# Reference-Sampled Boltzmann Projection for KL-Regularized RLVR: Target-Matched Weighted SFT, Finite One-Shot Gaps, and Policy Mirror Descent

arXiv: https://arxiv.org/abs/2605.02469 · HTML: https://mlsanity.com/paper/2605.02469/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 45 of its 59 references that are in the map.
- First posted on arXiv 4 May 2026.

## Builds on (most cited first, 30 of 45)

- [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)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [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)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Deep reinforcement learning from human preferences](https://mlsanity.com/paper/1706.03741/) (2017)
- [FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness](https://mlsanity.com/paper/2205.14135/) (2022)
- [QLoRA: Efficient Finetuning of Quantized LLMs](https://mlsanity.com/paper/2305.14314/) (2023)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [Learning to summarize from human feedback](https://mlsanity.com/paper/2009.01325/) (2020)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [Fine-Tuning Language Models from Human Preferences](https://mlsanity.com/paper/1909.08593/) (2019)
- [HybridFlow: A Flexible and Efficient RLHF Framework](https://mlsanity.com/paper/2409.19256/) (2024)
- [Conservative Q-Learning for Offline Reinforcement Learning](https://mlsanity.com/paper/2006.04779/) (2020)
- [Decision Transformer: Reinforcement Learning via Sequence Modeling](https://mlsanity.com/paper/2106.01345/) (2021)
- [Kimi k1.5: Scaling Reinforcement Learning with LLMs](https://mlsanity.com/paper/2501.12599/) (2025)
- [Tulu 3: Pushing Frontiers in Open Language Model Post-Training](https://mlsanity.com/paper/2411.15124/) (2024)
- [Does Reinforcement Learning Really Incentivize Reasoning Capacity in LLMs Beyond the Base Model?](https://mlsanity.com/paper/2504.13837/) (2025)
- [SimPO: Simple Preference Optimization with a Reference-Free Reward](https://mlsanity.com/paper/2405.14734/) (2024)
- [STaR: Bootstrapping Reasoning With Reasoning](https://mlsanity.com/paper/2203.14465/) (2022)
- [Advantage-Weighted Regression: Simple and Scalable Off-Policy Reinforcement Learning](https://mlsanity.com/paper/1910.00177/) (2019)
- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [AWAC: Accelerating Online Reinforcement Learning with Offline Datasets](https://mlsanity.com/paper/2006.09359/) (2020)
- [ORPO: Monolithic Preference Optimization without Reference Model](https://mlsanity.com/paper/2403.07691/) (2024)
- [SimpleRL-Zoo: Investigating and Taming Zero Reinforcement Learning for Open Base Models in the Wild](https://mlsanity.com/paper/2503.18892/) (2025)

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

- [Listwise Policy Optimization: Group-based RLVR as Target-Projection on the LLM Response Simplex](https://mlsanity.com/paper/2605.06139/) (2026)

## Abstract (arXiv)

Online reinforcement learning with verifiable rewards (RLVR) turns checkable outcomes into a scalable training signal, but it keeps rollout generation, verifier scoring, and reference-policy evaluations on the optimization path. Static weighted supervised fine-tuning (SFT) on precomputed rollouts seems to remove this bottleneck, yet a weighted likelihood is not specified by rewards alone: its sampler and weights induce the policy being fit. This paper identifies the reference-sampled weighted-SFT objective whose induced policy equals the fixed-reference KL-regularized RLVR optimizer. The optimizer is the standard Boltzmann target policy, obtained by exponentially tilting the reference policy by verifier reward. Matching a weighted-SFT induced policy to this target forces density-ratio weights; in the reference-sampled subclass, this reduces uniquely, up to prompt scaling, to the prompt-normalized Boltzmann weight exp(r(x,y)/β)/Z(x). BOLT, a Boltzmann-Targeted SFT procedure, is the empirical estimator of this projection. The finite one-shot analysis separates the exact stored-support price βlog(1/π^*(SNmid x)) from partition estimation, effective-sample-size variance, generalization, optimization, and approximation errors. This decomposition explains why extra SFT epochs cannot repair missing reference-policy coverage and exposes the temperature--coverage--variance frontier. When coverage needs adaptive sampling, refreshed Boltzmann projections become KL policy mirror descent; finite inner solves enter as additive drift from the exact mirror step. Single-run Qwen experiments provide projection evidence for the target-matched weight, one-shot saturation, refreshed-sampler gains, and optimization-time savings, within the stated single-run scope.