# Measuring Five-Nines Reliability: Sample-Efficient LLM Evaluation in Saturated Benchmarks

arXiv: https://arxiv.org/abs/2605.11209 · HTML: https://mlsanity.com/paper/2605.11209/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 34 references that are in the map.
- First posted on arXiv 11 May 2026.

## Builds on (most cited first, 20 of 20)

- [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)
- [GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding](https://mlsanity.com/paper/1804.07461/) (2018)
- [HellaSwag: Can a Machine Really Finish Your Sentence?](https://mlsanity.com/paper/1905.07830/) (2019)
- [SWE-bench: Can Language Models Resolve Real-World GitHub Issues?](https://mlsanity.com/paper/2310.06770/) (2023)
- [GPQA: A Graduate-Level Google-Proof Q&A Benchmark](https://mlsanity.com/paper/2311.12022/) (2023)
- [MMMU: A Massive Multi-discipline Multimodal Understanding and Reasoning Benchmark for Expert AGI](https://mlsanity.com/paper/2311.16502/) (2023)
- [gpt-oss-120b & gpt-oss-20b Model Card](https://mlsanity.com/paper/2508.10925/) (2025)
- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [Terminal-Bench: Benchmarking Agents on Hard, Realistic Tasks in Command Line Interfaces](https://mlsanity.com/paper/2601.11868/) (2026)
- [GSM-Symbolic: Understanding the Limitations of Mathematical Reasoning in Large Language Models](https://mlsanity.com/paper/2410.05229/) (2024)
- [MMMU-Pro: A More Robust Multi-discipline Multimodal Understanding Benchmark](https://mlsanity.com/paper/2409.02813/) (2024)
- [Humanity's Last Exam](https://mlsanity.com/paper/2501.14249/) (2025)
- [tinyBenchmarks: evaluating LLMs with fewer examples](https://mlsanity.com/paper/2402.14992/) (2024)
- [Anchor Points: Benchmarking Models with Much Fewer Examples](https://mlsanity.com/paper/2309.08638/) (2023)
- [Efficient Benchmarking of Language Models](https://mlsanity.com/paper/2308.11696/) (2023)
- [Fluid Language Model Benchmarking](https://mlsanity.com/paper/2509.11106/) (2025)
- [Reliable and Efficient Amortized Model-based Evaluation](https://mlsanity.com/paper/2503.13335/) (2025)
- [Efficient Evaluation of LLM Performance with Statistical Guarantees](https://mlsanity.com/paper/2601.20251/) (2026)
- [Rare-Event Simulation for Neural Network and Random Forest Predictors](https://mlsanity.com/paper/2010.04890/) (2020)

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

- [What Makes a Terminal-Bench Task Hard? Separating Genuine Hardness from Fake-Hardness on an Adjudicated Agentic Corpus](https://mlsanity.com/paper/2609.26826/) (2026)
- [Are We Testing Too Much? A Survey of Benchmark Compression for Efficient Evaluation of Large Language Models](https://mlsanity.com/paper/2610.18248/) (2026)

## Abstract (arXiv)

While existing benchmarks demonstrate the near-perfect performance of large language models (LLMs) on various tasks, this apparent saturation often obscures the need for rigorous evaluation of their reliability. In real-world deployment, however, achieving extremely high reliability (e.g., "five-nines" (99.999%) vs. "three-nines" (99.9%)) is fundamentally critical, as this gap results in an order-of-magnitude increase in failures, which is catastrophic in reliability-critical applications. Still, estimating such a rare failure probability with tight confidence bounds requires prohibitively large LLM inference sizes, making standard Monte Carlo evaluation infeasible under limited compute budgets. In this paper, we observe that LLM failures exhibit strong systematic patterns: across broad parameterized input spaces, a small subset of inputs disproportionately accounts for the majority of failures. Leveraging this observation, we propose to learn a sampling distribution concentrated on failure-prone inputs via the cross-entropy method (CEM). We evaluate our framework on three LLMs, Qwen2.5-Math-7B-Instruct, gpt-oss-20b-low, and Gemini 2.5 Flash Lite, across parameterized GSM8K templates and achieve up to 156.22x reduction in required inferences compared to naive uniform sampling. Our estimates reveal that models with indistinguishable accuracy on standard benchmarks can differ substantially in estimated failure rates, underscoring that reliability is a distinct and measurable axis of model quality. Our simple yet practical framework enables the evaluation of extreme reliability in LLMs, a distinct and underexplored dimension of evaluation beyond existing benchmarks, for their growing use in reliability-sensitive applications.