# Low-probability Tokens Sustain Exploration in Reinforcement Learning with Verifiable Reward

arXiv: https://arxiv.org/abs/2510.03222 · HTML: https://mlsanity.com/paper/2510.03222/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 31 of its 39 references that are in the map.
- First posted on arXiv 3 Oct 2025.

## Placed in

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

## Builds on (most cited first, 30 of 31)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [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)
- [GPQA: A Graduate-Level Google-Proof Q&A Benchmark](https://mlsanity.com/paper/2311.12022/) (2023)
- [HybridFlow: A Flexible and Efficient RLHF Framework](https://mlsanity.com/paper/2409.19256/) (2024)
- [LiveCodeBench: Holistic and Contamination Free Evaluation of Large Language Models for Code](https://mlsanity.com/paper/2403.07974/) (2024)
- [Language Models (Mostly) Know What They Know](https://mlsanity.com/paper/2207.05221/) (2022)
- [Solving Quantitative Reasoning Problems with Language Models](https://mlsanity.com/paper/2206.14858/) (2022)
- [Kimi k1.5: Scaling Reinforcement Learning with LLMs](https://mlsanity.com/paper/2501.12599/) (2025)
- [OlympiadBench: A Challenging Benchmark for Promoting AGI with Olympiad-Level Bilingual Multimodal Scientific Problems](https://mlsanity.com/paper/2402.14008/) (2024)
- [Beyond the 80/20 Rule: High-Entropy Minority Tokens Drive Effective Reinforcement Learning for LLM Reasoning](https://mlsanity.com/paper/2506.01939/) (2025)
- [The Entropy Mechanism of Reinforcement Learning for Reasoning Language Models](https://mlsanity.com/paper/2505.22617/) (2025)
- [Cognitive Behaviors that Enable Self-Improving Reasoners, or, Four Habits of Highly Effective STaRs](https://mlsanity.com/paper/2503.01307/) (2025)
- [Reasoning with Exploration: An Entropy Perspective](https://mlsanity.com/paper/2506.14758/) (2025)
- [AReaL: A Large-Scale Asynchronous Reinforcement Learning System for Language Reasoning](https://mlsanity.com/paper/2505.24298/) (2025)
- [Chain-of-Thought Reasoning Without Prompting](https://mlsanity.com/paper/2402.10200/) (2024)
- [Learning to Reason without External Rewards](https://mlsanity.com/paper/2505.19590/) (2025)
- [The Unreasonable Effectiveness of Entropy Minimization in LLM Reasoning](https://mlsanity.com/paper/2505.15134/) (2025)
- [Turning Up the Heat: Min-p Sampling for Creative and Coherent LLM Outputs](https://mlsanity.com/paper/2407.01082/) (2024)
- [Confidence Improves Self-Consistency in LLMs](https://mlsanity.com/paper/2502.06233/) (2025)
- [Geometric-Mean Policy Optimization](https://mlsanity.com/paper/2507.20673/) (2025)
- [Demystifying Reasoning Dynamics with Mutual Information: Thinking Tokens are Information Peaks in LLM Reasoning](https://mlsanity.com/paper/2506.02867/) (2025)
- [Confidence Matters: Revisiting Intrinsic Self-Correction Capabilities of Large Language Models](https://mlsanity.com/paper/2402.12563/) (2024)
- [Do Not Let Low-Probability Tokens Over-Dominate in RL for LLMs](https://mlsanity.com/paper/2505.12929/) (2025)
- [Klear-Reasoner: Advancing Reasoning Capability via Gradient-Preserving Clipping Policy Optimization](https://mlsanity.com/paper/2508.07629/) (2025)
- [One-shot Entropy Minimization](https://mlsanity.com/paper/2505.20282/) (2025)
- [SCP-116K: A High-Quality Problem-Solution Dataset and a Generalized Pipeline for Automated Extraction in the Higher Education Science Domain](https://mlsanity.com/paper/2501.15587/) (2025)
- [Why Distillation can Outperform Zero-RL: The Role of Flexible Reasoning](https://mlsanity.com/paper/2505.21067/) (2025)
- [Adaptive Termination for Multi-round Parallel Reasoning: An Universal Semantic Entropy-Guided Framework](https://mlsanity.com/paper/2507.06829/) (2025)

## Built on by (most cited first, 25 of 25)

- [Rethinking Sample Polarity in Reinforcement Learning with Verifiable Rewards](https://mlsanity.com/paper/2512.21625/) (2025)
- [ProFit: Leveraging High-Value Signals in SFT via Probability-Guided Token Selection](https://mlsanity.com/paper/2601.09195/) (2026)
- [The Flexibility Trap: Rethinking the Value of Arbitrary Order in Diffusion Language Models](https://mlsanity.com/paper/2601.15165/) (2026)
- [Rewarding the Rare: Uniqueness-Aware RL for Creative Problem Solving in LLMs](https://mlsanity.com/paper/2601.08763/) (2026)
- [MASPO: Unifying Gradient Utilization, Probability Mass, and Signal Reliability for Robust and Sample-Efficient LLM Reasoning](https://mlsanity.com/paper/2602.17550/) (2026)
- [STAPO: Stabilizing Reinforcement Learning for LLMs by Silencing Rare Spurious Tokens](https://mlsanity.com/paper/2602.15620/) (2026)
- [Anchored Policy Optimization: Mitigating Exploration Collapse Via Support-Constrained Rectification](https://mlsanity.com/paper/2602.05717/) (2026)
- [A Tale of Two Temperatures: Simple, Efficient, and Diverse Sampling from Diffusion Language Models](https://mlsanity.com/paper/2604.09921/) (2026)
- [The Art of Efficient Reasoning: Data, Reward, and Optimization](https://mlsanity.com/paper/2602.20945/) (2026)
- [Low-rank Optimization Trajectories Modeling for LLM RLVR Acceleration](https://mlsanity.com/paper/2604.11446/) (2026)
- [Clipping-Free Policy Optimization for Large Language Models](https://mlsanity.com/paper/2601.22801/) (2026)
- [STARE: Surprisal-Guided Token-Level Advantage Reweighting for Policy Entropy Stability](https://mlsanity.com/paper/2606.19236/) (2026)
- [Anti-Length Shift: Dynamic Outlier Truncation for Training Efficient Reasoning Models](https://mlsanity.com/paper/2601.03969/) (2026)
- [LAD: Learning Advantage Distribution for Reasoning](https://mlsanity.com/paper/2602.20132/) (2026)
- [How You Begin is How You Reason: Driving Exploration in RLVR via Prefix-Tuned Priors](https://mlsanity.com/paper/2605.08817/) (2026)
- [Clipping Bottleneck: Stabilizing RLVR via Stochastic Recovery of Near-Boundary Signals](https://mlsanity.com/paper/2605.22703/) (2026)
- [When Implausible Tokens Get Reinforced: Tail-Aware Credit Calibration for LLM Reinforcement Learning](https://mlsanity.com/paper/2607.07976/) (2026)
- [Supervised Fine-Tuning Needs to Unlock the Potential of Token Priority](https://mlsanity.com/paper/2602.01227/) (2026)
- [Free Energy-Driven Reinforcement Learning with Adaptive Advantage Shaping for Unsupervised Reasoning in LLMs](https://mlsanity.com/paper/2605.04065/) (2026)
- [Adapt to Thrive! Adaptive Power-Mean Policy Optimization for Improved LLM Reasoning](https://mlsanity.com/paper/2605.04066/) (2026)
- [When RL Suppresses Its Own Vocabulary: Recovering Reasoning Diversity in Puzzle-to-Math Transfer](https://mlsanity.com/paper/2605.29190/) (2026)
- [ACPO: Asymmetric Credit Policy Optimization via Mode-Local Entropy Surrogate](https://mlsanity.com/paper/2607.03126/) (2026)
- [A First-Principles Theory of Slow Thinking and Active Perception](https://mlsanity.com/paper/2607.08196/) (2026)
- [Video-HopChain: Multi-Hop Questions and Confidence-Gated Exploration for Video Reasoning Models](https://mlsanity.com/paper/2609.25773/) (2026)
- [Surprising Success, Repeated Failure: Entropy-Guided Credit Assignment for Exploration in LLM Reasoning](https://mlsanity.com/paper/2609.33781/) (2026)

## Abstract (arXiv)

Reinforcement Learning with Verifiable Rewards (RLVR) has propelled Large Language Models in complex reasoning, yet its scalability is often hindered by a training bottleneck where performance plateaus as policy entropy collapses, signaling a loss of exploration. Previous methods typically address this by maintaining high policy entropy, yet the precise mechanisms that govern meaningful exploration have remained underexplored. Our analysis suggests that an unselective focus on entropy risks amplifying irrelevant tokens and destabilizing training. This paper investigates the exploration dynamics within RLVR and identifies a key issue: the gradual elimination of valuable low-probability exploratory tokens, which we term textbftextitreasoning sparks. We find that while abundant in pre-trained models, these sparks are systematically extinguished during RLVR due to over-penalization, leading to a degeneracy in exploration. To address this, we introduce Low-probability Regularization (Lp-Reg). Its core mechanism regularizes the policy towards a heuristic proxy distribution. This proxy is constructed by filtering out presumed noise tokens and re-normalizing the distribution over the remaining candidates. The result is a less-noisy proxy where the probability of textitreasoning sparks is amplified, which then serves as a soft regularization target to shield these valuable tokens from elimination via KL divergence. Experiments show that Lp-Reg enables stable on-policy RL, sustaining continuous scaling across 3,000 training steps and 81,204 GPU-hours, where baseline entropy-control methods collapse. This sustained exploration leads to state-of-the-art performance, achieving a 60.17% average accuracy on five math benchmarks, an improvement of 2.66% over prior methods. Code is available at https://github.com/CarlanLark/Lp-Reg.