# Certain Head, Uncertain Tail: Expert-Sample for Test-Time Scaling in Fine-Grained MoE

arXiv: https://arxiv.org/abs/2602.02443 · HTML: https://mlsanity.com/paper/2602.02443/

## Key facts

- Cited by 2 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 22 of its 33 references that are in the map.
- First posted on arXiv 2 Feb 2026.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)
- Mixture-of-experts / [Expert routing strategies](https://mlsanity.com/moe/expert-routing-strategies/)

## Builds on (most cited first, 22 of 22)

- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [GPT-4o System Card](https://mlsanity.com/paper/2410.21276/) (2024)
- [Tree of Thoughts: Deliberate Problem Solving with Large Language Models](https://mlsanity.com/paper/2305.10601/) (2023)
- [GPQA: A Graduate-Level Google-Proof Q&A Benchmark](https://mlsanity.com/paper/2311.12022/) (2023)
- [Mixtral of Experts](https://mlsanity.com/paper/2401.04088/) (2024)
- [gpt-oss-120b & gpt-oss-20b Model Card](https://mlsanity.com/paper/2508.10925/) (2025)
- [DeepSeek-V2: A Strong, Economical, and Efficient Mixture-of-Experts Language Model](https://mlsanity.com/paper/2405.04434/) (2024)
- [Large Language Monkeys: Scaling Inference Compute with Repeated Sampling](https://mlsanity.com/paper/2407.21787/) (2024)
- [Direct Language Model Alignment from Online AI Feedback](https://mlsanity.com/paper/2402.04792/) (2024)
- [Rewarding Chatbots for Real-World Engagement with Millions of Users](https://mlsanity.com/paper/2303.06135/) (2023)
- [MoE++: Accelerating Mixture-of-Experts Methods with Zero-Computation Experts](https://mlsanity.com/paper/2410.07348/) (2024)
- [EDT: Improving Large Language Models' Generation by Entropy-based Dynamic Temperature Sampling](https://mlsanity.com/paper/2403.14541/) (2024)
- [Wait, Wait, Wait... Why Do Reasoning Models Loop?](https://mlsanity.com/paper/2512.12895/) (2025)
- [Diversity of Thought Improves Reasoning Abilities of LLMs](https://mlsanity.com/paper/2310.07088/) (2023)
- [Every FLOP Counts: Scaling a 300B Mixture-of-Experts LING LLM without Premium GPUs](https://mlsanity.com/paper/2503.05139/) (2025)
- [Block Rotation is All You Need for MXFP4 Quantization](https://mlsanity.com/paper/2511.04214/) (2025)
- [DartQuant: Efficient Rotational Distribution Calibration for LLM Quantization](https://mlsanity.com/paper/2511.04063/) (2025)
- [Two Experts Are All You Need for Steering Thinking: Reinforcing Cognitive Effort in MoE Reasoning Models Without Additional Training](https://mlsanity.com/paper/2505.14681/) (2025)
- [Ban&Pick: Enhancing Performance and Efficiency of MoE-LLMs via Smarter Routing](https://mlsanity.com/paper/2509.06346/) (2025)

## Built on by (most cited first, 2 of 2)

- [When Are Experts Misrouted? Counterfactual Routing Analysis in Mixture-of-Experts Language Models](https://mlsanity.com/paper/2605.07260/) (2026)
- [Don't Blame the Large Language Model: How Agent Harness Evolution Shapes Coding Agent Quality](https://mlsanity.com/paper/2607.03691/) (2026)

## Abstract (arXiv)

Test-time scaling improves LLM performance by generating multiple candidate solutions, yet token-level sampling requires temperature tuning that trades off diversity against stability. Fine-grained MoE, featuring hundreds of well-trained experts per layer and multi-expert activation per token, offers an unexplored alternative through its rich routing space. We empirically characterize fine-grained MoE routing and uncover an informative pattern: router scores exhibit a certain head of high-confidence experts followed by an uncertain tail of low-confidence candidates. While single-run greedy accuracy remains stable when fewer experts are activated, multi-sample pass@n degrades significantly-suggesting that the certain head governs core reasoning capability while the uncertain tail correlates with reasoning diversity. Motivated by these findings, we propose Expert-Sample, a training-free method that preserves high-confidence selections while injecting controlled stochasticity into the uncertain tail, enabling diverse generation without destabilizing outputs. Evaluated on multiple fine-grained MoE models across math, knowledge reasoning, and code tasks, Expert-Sample consistently improves pass@n and verification-based accuracy. On Qwen3-30B-A3B-Instruct evaluated on GPQA-Diamond with 32 parallel samples, pass@32 rises from 85.4% to 91.9%, and accuracy improves from 59.1% to 62.6% with Best-of-N verification.