# Sample What You Say: Aligning Language Models to Sample the Distributions They State

arXiv: https://arxiv.org/abs/2609.34929 · HTML: https://mlsanity.com/paper/2609.34929/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in LLM preference optimization methods, one of 10 directions in LLM alignment / RLHF / preference optimisation.
- Builds on 48 of its 61 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Placed in

- LLM alignment / RLHF / preference optimisation / [LLM preference optimization methods](https://mlsanity.com/alignment/llm-preference-optimization-methods/)

## Builds on (most cited first, 30 of 48)

- [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)
- [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)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Efficient Memory Management for Large Language Model Serving with PagedAttention](https://mlsanity.com/paper/2309.06180/) (2023)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [Understanding R1-Zero-Like Training: A Critical Perspective](https://mlsanity.com/paper/2503.20783/) (2025)
- [Instruction-Following Evaluation for Large Language Models](https://mlsanity.com/paper/2311.07911/) (2023)
- [Whose Opinions Do Language Models Reflect?](https://mlsanity.com/paper/2303.17548/) (2023)
- [The Entropy Mechanism of Reinforcement Learning for Reasoning Language Models](https://mlsanity.com/paper/2505.22617/) (2025)
- [Towards Measuring the Representation of Subjective Global Opinions in Language Models](https://mlsanity.com/paper/2306.16388/) (2023)
- [Understanding the Effects of RLHF on LLM Generalisation and Diversity](https://mlsanity.com/paper/2310.06452/) (2023)
- [Does Writing with Language Models Reduce Content Diversity?](https://mlsanity.com/paper/2309.05196/) (2023)
- [Artificial Hivemind: The Open-Ended Homogeneity of Language Models (and Beyond)](https://mlsanity.com/paper/2510.22954/) (2025)
- [Distributional Preference Learning: Understanding and Accounting for Hidden Context in RLHF](https://mlsanity.com/paper/2312.08358/) (2023)
- [A Distributional Approach to Controlled Text Generation](https://mlsanity.com/paper/2012.11635/) (2020)
- [Verbalized Sampling: How to Mitigate Mode Collapse and Unlock LLM Diversity](https://mlsanity.com/paper/2510.01171/) (2025)
- [Self-Improvement in Language Models: The Sharpening Mechanism](https://mlsanity.com/paper/2412.01951/) (2024)
- [Amortizing intractable inference in large language models](https://mlsanity.com/paper/2310.04363/) (2023)
- [Probabilistic Inference in Language Models via Twisted Sequential Monte Carlo](https://mlsanity.com/paper/2404.17546/) (2024)
- [Pass@K Policy Optimization: Solving Harder Reinforcement Learning Problems](https://mlsanity.com/paper/2505.15201/) (2025)
- [Rewarding the Unlikely: Lifting GRPO Beyond Distribution Sharpening](https://mlsanity.com/paper/2506.02355/) (2025)
- [Jointly Reinforcing Diversity and Quality in Language Model Generations](https://mlsanity.com/paper/2509.02534/) (2025)
- [NoveltyBench: Evaluating Language Models for Humanlike Diversity](https://mlsanity.com/paper/2504.05228/) (2025)
- [Benchmarking Distributional Alignment of Large Language Models](https://mlsanity.com/paper/2411.05403/) (2024)
- [One fish, two fish, but not the whole sea: Alignment reduces language models' conceptual diversity](https://mlsanity.com/paper/2411.04427/) (2024)
- [Base Models Beat Aligned Models at Randomness and Creativity](https://mlsanity.com/paper/2505.00047/) (2025)
- [Forcing Diffuse Distributions out of Language Models](https://mlsanity.com/paper/2404.10859/) (2024)

## Abstract (arXiv)

Language models are increasingly used to sample from a specified distribution, for instance, to simulate survey respondents or generate synthetic data. Instruction-tuned models can state such a distribution correctly and still fail to sample from it. Prompting and changes to decoding reduce this mismatch only partly, which motivates training with policy optimization. Group relative policy optimization (GRPO) is a natural fit for this problem because it already samples a group of rollouts per prompt, and the group's empirical distribution can be compared with the target. However, scoring the group as a whole gives every rollout the same reward. Group-relative centering then sets all advantages to zero, and the model receives no learning signal. To give each rollout its own signal, we introduce the witness advantage, a per-rollout advantage derived from maximum mean discrepancy (MMD). It trains a model to match a target distribution over a finite set of outcomes. The MMD between the model's distribution and the target has a witness function that measures how over- or under-produced each outcome is. Each rollout's advantage estimates the negative witness at its outcome, so a rollout is rewarded for an outcome the group under-produces and penalized for one it over-produces. The witness advantage is computed in closed form from the group's outcome counts, and we use it as the reward in GRPO. On unseen target distributions, training with the witness advantage substantially reduces the total variation distance to the target while largely preserving the model's general capabilities.