# BandPO: Bridging Trust Regions and Ratio Clipping via Probability-Aware Bounds for LLM Reinforcement Learning

arXiv: https://arxiv.org/abs/2603.04918 · HTML: https://mlsanity.com/paper/2603.04918/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 26 references that are in the map.
- First posted on arXiv 5 Mar 2026.

## Builds on (most cited first, 20 of 20)

- [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)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [HybridFlow: A Flexible and Efficient RLHF Framework](https://mlsanity.com/paper/2409.19256/) (2024)
- [High-Dimensional Continuous Control Using Generalized Advantage Estimation](https://mlsanity.com/paper/1506.02438/) (2015)
- [Dota 2 with Large Scale Deep Reinforcement Learning](https://mlsanity.com/paper/1912.06680/) (2019)
- [The Surprising Effectiveness of PPO in Cooperative, Multi-Agent Games](https://mlsanity.com/paper/2103.01955/) (2021)
- [Learning Dexterous In-Hand Manipulation](https://mlsanity.com/paper/1808.00177/) (2018)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [The Entropy Mechanism of Reinforcement Learning for Reasoning Language Models](https://mlsanity.com/paper/2505.22617/) (2025)
- [Emergent Tool Use From Multi-Agent Autocurricula](https://mlsanity.com/paper/1909.07528/) (2019)
- [REINFORCE++: Stabilizing Critic-Free Policy Optimization with Global Advantage Normalization](https://mlsanity.com/paper/2501.03262/) (2025)
- [Spurious Rewards: Rethinking Training Signals in RLVR](https://mlsanity.com/paper/2506.10947/) (2025)
- [ReMax: A Simple, Effective, and Efficient Reinforcement Learning Method for Aligning Large Language Models](https://mlsanity.com/paper/2310.10505/) (2023)
- [Mastering Complex Control in MOBA Games with Deep Reinforcement Learning](https://mlsanity.com/paper/1912.09729/) (2019)
- [Trust Region-Guided Proximal Policy Optimization](https://mlsanity.com/paper/1901.10314/) (2019)
- [An Adaptive Clipping Approach for Proximal Policy Optimization](https://mlsanity.com/paper/1804.06461/) (2018)
- [Clip-Low Increases Entropy and Clip-High Decreases Entropy in Reinforcement Learning of Large Language Models](https://mlsanity.com/paper/2509.26114/) (2025)
- [Trust Region Policy Optimization with Optimal Transport Discrepancies: Duality and Algorithm for Continuous Actions](https://mlsanity.com/paper/2210.11137/) (2022)
- [Decaying Clipping Range in Proximal Policy Optimization](https://mlsanity.com/paper/2102.10456/) (2021)

## Built on by (most cited first, 3 of 3)

- [UP: Unbounded Positive Asymmetric Optimization for Breaking the Exploration-Stability Dilemma](https://mlsanity.com/paper/2607.06987/) (2026)
- [Deconstructing Off-Policy Ratios: Entropy-Normalized Trust Regions for Asynchronous Reinforcement Learning](https://mlsanity.com/paper/2607.22186/) (2026)
- [BRACE: Anchored Bellman-Residual Correction for Stale Critics in Asynchronous RL](https://mlsanity.com/paper/2609.09783/) (2026)

## Abstract (arXiv)

Proximal constraints are fundamental to the stability of the Large Language Model reinforcement learning. While the canonical clipping mechanism in PPO serves as an efficient surrogate for trust regions, we identify a critical bottleneck: fixed bounds strictly constrain the upward update margin of low-probability actions, disproportionately suppressing high-advantage tail strategies and inducing rapid entropy collapse. To address this, we introduce Band-constrained Policy Optimization (BandPO). BandPO replaces canonical clipping with Band, a unified theoretical operator that projects trust regions defined by f-divergences into dynamic, probability-aware clipping intervals. Theoretical analysis confirms that Band effectively resolves this exploration bottleneck. We formulate this mapping as a convex optimization problem, guaranteeing a globally optimal numerical solution while deriving closed-form solutions for specific divergences. Extensive experiments across diverse models and datasets demonstrate that BandPO consistently outperforms canonical clipping and Clip-Higher, while robustly mitigating entropy collapse.