# Scalable Best-of-N Selection for Large Language Models via Self-Certainty

arXiv: https://arxiv.org/abs/2502.18581 · HTML: https://mlsanity.com/paper/2502.18581/

## Key facts

- Cited by 187 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 27 of its 36 references that are in the map.
- First posted on arXiv 25 Feb 2025.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)
- LLM alignment / RLHF / preference optimisation / [Process reward models for reasoning](https://mlsanity.com/alignment/process-reward-models-for-reasoning/)

## Builds on (most cited first, 27 of 27)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [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)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Training a Helpful and Harmless Assistant with Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2204.05862/) (2022)
- [LiveCodeBench: Holistic and Contamination Free Evaluation of Large Language Models for Code](https://mlsanity.com/paper/2403.07974/) (2024)
- [Scaling LLM Test-Time Compute Optimally can be More Effective than Scaling Model Parameters](https://mlsanity.com/paper/2408.03314/) (2024)
- [Qwen2.5-Coder Technical Report](https://mlsanity.com/paper/2409.12186/) (2024)
- [Math-Shepherd: Verify and Reinforce LLMs Step-by-step without Human Annotations](https://mlsanity.com/paper/2312.08935/) (2023)
- [Solving math word problems with process- and outcome-based feedback](https://mlsanity.com/paper/2211.14275/) (2022)
- [The Lessons of Developing Process Reward Models in Mathematical Reasoning](https://mlsanity.com/paper/2501.07301/) (2025)
- [CRUXEval: A Benchmark for Code Reasoning, Understanding and Execution](https://mlsanity.com/paper/2401.03065/) (2024)
- [A Survey of Confidence Estimation and Calibration in Large Language Models](https://mlsanity.com/paper/2311.08298/) (2023)
- [Reward Model Ensembles Help Mitigate Overoptimization](https://mlsanity.com/paper/2310.02743/) (2023)
- [Universal Self-Consistency for Large Language Model Generation](https://mlsanity.com/paper/2311.17311/) (2023)
- [Helping or Herding? Reward Model Ensembles Mitigate but do not Eliminate Reward Hacking](https://mlsanity.com/paper/2312.09244/) (2023)
- [Quantifying Uncertainty in Answers from any Language Model and Enhancing their Trustworthiness](https://mlsanity.com/paper/2308.16175/) (2023)
- [Towards System 2 Reasoning in LLMs: Learning How to Think With Meta Chain-of-Thought](https://mlsanity.com/paper/2501.04682/) (2025)
- [Trading Off Diversity and Quality in Natural Language Generation](https://mlsanity.com/paper/2004.10450/) (2020)
- [Self-Evaluation Improves Selective Generation in Large Language Models](https://mlsanity.com/paper/2312.09300/) (2023)
- [Internal Consistency and Self-Feedback in Large Language Models: A Survey](https://mlsanity.com/paper/2407.14507/) (2024)
- [Can Perplexity Reflect Large Language Model's Ability in Long Text Understanding?](https://mlsanity.com/paper/2405.06105/) (2024)
- [TrustScore: Reference-Free Evaluation of LLM Response Trustworthiness](https://mlsanity.com/paper/2402.12545/) (2024)

## Built on by (most cited first, 30 of 187)

- [Towards Reasoning Era: A Survey of Long Chain-of-Thought for Reasoning Large Language Models](https://mlsanity.com/paper/2503.09567/) (2025)
- [A Survey on Test-Time Scaling in Large Language Models: What, How, Where, and How Well?](https://mlsanity.com/paper/2503.24235/) (2025)
- [Learning to Reason without External Rewards](https://mlsanity.com/paper/2505.19590/) (2025)
- [Deep Think with Confidence](https://mlsanity.com/paper/2508.15260/) (2025)
- [Reasoning Models Can Be Effective Without Thinking](https://mlsanity.com/paper/2504.09858/) (2025)
- [A Survey of Self-Evolving Agents: What, When, How, and Where to Evolve on the Path to Artificial Super Intelligence](https://mlsanity.com/paper/2507.21046/) (2025)
- [Beyond Binary Rewards: Training LMs to Reason About Their Uncertainty](https://mlsanity.com/paper/2507.16806/) (2025)
- [Scaling Text-to-Image Diffusion Transformers with Representation Autoencoders](https://mlsanity.com/paper/2601.16208/) (2026)
- [Reinforcement Learning Meets Large Language Models: A Survey of Advancements and Applications Across the LLM Lifecycle](https://mlsanity.com/paper/2509.16679/) (2025)
- [Unlocking Exploration in RLVR: Uncertainty-aware Advantage Shaping for Deeper Reasoning](https://mlsanity.com/paper/2510.10649/) (2025)
- [Reinforcing Video Reasoning with Focused Thinking](https://mlsanity.com/paper/2505.24718/) (2025)
- [AlphaOne: Reasoning Models Thinking Slow and Fast at Test Time](https://mlsanity.com/paper/2505.24863/) (2025)
- [TokUR: Token-Level Uncertainty Estimation for Large Language Model Reasoning](https://mlsanity.com/paper/2505.11737/) (2025)
- [Verifier-free Test-Time Sampling for Vision-Language-Action Models](https://mlsanity.com/paper/2510.05681/) (2025)
- [A Survey on Agentic Multimodal Large Language Models](https://mlsanity.com/paper/2510.10991/) (2025)
- [Think Deep, Not Just Long: Measuring LLM Reasoning Effort via Deep-Thinking Tokens](https://mlsanity.com/paper/2602.13517/) (2026)
- [Uncertainty Quantification in LLM Agents: Foundations, Emerging Challenges, and Opportunities](https://mlsanity.com/paper/2602.05073/) (2026)
- [Shop-R1: Rewarding LLMs to Simulate Human Behavior in Online Shopping via Reinforcement Learning](https://mlsanity.com/paper/2507.17842/) (2025)
- [Let's (not) just put things in Context: Test-Time Training for Long-Context LLMs](https://mlsanity.com/paper/2512.13898/) (2025)
- [Generative AI Act II: Test Time Scaling Drives Cognition Engineering](https://mlsanity.com/paper/2504.13828/) (2025)
- [Accurate Failure Prediction in Agents Does Not Imply Effective Failure Prevention](https://mlsanity.com/paper/2602.03338/) (2026)
- [AgentV-RL: Scaling Reward Modeling with Agentic Verifier](https://mlsanity.com/paper/2604.16004/) (2026)
- [Scaling Test-Time Compute for Agentic Coding](https://mlsanity.com/paper/2604.16529/) (2026)
- [Know When to Explore: Difficulty-Aware Certainty as a Guide for LLM Reinforcement Learning](https://mlsanity.com/paper/2509.00125/) (2025)
- [Certified Self-Consistency: Statistical Guarantees and Test-Time Training for Reliable Reasoning in LLMs](https://mlsanity.com/paper/2510.17472/) (2025)
- [From Efficiency to Adaptivity: A Deeper Look at Adaptive Reasoning in Large Language Models](https://mlsanity.com/paper/2511.10788/) (2025)
- [Reliability-Aware Adaptive Self-Consistency for Efficient Sampling in LLM Reasoning](https://mlsanity.com/paper/2601.02970/) (2026)
- [Adaptive Termination for Multi-round Parallel Reasoning: An Universal Semantic Entropy-Guided Framework](https://mlsanity.com/paper/2507.06829/) (2025)
- [PiCSAR: Probabilistic Confidence Selection And Ranking for Reasoning Chains](https://mlsanity.com/paper/2508.21787/) (2025)
- [Does higher interpretability imply better utility? A Pairwise Analysis on Sparse Autoencoders](https://mlsanity.com/paper/2510.03659/) (2025)

## Abstract (arXiv)

Best-of-N selection is a key technique for improving the reasoning performance of Large Language Models (LLMs) through increased test-time computation. Current state-of-the-art methods often employ computationally intensive reward models for response evaluation and selection. Reward-free alternatives, like self-consistency and universal self-consistency, are limited in their ability to handle open-ended generation tasks or scale effectively. To address these limitations, we propose self-certainty, a novel and efficient metric that leverages the inherent probability distribution of LLM outputs to estimate response quality without requiring external reward models. We hypothesize that higher distributional self-certainty, aggregated across multiple samples, correlates with improved response accuracy, as it reflects greater confidence in the generated output. Through extensive experiments on various reasoning tasks, we demonstrate that self-certainty (1) scales effectively with increasing sample size N, akin to reward models but without the computational overhead; (2) complements chain-of-thought, improving reasoning performance beyond greedy decoding; and (3) generalizes to open-ended tasks where traditional self-consistency methods fall short. Our findings establish self-certainty as a practical and efficient way for improving LLM reasoning capabilities. The code is available at https://github.com/backprop07/Self-Certainty