# RSPO: Risk-Seeking Policy Optimization for Pass@k and Max@k Metrics in Large Language Models

arXiv: https://arxiv.org/abs/2508.01174 · HTML: https://mlsanity.com/paper/2508.01174/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 34 references that are in the map.
- First posted on arXiv 2 Aug 2025.

## Builds on (most cited first, 15 of 15)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [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)
- [HybridFlow: A Flexible and Efficient RLHF Framework](https://mlsanity.com/paper/2409.19256/) (2024)
- [Solving Quantitative Reasoning Problems with Language Models](https://mlsanity.com/paper/2206.14858/) (2022)
- [Does Reinforcement Learning Really Incentivize Reasoning Capacity in LLMs Beyond the Base Model?](https://mlsanity.com/paper/2504.13837/) (2025)
- [Large Language Models: A Survey](https://mlsanity.com/paper/2402.06196/) (2024)
- [BoNBoN Alignment for Large Language Models and the Sweetness of Best-of-n Sampling](https://mlsanity.com/paper/2406.00832/) (2024)
- [Reasoning or Memorization? Unreliable Results of Reinforcement Learning Due to Data Contamination](https://mlsanity.com/paper/2507.10532/) (2025)
- [Are Your LLMs Capable of Stable Reasoning?](https://mlsanity.com/paper/2412.13147/) (2024)
- [A Survey on Large Language Models for Mathematical Reasoning](https://mlsanity.com/paper/2506.08446/) (2025)
- [Learning from negative feedback, or positive feedback or both](https://mlsanity.com/paper/2410.04166/) (2024)

## Built on by (most cited first, 5 of 5)

- [Beyond the Sampled Token: Preserving Candidate Support in RLVR](https://mlsanity.com/paper/2510.14807/) (2025)
- [Leveraging Error Diversity in Group Rollouts for Reinforcement Learning](https://mlsanity.com/paper/2605.17333/) (2026)
- [Boosting LLM Reasoning via Human-Inspired Reward Shaping](https://mlsanity.com/paper/2602.04265/) (2026)
- [Rank-Conditioned Sample Reuse for the Plackett--Luce Best-of-K Objective](https://mlsanity.com/paper/2607.11146/) (2026)
- [GVPO++: Group Variance Policy Optimization for LLM Post-Training and On-Policy Distillation](https://mlsanity.com/paper/2609.21432/) (2026)

## Abstract (arXiv)

Current large language model post-training optimizes a risk-neutral objective that maximizes expected reward, yet evaluation relies heavily on risk-seeking metrics like Pass@k (at least one success in k trials) and Max@k (maximum reward across k responses). This mismatch in risk preferences can inevitably lead to suboptimal performance. To bridge this gap, we propose Risk-Seeking Policy Optimization (RSPO), a novel method that directly targets Pass@k and Max@k during training. A key challenge in optimizing these metrics is the "hitchhiking" problem: low-reward responses are inadvertently reinforced if they co-occur with a high-reward response within a sample of k generations, resulting in inefficient optimization. RSPO addresses this problem by leveraging the closed-form probability that a given response is the maximum among k samplings. Despite the complexity of nested gradients over multiple responses, RSPO produces efficient, unbiased gradient estimators for both metrics. We validate our approach with both rigorous theoretical analysis and comprehensive experimental results.