# Latent Poincar'e Shaping for Agentic Reinforcement Learning

arXiv: https://arxiv.org/abs/2602.09375 · HTML: https://mlsanity.com/paper/2602.09375/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for LLM agents, one of 16 directions in LLM agents and tool use.
- Builds on 24 of its 38 references that are in the map.
- First posted on arXiv 10 Feb 2026.

## Placed in

- LLM agents and tool use / [Reinforcement learning for LLM agents](https://mlsanity.com/agents/reinforcement-learning-for-llm-agents/)

## Builds on (most cited first, 24 of 24)

- [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)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [ReAct: Synergizing Reasoning and Acting in Language Models](https://mlsanity.com/paper/2210.03629/) (2022)
- [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)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [Root Mean Square Layer Normalization](https://mlsanity.com/paper/1910.07467/) (2019)
- [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)
- [Reasoning with Language Model is Planning with World Model](https://mlsanity.com/paper/2305.14992/) (2023)
- [Scheduled Sampling for Sequence Prediction with Recurrent Neural Networks](https://mlsanity.com/paper/1506.03099/) (2015)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Sequence Level Training with Recurrent Neural Networks](https://mlsanity.com/paper/1511.06732/) (2015)
- [Beyond the 80/20 Rule: High-Entropy Minority Tokens Drive Effective Reinforcement Learning for LLM Reasoning](https://mlsanity.com/paper/2506.01939/) (2025)
- [The Lessons of Developing Process Reward Models in Mathematical Reasoning](https://mlsanity.com/paper/2501.07301/) (2025)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [ReST-MCTS*: LLM Self-Training via Process Reward Guided Tree Search](https://mlsanity.com/paper/2406.03816/) (2024)
- [rStar-Math: Small LLMs Can Master Math Reasoning with Self-Evolved Deep Thinking](https://mlsanity.com/paper/2501.04519/) (2025)
- [Alphazero-like Tree-Search can Guide Large Language Model Decoding and Training](https://mlsanity.com/paper/2309.17179/) (2023)
- [Reinforcement Learning with Verifiable Rewards Implicitly Incentivizes Correct Reasoning in Base LLMs](https://mlsanity.com/paper/2506.14245/) (2025)
- [Monte Carlo Tree Search Boosts Reasoning via Iterative Preference Learning](https://mlsanity.com/paper/2405.00451/) (2024)
- [Evaluating the Performance of Large Language Models on GAOKAO Benchmark](https://mlsanity.com/paper/2305.12474/) (2023)
- [Revisiting Overthinking in Long Chain-of-Thought from the Perspective of Self-Doubt](https://mlsanity.com/paper/2505.23480/) (2025)

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

- [ARLArena: A Unified Framework for Stable Agentic Reinforcement Learning](https://mlsanity.com/paper/2602.21534/) (2026)

## Abstract (arXiv)

We propose LaPha, a method for training AlphaZero-like LLM agents in a Poincar'e latent space. Under LaPha, the search process can be visualized as a tree rooted at the prompt and growing outward from the origin toward the boundary of the Poincar'e ball, where negative curvature provides exponentially increasing capacity with radius. Using hyperbolic geodesic distance to rule-verified correctness, we define a node potential and assign dense process rewards by potential differences. We further attach a lightweight value head on the same shared latent space, enabling self-guided test-time scaling with almost no additional overhead. On MATH-500, LaPha improves Qwen2.5-Math-1.5B from 66.0% to 88.2%. With value-head-guided search, LaPha-1.5B reaches 56.7% accuracy on AIME'24, and LaPha-7B further achieves 60.0% on AIME'24 and 53.3% on AIME'25.