# When Errors Can Be Beneficial: A Categorization of Imperfect Rewards for Policy Gradient

arXiv: https://arxiv.org/abs/2604.25872 · HTML: https://mlsanity.com/paper/2604.25872/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Reward modeling and RLHF, one of 10 directions in LLM alignment / RLHF / preference optimisation.
- Builds on 29 of its 99 references that are in the map.
- First posted on arXiv 28 Apr 2026.

## Placed in

- LLM alignment / RLHF / preference optimisation / [Reward modeling and RLHF](https://mlsanity.com/alignment/reward-modeling-and-rlhf/)
- Deep reinforcement learning methods / [RL training of language and agent models](https://mlsanity.com/deeprl/rl-training-of-language-and-agent-models/)

## Builds on (most cited first, 29 of 29)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [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)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [HuggingFace's Transformers: State-of-the-art Natural Language Processing](https://mlsanity.com/paper/1910.03771/) (2019)
- [Fine-Tuning Language Models from Human Preferences](https://mlsanity.com/paper/1909.08593/) (2019)
- [Concrete Problems in AI Safety](https://mlsanity.com/paper/1606.06565/) (2016)
- [Dota 2 with Large Scale Deep Reinforcement Learning](https://mlsanity.com/paper/1912.06680/) (2019)
- [Tulu 3: Pushing Frontiers in Open Language Model Post-Training](https://mlsanity.com/paper/2411.15124/) (2024)
- [Interpretable Preferences via Multi-Objective Reward Modeling and Mixture-of-Experts](https://mlsanity.com/paper/2406.12845/) (2024)
- [2 OLMo 2 Furious](https://mlsanity.com/paper/2501.00656/) (2024)
- [Is DPO Superior to PPO for LLM Alignment? A Comprehensive Study](https://mlsanity.com/paper/2404.10719/) (2024)
- [Skywork-Reward-V2: Scaling Preference Data Curation via Human-AI Synergy](https://mlsanity.com/paper/2507.01352/) (2025)
- [RewardBench 2: Advancing Reward Model Evaluation](https://mlsanity.com/paper/2506.01937/) (2025)
- [Olmo 3](https://mlsanity.com/paper/2512.13961/) (2025)
- [Understanding the performance gap between online and offline alignment algorithms](https://mlsanity.com/paper/2405.08448/) (2024)
- [Back to Basics: Revisiting REINFORCE Style Optimization for Learning from Human Feedback in LLMs](https://mlsanity.com/paper/2402.14740/) (2024)
- [Ray Interference: a Source of Plateaus in Deep Reinforcement Learning](https://mlsanity.com/paper/1904.11455/) (2019)
- [InfAlign: Inference-aware language model alignment](https://mlsanity.com/paper/2412.19792/) (2024)
- [From Accuracy to Robustness: A Study of Rule- and Model-based Verifiers in Mathematical Reasoning](https://mlsanity.com/paper/2505.22203/) (2025)
- [Hybrid Reinforcement: When Reward Is Sparse, It's Better to Be Dense](https://mlsanity.com/paper/2510.07242/) (2025)
- [Elementary Analysis of Policy Gradient Methods](https://mlsanity.com/paper/2404.03372/) (2024)
- [Neural Replicator Dynamics](https://mlsanity.com/paper/1906.00190/) (2019)
- [On the Emergence of Implicit Curriculum in RLVR Learning Dynamics](https://mlsanity.com/paper/2602.14872/) (2026)
- [Outcome-Based RL Provably Leads Transformers to Reason, but Only With the Right Data](https://mlsanity.com/paper/2601.15158/) (2026)
- [Rethinking the Global Convergence of Softmax Policy Gradient with Linear Function Approximation: The Case of Multi-Armed Bandits](https://mlsanity.com/paper/2505.03155/) (2025)
- [Reward Engineering for Software Tasks: A Survey of Reinforcement Learning Approaches](https://mlsanity.com/paper/2601.19100/) (2026)

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

- [Multimodal Reward Hacking in Reinforcement Learning](https://mlsanity.com/paper/2607.09492/) (2026)

## Abstract (arXiv)

Training language models via reinforcement learning often relies on imperfect proxy rewards, since ground truth rewards that precisely define the intended behavior are rarely available. Standard metrics for assessing the quality of proxy rewards, such as ranking accuracy, treat incorrect rewards as strictly harmful. In this work, however, we highlight that not all deviations from the ground truth are equal. By theoretically analyzing which outputs attract probability during policy gradient optimization, we categorize reward errors according to their effect on the increase in ground truth reward. The analysis establishes that reward errors, though conventionally viewed as harmful, can also be benign or even beneficial by preventing the policy from stalling around outputs with mediocre ground truth reward. We then present two practical implications of our theory. First, for reinforcement learning from human feedback (RLHF), we develop reward model evaluation metrics that account for the harmfulness of reward errors. Compared to standard ranking accuracy, these metrics typically correlate better with the performance of a language model after RLHF, yet gaps remain in robustly evaluating reward models. Second, we provide insights for reward design in settings with verifiable rewards. A key theme underlying our results is that the effectiveness of a proxy reward function depends heavily on its interaction with the initial policy and learning algorithm.