# Learning to Reason with Mixture of Tokens

arXiv: https://arxiv.org/abs/2509.21482 · HTML: https://mlsanity.com/paper/2509.21482/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 25 of its 32 references that are in the map.
- First posted on arXiv 25 Sep 2025.

## Placed in

- LLM reasoning / [Reinforcement learning for reasoning](https://mlsanity.com/lr/reinforcement-learning-for-reasoning/)

## Builds on (most cited first, 25 of 25)

- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [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)
- [Understanding R1-Zero-Like Training: A Critical Perspective](https://mlsanity.com/paper/2503.20783/) (2025)
- [Does Reinforcement Learning Really Incentivize Reasoning Capacity in LLMs Beyond the Base Model?](https://mlsanity.com/paper/2504.13837/) (2025)
- [Group Sequence Policy Optimization](https://mlsanity.com/paper/2507.18071/) (2025)
- [Training Large Language Models to Reason in a Continuous Latent Space](https://mlsanity.com/paper/2412.06769/) (2024)
- [Beyond the 80/20 Rule: High-Entropy Minority Tokens Drive Effective Reinforcement Learning for LLM Reasoning](https://mlsanity.com/paper/2506.01939/) (2025)
- [Scaling up Test-Time Compute with Latent Reasoning: A Recurrent Depth Approach](https://mlsanity.com/paper/2502.05171/) (2025)
- [Absolute Zero: Reinforced Self-play Reasoning with Zero Data](https://mlsanity.com/paper/2505.03335/) (2025)
- [The Illusion of Thinking: Understanding the Strengths and Limitations of Reasoning Models via the Lens of Problem Complexity](https://mlsanity.com/paper/2506.06941/) (2025)
- [Reinforcement Learning with Verifiable Rewards Implicitly Incentivizes Correct Reasoning in Base LLMs](https://mlsanity.com/paper/2506.14245/) (2025)
- [From Explicit CoT to Implicit CoT: Learning to Internalize CoT Step by Step](https://mlsanity.com/paper/2405.14838/) (2024)
- [ProRL: Prolonged Reinforcement Learning Expands Reasoning Boundaries in Large Language Models](https://mlsanity.com/paper/2505.24864/) (2025)
- [REASONING GYM: Reasoning Environments for Reinforcement Learning with Verifiable Rewards](https://mlsanity.com/paper/2505.24760/) (2025)
- [Soft Thinking: Unlocking the Reasoning Potential of LLMs in Continuous Concept Space](https://mlsanity.com/paper/2505.15778/) (2025)
- [SoftCoT: Soft Chain-of-Thought for Efficient Reasoning with LLMs](https://mlsanity.com/paper/2502.12134/) (2025)
- [Token Assorted: Mixing Latent and Text Tokens for Improved Language Model Reasoning](https://mlsanity.com/paper/2502.03275/) (2025)
- [Winning Gold at IMO 2025 with a Model-Agnostic Verification-and-Refinement Pipeline](https://mlsanity.com/paper/2507.15855/) (2025)
- [Reasoning by Superposition: A Theoretical Perspective on Chain of Continuous Thought](https://mlsanity.com/paper/2505.12514/) (2025)
- [Soft Tokens, Hard Truths](https://mlsanity.com/paper/2509.19170/) (2025)
- [AlphaOne: Reasoning Models Thinking Slow and Fast at Test Time](https://mlsanity.com/paper/2505.24863/) (2025)
- [Text Generation Beyond Discrete Token Sampling](https://mlsanity.com/paper/2505.14827/) (2025)
- [HDFlow: Enhancing LLM Complex Problem-Solving with Hybrid Thinking and Dynamic Workflows](https://mlsanity.com/paper/2409.17433/) (2024)
- [Learning Refined Document Representations for Dense Retrieval via Deliberate Thinking](https://mlsanity.com/paper/2502.12974/) (2025)

## Built on by (most cited first, 3 of 3)

- [SofT-GRPO: Surpassing Discrete-Token LLM Reinforcement Learning via Gumbel-Reparameterized Soft-Thinking Policy Optimization](https://mlsanity.com/paper/2511.06411/) (2025)
- [OPPO: Bayesian Value Recursion for Token-Level Credit Assignment in LLM Reasoning](https://mlsanity.com/paper/2605.21851/) (2026)
- [SuperThoughts: Reasoning Tokens in Superposition](https://mlsanity.com/paper/2606.13862/) (2026)

## Abstract (arXiv)

Reinforcement learning with verifiable rewards (RLVR) has become a leading approach for improving large language model (LLM) reasoning capabilities. Most current methods follow variants of Group Relative Policy Optimization, which samples multiple reasoning completions, scores them relative to each other, and adjusts the policy accordingly. However, these approaches invariably sample discrete tokens at each reasoning step, discarding the rich distributional information in the model's probability distribution over candidate tokens. While preserving and utilizing this distributional information has proven beneficial in non-RL settings, current RLVR methods seem to be unnecessarily constraining the reasoning search space by not using this information. To address this limitation, we investigate mixture-of-token generation (MoT-G) in RLVR. We present a unified framework that generalizes existing MoT-G approaches, including existing training-free methods that construct mixture embeddings as weighted sums over token embeddings, and extend RLVR to operate directly in this continuous mixture space for generating chain-of-thought. Evaluating two MoT-G variants on Reasoning-Gym, a suite of reasoning-intensive language tasks, we find that MoT--G methods achieve substantial improvements (5--35 % gains on 7 out of 10 tasks) compared to standard decoding with the Qwen2.5-1.5B model, while reaching comparable accuracy with half the number of trajectories, suggesting improved training efficiency. Through comprehensive hidden-state and token-level analyses, we provide evidence that MoT--G's benefits may stem from its ability to maintain higher hidden-state entropy throughout the reasoning process and promote exploration in token space.