# Correlated Proxies: A New Definition and Improved Mitigation for Reward Hacking

arXiv: https://arxiv.org/abs/2403.03185 · HTML: https://mlsanity.com/paper/2403.03185/

## Key facts

- Cited by 50 papers in the map (data through 14 Oct 2026).
- Placed in Reward hacking and misspecification, one of 10 directions in LLM alignment / RLHF / preference optimisation.
- Builds on 64 of its 92 references that are in the map.
- First posted on arXiv 5 Mar 2024.

## Placed in

- LLM alignment / RLHF / preference optimisation / [Reward hacking and misspecification](https://mlsanity.com/alignment/reward-hacking-and-misspecification/)

## Builds on (most cited first, 30 of 64)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Training a Helpful and Harmless Assistant with Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2204.05862/) (2022)
- [Learning to summarize from human feedback](https://mlsanity.com/paper/2009.01325/) (2020)
- [Pythia: A Suite for Analyzing Large Language Models Across Training and Scaling](https://mlsanity.com/paper/2304.01373/) (2023)
- [Concrete Problems in AI Safety](https://mlsanity.com/paper/1606.06565/) (2016)
- [Generative Adversarial Imitation Learning](https://mlsanity.com/paper/1606.03476/) (2016)
- [Off-Policy Deep Reinforcement Learning without Exploration](https://mlsanity.com/paper/1812.02900/) (2018)
- [Scaling Laws for Reward Model Overoptimization](https://mlsanity.com/paper/2210.10760/) (2022)
- [AlpacaFarm: A Simulation Framework for Methods that Learn from Human Feedback](https://mlsanity.com/paper/2305.14387/) (2023)
- [Improving alignment of dialogue agents via targeted human judgements](https://mlsanity.com/paper/2209.14375/) (2022)
- [Scalable agent alignment via reward modeling: a research direction](https://mlsanity.com/paper/1811.07871/) (2018)
- [Algorithmic decision making and the cost of fairness](https://mlsanity.com/paper/1701.08230/) (2017)
- [The Effects of Reward Misspecification: Mapping and Mitigating Misaligned Models](https://mlsanity.com/paper/2201.03544/) (2022)
- [Reward learning from human preferences and demonstrations in Atari](https://mlsanity.com/paper/1811.06521/) (2018)
- [PEBBLE: Feedback-Efficient Interactive Reinforcement Learning via Relabeling Experience and Unsupervised Pre-training](https://mlsanity.com/paper/2106.05091/) (2021)
- [MINT: Evaluating LLMs in Multi-turn Interaction with Tools and Language Feedback](https://mlsanity.com/paper/2309.10691/) (2023)
- [Bridging Offline Reinforcement Learning and Imitation Learning: A Tale of Pessimism](https://mlsanity.com/paper/2103.12021/) (2021)
- [Bellman-consistent Pessimism for Offline Reinforcement Learning](https://mlsanity.com/paper/2106.06926/) (2021)
- [The Alignment Problem from a Deep Learning Perspective](https://mlsanity.com/paper/2209.00626/) (2022)
- [Inverse Reward Design](https://mlsanity.com/paper/1711.02827/) (2017)
- [Safe Exploration in Continuous Action Spaces](https://mlsanity.com/paper/1801.08757/) (2018)
- [Provably Efficient Maximum Entropy Exploration](https://mlsanity.com/paper/1812.02690/) (2018)
- [Reward Model Ensembles Help Mitigate Overoptimization](https://mlsanity.com/paper/2310.02743/) (2023)
- [Defining and Characterizing Reward Hacking](https://mlsanity.com/paper/2209.13085/) (2022)
- [Efficient Exploration via State Marginal Matching](https://mlsanity.com/paper/1906.05274/) (2019)
- [Beyond Reverse KL: Generalizing Direct Preference Optimization with Diverse Divergence Constraints](https://mlsanity.com/paper/2309.16240/) (2023)
- [Categorizing Variants of Goodhart's Law](https://mlsanity.com/paper/1803.04585/) (2018)

## Built on by (most cited first, 30 of 50)

- [Monitoring Reasoning Models for Misbehavior and the Risks of Promoting Obfuscation](https://mlsanity.com/paper/2503.11926/) (2025)
- [Curriculum Reinforcement Learning from Easy to Hard Tasks Improves LLM Reasoning](https://mlsanity.com/paper/2506.06632/) (2025)
- [A Comprehensive Survey of Reward Models: Taxonomy, Applications, Challenges, and Future](https://mlsanity.com/paper/2504.12328/) (2025)
- [Inference-Time Reward Hacking in Large Language Models](https://mlsanity.com/paper/2506.19248/) (2025)
- [Reward Hacking in the Era of Large Models: Mechanisms, Emergent Misalignment, Challenges](https://mlsanity.com/paper/2604.13602/) (2026)
- [CHiP: Cross-modal Hierarchical Direct Preference Optimization for Multimodal LLMs](https://mlsanity.com/paper/2501.16629/) (2025)
- [SpecBench: Measuring Reward Hacking in Long-Horizon Coding Agents](https://mlsanity.com/paper/2605.21384/) (2026)
- [GARDO: Reinforcing Diffusion Models without Reward Hacking](https://mlsanity.com/paper/2512.24138/) (2025)
- [Adaptive Rollout Allocation for Online Reinforcement Learning with Verifiable Rewards](https://mlsanity.com/paper/2602.01601/) (2026)
- [Countdown-Code: A Testbed for Studying The Emergence and Generalization of Reward Hacking in RLVR](https://mlsanity.com/paper/2603.07084/) (2026)
- [Alignment and Safety in Large Language Models: Safety Mechanisms, Training Paradigms, and Emerging Challenges](https://mlsanity.com/paper/2507.19672/) (2025)
- [Recontextualization Mitigates Specification Gaming without Modifying the Specification](https://mlsanity.com/paper/2512.19027/) (2025)
- [Flattery, Fluff, and Fog: Diagnosing and Mitigating Idiosyncratic Biases in Preference Models](https://mlsanity.com/paper/2506.05339/) (2025)
- [Doubly Robust Alignment for Large Language Models](https://mlsanity.com/paper/2506.01183/) (2025)
- [Bradley-Terry and Multi-Objective Reward Modeling Are Complementary](https://mlsanity.com/paper/2507.07375/) (2025)
- [Hummer: Towards Limited Competitive Preference Dataset](https://mlsanity.com/paper/2405.11647/) (2024)
- [Towards a Sharp Analysis of Offline Policy Learning for f-Divergence-Regularized Contextual Bandits](https://mlsanity.com/paper/2502.06051/) (2025)
- [VRPO: Rethinking Value Modeling for Robust RL under Noisy Supervision in LLM Post-Training](https://mlsanity.com/paper/2508.03058/) (2025)
- [The Perils of Optimizing Learned Reward Functions: Low Training Error Does Not Guarantee Low Regret](https://mlsanity.com/paper/2406.15753/) (2024)
- [Efficient RLVR Training via Weighted Mutual Information Data Selection](https://mlsanity.com/paper/2603.01907/) (2026)
- [Shallow Preference Signals: Large Language Model Aligns Even Better with Truncated Data?](https://mlsanity.com/paper/2505.17122/) (2025)
- [Improved techniques for fine-tuning flow models via adjoint matching: a deterministic control pipeline](https://mlsanity.com/paper/2605.06583/) (2026)
- [Imperfect World Models are Exploitable](https://mlsanity.com/paper/2605.15960/) (2026)
- [Reward Hacking in Language Model Agents: Revisiting AI Safety Gridworlds](https://mlsanity.com/paper/2606.15385/) (2026)
- [Debate Training Reduces Reward Hacking in RLAIF](https://mlsanity.com/paper/2608.17776/) (2026)
- [CANDERE-COACH: Reinforcement Learning from Noisy Feedback](https://mlsanity.com/paper/2409.15521/) (2024)
- [Best-of-Tails: Bridging Optimism and Pessimism in Inference-Time Alignment](https://mlsanity.com/paper/2603.06797/) (2026)
- [From Curiosity to Caution: Mitigating Reward Hacking for Best-of-N with Pessimism](https://mlsanity.com/paper/2604.04648/) (2026)
- [CoAct: Co-Active LLM Preference Learning with Human-AI Synergy](https://mlsanity.com/paper/2604.17501/) (2026)
- [Engineering Reliable Coding Agents: Evaluating and Operating the System Around the Model](https://mlsanity.com/paper/2608.13867/) (2026)

## Abstract (arXiv)

Because it is difficult to precisely specify complex objectives, reinforcement learning policies are often optimized using proxy reward functions that only approximate the true goal. However, optimizing proxy rewards frequently leads to reward hacking: the optimized reward function ceases to be a good proxy and the resulting policy performs poorly with respect to the unspecified true reward. Principled solutions to reward hacking have been impeded by the lack of a good definition for the problem. To address this gap, we introduce a definition of reward hacking based on the correlation between proxy and true rewards for states and actions seen by a "reference policy" that breaks down under optimization. We show that this definition captures reward hacking behavior across several realistic settings, including in reinforcement learning from human feedback (RLHF). Using our formulation, we show theoretically that regularization to the reference policy can effectively prevent reward hacking. While the current practice in RLHF applies a KL penalty between action distributions for this purpose, our theory suggests regularizing the chi2 divergence between the policies' occupancy measures can be more effective. We intuitively show the benefits of this type of regularization and demonstrate that it better mitigates reward hacking in practice across four realistic settings, including RLHF. Our code is available at https://github.com/cassidylaidlaw/orpo.