Adaptive Negative Reinforcement for LLM Reasoning:Dynamically Balancing Correction and Diversity in RLVR

Reinforcement learning with verifiable rewards (RLVR) has become a highly effective method for improving the reasoning abilities of Large Language Models (LLMs). Recent research shows that Negative Sample Reinforcement (NSR) -- which focuses on penalizing incorrect steps rather than simply rewarding correct ones -- can match or even exceed the performance of more complex frameworks like PPO and GRPO across the entire Pass@k spectrum. However, current NSR techniques usually apply a fixed penalty throughout the training process and treat every incorrect response with the same weight. To address these limitations, we propose two extensions to the NSR framework: Adaptive Negative Sample Reinforcement. Rather than using a fixed update rule, A-NSR uses time-dependent scheduling functions. In the initial training phases, the system focuses heavily on correcting errors to stabilize the model. As training continues, it shifts toward more subtle and controlled updates. We also introduce Confidence-Weighted Negative Reinforcement, which operates on the principle that different mistakes carry different levels of importance. CW-NSR assigns specific penalty weights based on the model's normalized sequence likelihood. If the model is highly confident in a wrong path, it receives a larger penalty and for uncertain errors -- where the model is effectively exploring -- are penalized less strictly. Our formal analysis shows how these mechanisms govern token-level updates, allowing the model to leverage prior-guided probability redistribution while providing a natural defense against overfitting. We evaluated these methods on difficult reasoning datasets, including MATH, AIME 2025, and AMC23, using the Qwen2.5-Math-1.5B architecture.