# Best-of-∞ -- Asymptotic Performance of Test-Time LLM Ensembling

arXiv: https://arxiv.org/abs/2509.21091 · HTML: https://mlsanity.com/paper/2509.21091/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 50 of its 77 references that are in the map.
- First posted on arXiv 25 Sep 2025.

## Builds on (most cited first, 30 of 50)

- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [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)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [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)
- [Large Language Models are Zero-Shot Reasoners](https://mlsanity.com/paper/2205.11916/) (2022)
- [Gemini 2.5: Pushing the Frontier with Advanced Reasoning, Multimodality, Long Context, and Next Generation Agentic Capabilities](https://mlsanity.com/paper/2507.06261/) (2025)
- [GPQA: A Graduate-Level Google-Proof Q&A Benchmark](https://mlsanity.com/paper/2311.12022/) (2023)
- [BLOOM: A 176B-Parameter Open-Access Multilingual Language Model](https://mlsanity.com/paper/2211.05100/) (2022)
- [Pythia: A Suite for Analyzing Large Language Models Across Training and Scaling](https://mlsanity.com/paper/2304.01373/) (2023)
- [s1: Simple test-time scaling](https://mlsanity.com/paper/2501.19393/) (2025)
- [Large Language Monkeys: Scaling Inference Compute with Repeated Sampling](https://mlsanity.com/paper/2407.21787/) (2024)
- [Solving math word problems with process- and outcome-based feedback](https://mlsanity.com/paper/2211.14275/) (2022)
- [The Falcon Series of Open Language Models](https://mlsanity.com/paper/2311.16867/) (2023)
- [LIMO: Less is More for Reasoning](https://mlsanity.com/paper/2502.03387/) (2025)
- [Skywork-Reward: Bag of Tricks for Reward Modeling in LLMs](https://mlsanity.com/paper/2410.18451/) (2024)
- [Alphazero-like Tree-Search can Guide Large Language Model Decoding and Training](https://mlsanity.com/paper/2309.17179/) (2023)
- [Inference Scaling Laws: An Empirical Analysis of Compute-Optimal Inference for Problem-Solving with Language Models](https://mlsanity.com/paper/2408.00724/) (2024)
- [RLHF Workflow: From Reward Modeling to Online RLHF](https://mlsanity.com/paper/2405.07863/) (2024)
- [Optimal Best Arm Identification with Fixed Confidence](https://mlsanity.com/paper/1602.04589/) (2016)
- [Learning to Reason without External Rewards](https://mlsanity.com/paper/2505.19590/) (2025)
- [Deep Think with Confidence](https://mlsanity.com/paper/2508.15260/) (2025)
- [More Agents Is All You Need](https://mlsanity.com/paper/2402.05120/) (2024)
- [Secrets of RLHF in Large Language Models Part II: Reward Modeling](https://mlsanity.com/paper/2401.06080/) (2024)
- [RM-R1: Reward Modeling as Reasoning](https://mlsanity.com/paper/2505.02387/) (2025)
- [Phi-4-reasoning Technical Report](https://mlsanity.com/paper/2504.21318/) (2025)
- [Generative Reward Models](https://mlsanity.com/paper/2410.12832/) (2024)
- [Large Language Model Cascades with Mixture of Thoughts Representations for Cost-efficient Reasoning](https://mlsanity.com/paper/2310.03094/) (2023)

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

- [Re-FORC: Adaptive Reward Prediction for Efficient Chain-of-Thought Reasoning](https://mlsanity.com/paper/2511.02130/) (2025)
- [Test-Time Compute Games](https://mlsanity.com/paper/2601.21839/) (2026)
- [Strategic Self-Consistency](https://mlsanity.com/paper/2609.30352/) (2026)

## Abstract (arXiv)

We study best-of-N for large language models (LLMs) where the selection is based on majority voting. In particular, we analyze the limit N → ∞, which we denote as boinflower. While this approach achieves impressive performance in the limit, it requires an infinite test-time budget. To address this, we propose an adaptive generation scheme that selects N based on answer agreement, thereby efficiently allocating inference-time computation. Beyond adaptivity, we extend the framework to weighted ensembles of multiple LLMs, showing that such mixtures can outperform any individual model. The optimal ensemble weighting is formulated and efficiently computed as a mixed-integer linear program. Extensive experiments demonstrate the effectiveness of our approach.