# Sparse but Critical: A Token-Level Analysis of Distributional Shifts in RLVR Fine-Tuning of LLMs

arXiv: https://arxiv.org/abs/2603.22446 · HTML: https://mlsanity.com/paper/2603.22446/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 36 of its 40 references that are in the map.
- First posted on arXiv 23 Mar 2026.

## Placed in

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

## Builds on (most cited first, 30 of 36)

- [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)
- [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)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [Deep reinforcement learning from human preferences](https://mlsanity.com/paper/1706.03741/) (2017)
- [The Curious Case of Neural Text Degeneration](https://mlsanity.com/paper/1904.09751/) (2019)
- [GPQA: A Graduate-Level Google-Proof Q&A Benchmark](https://mlsanity.com/paper/2311.12022/) (2023)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [HybridFlow: A Flexible and Efficient RLHF Framework](https://mlsanity.com/paper/2409.19256/) (2024)
- [Fast Inference from Transformers via Speculative Decoding](https://mlsanity.com/paper/2211.17192/) (2022)
- [Understanding R1-Zero-Like Training: A Critical Perspective](https://mlsanity.com/paper/2503.20783/) (2025)
- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [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)
- [Group Sequence Policy Optimization](https://mlsanity.com/paper/2507.18071/) (2025)
- [SFT Memorizes, RL Generalizes: A Comparative Study of Foundation Model Post-training](https://mlsanity.com/paper/2501.17161/) (2025)
- [Beyond the 80/20 Rule: High-Entropy Minority Tokens Drive Effective Reinforcement Learning for LLM Reasoning](https://mlsanity.com/paper/2506.01939/) (2025)
- [SimpleRL-Zoo: Investigating and Taming Zero Reinforcement Learning for Open Base Models in the Wild](https://mlsanity.com/paper/2503.18892/) (2025)
- [The Entropy Mechanism of Reinforcement Learning for Reasoning Language Models](https://mlsanity.com/paper/2505.22617/) (2025)
- [Reinforcement Learning with Verifiable Rewards Implicitly Incentivizes Correct Reasoning in Base LLMs](https://mlsanity.com/paper/2506.14245/) (2025)
- [Reasoning with Exploration: An Entropy Perspective](https://mlsanity.com/paper/2506.14758/) (2025)
- [Spurious Rewards: Rethinking Training Signals in RLVR](https://mlsanity.com/paper/2506.10947/) (2025)
- [ProRL: Prolonged Reinforcement Learning Expands Reasoning Boundaries in Large Language Models](https://mlsanity.com/paper/2505.24864/) (2025)
- [RL's Razor: Why Online Reinforcement Learning Forgets Less](https://mlsanity.com/paper/2509.04259/) (2025)
- [Speculative Decoding with Big Little Decoder](https://mlsanity.com/paper/2302.07863/) (2023)
- [Does Math Reasoning Improve General LLM Capabilities? Understanding Transferability of LLM Reasoning](https://mlsanity.com/paper/2507.00432/) (2025)
- [Reasoning with Sampling: Your Base Model is Smarter Than You Think](https://mlsanity.com/paper/2510.14901/) (2025)
- [Critical Tokens Matter: Token-Level Contrastive Estimation Enhances LLM's Reasoning Capability](https://mlsanity.com/paper/2411.19943/) (2024)
- [The Path Not Taken: RLVR Provably Learns Off the Principals](https://mlsanity.com/paper/2511.08567/) (2025)

## Built on by (most cited first, 20 of 20)

- [Rebellious Student: Reversing Teacher Signals for Reasoning Exploration with Self-Distilled RLVR](https://mlsanity.com/paper/2605.10781/) (2026)
- [Cornerstones or Stumbling Blocks? Deciphering the Rock Tokens in On-Policy Distillation](https://mlsanity.com/paper/2605.09253/) (2026)
- [ParaVT: Taming the Tool Prior Paradox for Parallel Tool Use in Agentic Video Reinforcement Learning](https://mlsanity.com/paper/2605.20342/) (2026)
- [APPO: Agentic Procedural Policy Optimization](https://mlsanity.com/paper/2606.12384/) (2026)
- [Entropy Polarity in Reinforcement Fine-Tuning: Direction, Asymmetry, and Control](https://mlsanity.com/paper/2605.11775/) (2026)
- [Which Tokens Matter? Adaptive Token Selection for RLVR with the Relative Surprisal Index](https://mlsanity.com/paper/2606.31575/) (2026)
- [Not All Tokens Learn Alike: Attention Entropy Reveals Heterogeneous Signals in RL Reasoning](https://mlsanity.com/paper/2605.07660/) (2026)
- [One-Way Policy Optimization for Self-Evolving LLMs](https://mlsanity.com/paper/2605.22156/) (2026)
- [DART-SD: Diamond-topology Aware Retrieval and Tuning for Self-Distillation of Multi-Turn Tool-Calling Agents](https://mlsanity.com/paper/2608.18524/) (2026)
- [DelTA: Discriminative Token Credit Assignment for Reinforcement Learning from Verifiable Rewards](https://mlsanity.com/paper/2605.21467/) (2026)
- [Clipping Bottleneck: Stabilizing RLVR via Stochastic Recovery of Near-Boundary Signals](https://mlsanity.com/paper/2605.22703/) (2026)
- [Beyond Entropy: Learning from Token-Level Distributional Deviations for LLM Reasoning](https://mlsanity.com/paper/2606.19771/) (2026)
- [Verifier-Induced Support Reshaping in On-Policy Optimization](https://mlsanity.com/paper/2608.00220/) (2026)
- [Not only where, But when: Temporal Scheduling for RLVR](https://mlsanity.com/paper/2605.25381/) (2026)
- [ARCA: Adapter-Residual Credit Assignment When Token Signals Degenerate](https://mlsanity.com/paper/2606.00257/) (2026)
- [Tandem Reinforcement Learning with Verifiable Rewards](https://mlsanity.com/paper/2606.28166/) (2026)
- [ARMOR: Stabilizing On-Policy LLM RL with Off-Policy Anchor Samples](https://mlsanity.com/paper/2607.10481/) (2026)
- [Divergence Decoding: Training-Free Capability Fusion](https://mlsanity.com/paper/2607.27248/) (2026)
- [From Storage to Access: Verifiable Activation of Parametric Knowledge in LLMs via Explicit Priming and Implicit Reasoning](https://mlsanity.com/paper/2608.18581/) (2026)
- [Advancing Entropy-Level Credit Assignment in RLVR via Proximal Entropy Policy Optimization](https://mlsanity.com/paper/2609.39402/) (2026)

## Abstract (arXiv)

Reinforcement learning with verifiable rewards (RLVR) has significantly improved reasoning in large language models (LLMs), yet the token-level mechanisms underlying these improvements remain unclear. We present a systematic empirical study of RLVR's distributional effects organized around three main analyses: (1) token-level characterization of distributional shifts between base and RL models, (2) the impact of token-level distributional shifts on sequence-level reasoning performance through cross-sampling interventions, and (3) fine-grained mechanics of these shifts at the token level. We find that RL fine-tuning induces highly sparse and targeted changes, with only a small fraction of token distributions exhibiting meaningful divergence between the base and RL policies. We further characterize the structure and evolution of these shifts through analyses of token entropy, positional concentration, and reallocation of probability mass. To assess the functional importance of these sparse changes, we conduct cross-sampling experiments that selectively swap token choices between the base and RL models with varying intervention budgets. We show that inserting only a small fraction of RL-sampled tokens into base generations progressively recovers RL performance gains, while injecting a similarly small number of base token choices into otherwise RL-generated sequences collapses performance to base levels, isolating a small set of token-level decisions directly responsible for RLVR's performance gains. Finally, we explore divergence-weighted variants of the advantage signal as a diagnostic intervention, finding that they can yield improvements over baselines. Together, our results shed light on the distributional changes induced by RLVR and provide a fine-grained, token-level lens for understanding RLVR fine-tuning as a targeted refinement process.