhancing Visual Reflection in Vision-Language Models](https://mlsanity.com/paper/2509.12132/) (2025)
- [CROP: Contextual Region-Oriented Visual Token Pruning](https://mlsanity.com/paper/2505.21233/) (2025)
- [What and When to Distill: Selective Hindsight Distillation for Multi-Turn Agents](https://mlsanity.com/paper/2605.19447/) (2026)
- [Teaching Vision-Language Models to Ask: Resolving Ambiguity in Visual Questions](https://mlsanity.com/paper/2507.13773/) (2025)
- [Reinforcement Learning from Rich Feedback with Distributional DAgger](https://mlsanity.com/paper/2606.05152/) (2026)
- [VRPO: Rethinking Value Modeling for Robust RL under Noisy Supervision in LLM Post-Training](https://mlsanity.com/paper/2508.03058/) (2025)
- [Beyond Uniform Credit Assignment: Selective Eligibility Traces for RLVR](https://mlsanity.com/paper/2605.05965/) (2026)
- [Parallel Scaling Law: Unveiling Reasoning Generalization through A Cross-Linguistic Perspective](https://mlsanity.com/paper/2510.02272/) (2025)
- [Know What You Know: Metacognitive Entropy Calibration for Verifiable RL Reasoning](https://mlsanity.com/paper/2602.22751/) (2026)
- [Policy Improvement Reinforcement Learning](https://mlsanity.com/paper/2604.00860/) (2026)
- [Beyond Entropy: Learning from Token-Level Distributional Deviations for LLM Reasoning](https://mlsanity.com/paper/2606.19771/) (2026)
- [DFPO: Scaling Value Modeling via Distributional Flow towards Robust and Generalizable LLM Post-Training](https://mlsanity.com/paper/2602.05890/) (2026)
- [Guidance Contrastive Token Credit Assignment for Discrete Policy Optimization](https://mlsanity.com/paper/2605.29198/) (2026)
- [BALTO: Balanced Token-Level Policy Optimization for Hallucination Mitigation](https://mlsanity.com/paper/2606.15893/) (2026)

## Abstract (arXiv)

Recent advances have demonstrated that integrating reinforcement learning with rule-based rewards can significantly enhance the reasoning capabilities of large language models, even without supervised fine-tuning. However, prevalent reinforcement learning algorithms such as GRPO and its variants like DAPO, suffer from a coarse granularity issue when computing the advantage. Specifically, they compute rollout-level advantages that assign identical values to every token within a sequence, failing to capture token-specific contributions and hindering effective learning. To address this limitation, we propose Key-token Advantage Estimation (KTAE) - a novel algorithm that estimates fine-grained, token-level advantages without introducing additional models. KTAE leverages the correctness of sampled rollouts and applies statistical analysis to quantify the importance of individual tokens within a sequence to the final outcome. This quantified token-level importance is then combined with the rollout-level advantage to obtain a more fine-grained token-level advantage estimation. Empirical results show that models trained with GRPO+KTAE and DAPO+KTAE outperform baseline methods across five mathematical reasoning benchmarks. Notably, they achieve higher accuracy with shorter responses and even surpass R1-Distill-Qwen-1.5B using the same base model.