# Variational Reasoning for Language Models

arXiv: https://arxiv.org/abs/2509.22637 · HTML: https://mlsanity.com/paper/2509.22637/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 53 of its 78 references that are in the map.
- First posted on arXiv 26 Sep 2025.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)

## Builds on (most cited first, 30 of 53)

- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Gemini 2.5: Pushing the Frontier with Advanced Reasoning, Multimodality, Long Context, and Next Generation Agentic Capabilities](https://mlsanity.com/paper/2507.06261/) (2025)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [LlamaFactory: Unified Efficient Fine-Tuning of 100+ Language Models](https://mlsanity.com/paper/2403.13372/) (2024)
- [Search-R1: Training LLMs to Reason and Leverage Search Engines with Reinforcement Learning](https://mlsanity.com/paper/2503.09516/) (2025)
- [Understanding R1-Zero-Like Training: A Critical Perspective](https://mlsanity.com/paper/2503.20783/) (2025)
- [s1: Simple test-time scaling](https://mlsanity.com/paper/2501.19393/) (2025)
- [OpenAI o1 System Card](https://mlsanity.com/paper/2412.16720/) (2024)
- [Kimi k1.5: Scaling Reinforcement Learning with LLMs](https://mlsanity.com/paper/2501.12599/) (2025)
- [Does Reinforcement Learning Really Incentivize Reasoning Capacity in LLMs Beyond the Base Model?](https://mlsanity.com/paper/2504.13837/) (2025)
- [Solving math word problems with process- and outcome-based feedback](https://mlsanity.com/paper/2211.14275/) (2022)
- [Group Sequence Policy Optimization](https://mlsanity.com/paper/2507.18071/) (2025)
- [RAFT: Reward rAnked FineTuning for Generative Foundation Model Alignment](https://mlsanity.com/paper/2304.06767/) (2023)
- [SFT Memorizes, RL Generalizes: A Comparative Study of Foundation Model Post-training](https://mlsanity.com/paper/2501.17161/) (2025)
- [Eureka: Human-Level Reward Design via Coding Large Language Models](https://mlsanity.com/paper/2310.12931/) (2023)
- [Process Reinforcement through Implicit Rewards](https://mlsanity.com/paper/2502.01456/) (2025)
- [The Entropy Mechanism of Reinforcement Learning for Reasoning Language Models](https://mlsanity.com/paper/2505.22617/) (2025)
- [Importance Weighted Autoencoders](https://mlsanity.com/paper/1509.00519/) (2015)
- [RAGEN: Understanding Self-Evolution in LLM Agents via Multi-Turn Reinforcement Learning](https://mlsanity.com/paper/2504.20073/) (2025)
- [OpenThoughts: Data Recipes for Reasoning Models](https://mlsanity.com/paper/2506.04178/) (2025)
- [VAPO: Efficient and Reliable Reinforcement Learning for Advanced Reasoning Tasks](https://mlsanity.com/paper/2504.05118/) (2025)
- [SWE-smith: Scaling Data for Software Engineering Agents](https://mlsanity.com/paper/2504.21798/) (2025)
- [Reasoning with Exploration: An Entropy Perspective](https://mlsanity.com/paper/2506.14758/) (2025)
- [SWE-RL: Advancing LLM Reasoning via Reinforcement Learning on Open Software Evolution](https://mlsanity.com/paper/2502.18449/) (2025)
- [Logic-RL: Unleashing LLM Reasoning with Rule-Based Reinforcement Learning](https://mlsanity.com/paper/2502.14768/) (2025)
- [REINFORCE++: Stabilizing Critic-Free Policy Optimization with Global Advantage Normalization](https://mlsanity.com/paper/2501.03262/) (2025)

## Built on by (most cited first, 8 of 8)

- [Privileged Information Distillation for Language Models](https://mlsanity.com/paper/2602.04942/) (2026)
- [Learning from Language Feedback via Variational Policy Distillation](https://mlsanity.com/paper/2605.15113/) (2026)
- [Learning from Your Own Mistakes: Constructing Learnable Micro-Reflective Trajectories for Self-Distillation](https://mlsanity.com/paper/2606.18844/) (2026)
- [Coupled Variational Reinforcement Learning for Language Model General Reasoning](https://mlsanity.com/paper/2512.12576/) (2025)
- [Multimodal Continuous Reasoning via Asymmetric Mutual Variational Learning](https://mlsanity.com/paper/2607.00461/) (2026)
- [A First-Principles Theory of Slow Thinking and Active Perception](https://mlsanity.com/paper/2607.08196/) (2026)
- [Latent Memory Palace: Reasoning for Control as Autoregressive Variational Inference](https://mlsanity.com/paper/2607.08724/) (2026)
- [Predictive Divergence Masks for LLM RL](https://mlsanity.com/paper/2607.10848/) (2026)

## Abstract (arXiv)

We introduce a variational reasoning framework for language models that treats thinking traces as latent variables and optimizes them through variational inference. Starting from the evidence lower bound (ELBO), we extend it to a multi-trace objective for tighter bounds and propose a forward-KL formulation that stabilizes the training of the variational posterior. We further show that rejection sampling finetuning and binary-reward RL, including GRPO, can be interpreted as local forward-KL objectives, where an implicit weighting by model accuracy naturally arises from the derivation and reveals a previously unnoticed bias toward easier questions. We empirically validate our method on the Qwen 2.5 and Qwen 3 model families across a wide range of reasoning tasks. Overall, our work provides a principled probabilistic perspective that unifies variational inference with RL-style methods and yields stable objectives for improving the reasoning ability of language models. Our code is available at https://github.com/sail-sg/variational-reasoning.