# REAL Sampling: Boosting Factuality and Diversity of Open-Ended Generation via Asymptotic Entropy

arXiv: https://arxiv.org/abs/2406.07735 · HTML: https://mlsanity.com/paper/2406.07735/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 47 of its 63 references that are in the map.
- First posted on arXiv 11 Jun 2024.

## Builds on (most cited first, 30 of 47)

- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Tree of Thoughts: Deliberate Problem Solving with Large Language Models](https://mlsanity.com/paper/2305.10601/) (2023)
- [The Curious Case of Neural Text Degeneration](https://mlsanity.com/paper/1904.09751/) (2019)
- [Pythia: A Suite for Analyzing Large Language Models Across Training and Scaling](https://mlsanity.com/paper/2304.01373/) (2023)
- [A Survey on Hallucination in Large Language Models: Principles, Taxonomy, Challenges, and Open Questions](https://mlsanity.com/paper/2311.05232/) (2023)
- [Competition-Level Code Generation with AlphaCode](https://mlsanity.com/paper/2203.07814/) (2022)
- [FEVER: a large-scale dataset for Fact Extraction and VERification](https://mlsanity.com/paper/1803.05355/) (2018)
- [Inference-Time Intervention: Eliciting Truthful Answers from a Language Model](https://mlsanity.com/paper/2306.03341/) (2023)
- [Hierarchical Neural Story Generation](https://mlsanity.com/paper/1805.04833/) (2018)
- [A Diversity-Promoting Objective Function for Neural Conversation Models](https://mlsanity.com/paper/1510.03055/) (2015)
- [SelfCheckGPT: Zero-Resource Black-Box Hallucination Detection for Generative Large Language Models](https://mlsanity.com/paper/2303.08896/) (2023)
- [Discovering Latent Knowledge in Language Models Without Supervision](https://mlsanity.com/paper/2212.03827/) (2022)
- [The Internal State of an LLM Knows When It's Lying](https://mlsanity.com/paper/2304.13734/) (2023)
- [Contrastive Decoding: Open-ended Text Generation as Optimization](https://mlsanity.com/paper/2210.15097/) (2022)
- [DoLa: Decoding by Contrasting Layers Improves Factuality in Large Language Models](https://mlsanity.com/paper/2309.03883/) (2023)
- [Trusting Your Evidence: Hallucinate Less with Context-aware Decoding](https://mlsanity.com/paper/2305.14739/) (2023)
- [How Language Model Hallucinations Can Snowball](https://mlsanity.com/paper/2305.13534/) (2023)
- [A Comprehensive Survey of Hallucination Mitigation Techniques in Large Language Models](https://mlsanity.com/paper/2401.01313/) (2024)
- [A Stitch in Time Saves Nine: Detecting and Mitigating Hallucinations of LLMs by Validating Low-Confidence Generation](https://mlsanity.com/paper/2307.03987/) (2023)
- [Factuality Enhanced Language Models for Open-Ended Text Generation](https://mlsanity.com/paper/2206.04624/) (2022)
- [Detecting Hallucinated Content in Conditional Neural Sequence Generation](https://mlsanity.com/paper/2011.02593/) (2020)
- [Controlling Linguistic Style Aspects in Neural Language Generation](https://mlsanity.com/paper/1707.02633/) (2017)
- [Emergent and Predictable Memorization in Large Language Models](https://mlsanity.com/paper/2304.11158/) (2023)
- [A Token-level Reference-free Hallucination Detection Benchmark for Free-form Text Generation](https://mlsanity.com/paper/2104.08704/) (2021)
- [The Troubling Emergence of Hallucination in Large Language Models -- An Extensive Definition, Quantification, and Prescriptive Remediations](https://mlsanity.com/paper/2310.04988/) (2023)
- [Truncation Sampling as Language Model Desmoothing](https://mlsanity.com/paper/2210.15191/) (2022)
- [Do Language Models Know When They're Hallucinating References?](https://mlsanity.com/paper/2305.18248/) (2023)

## Built on by (most cited first, 11 of 11)

- [A Survey on Uncertainty Quantification of Large Language Models: Taxonomy, Open Research Challenges, and Future Directions](https://mlsanity.com/paper/2412.05563/) (2024)
- [Sparse Logit Sampling: Accelerating Knowledge Distillation in LLMs](https://mlsanity.com/paper/2503.16870/) (2025)
- [Explaining and Improving Contrastive Decoding by Extrapolating the Probabilities of a Huge and Hypothetical LM](https://mlsanity.com/paper/2411.01610/) (2024)
- [Min-k Sampling: Decoupling Truncation from Temperature Scaling via Relative Logit Dynamics](https://mlsanity.com/paper/2604.11012/) (2026)
- [Generating Diverse Hypotheses for Inductive Reasoning](https://mlsanity.com/paper/2412.13422/) (2024)
- [A Survey on Proactive Defense Strategies Against Misinformation in Large Language Models](https://mlsanity.com/paper/2507.05288/) (2025)
- [Sample Smart, Not Hard: Correctness-First Decoding for Better Reasoning in LLMs](https://mlsanity.com/paper/2510.05987/) (2025)
- [On the Convergence of Moral Self-Correction in Large Language Models](https://mlsanity.com/paper/2510.07290/) (2025)
- [Beyond Temperature: Hyperfitting as a Late-Stage Geometric Expansion](https://mlsanity.com/paper/2605.22579/) (2026)
- [Breaking the Likelihood Trap: Variance-Calibrated Modulation for Large Language Model Decoding](https://mlsanity.com/paper/2606.22511/) (2026)
- [VOYAGER: A Training Free Approach for Generating Diverse Datasets using LLMs](https://mlsanity.com/paper/2512.12072/) (2025)

## Abstract (arXiv)

Decoding methods for large language models (LLMs) usually struggle with the tradeoff between ensuring factuality and maintaining diversity. For example, a higher p threshold in the nucleus (top-p) sampling increases the diversity but decreases the factuality, and vice versa. In this paper, we propose REAL (Residual Entropy from Asymptotic Line) sampling, a decoding method that achieves improved factuality and diversity over nucleus sampling by predicting an adaptive threshold of p. Specifically, REAL sampling predicts the step-wise likelihood of an LLM to hallucinate, and lowers the p threshold when an LLM is likely to hallucinate. Otherwise, REAL sampling increases the p threshold to boost the diversity. To predict the step-wise hallucination likelihood without supervision, we construct a Token-level Hallucination Forecasting (THF) model to predict the asymptotic entropy (i.e., inherent uncertainty) of the next token by extrapolating the next-token entropies from a series of LLMs with different sizes. If a LLM's entropy is higher than the asymptotic entropy (i.e., the LLM is more uncertain than it should be), the THF model predicts a high hallucination hazard, which leads to a lower p threshold in REAL sampling. In the FactualityPrompts benchmark, we demonstrate that REAL sampling based on a 70M THF model can substantially improve the factuality and diversity of 7B LLMs simultaneously, judged by both retrieval-based metrics and human evaluation. After combined with contrastive decoding, REAL sampling outperforms 9 sampling methods, and generates texts that are more factual than the greedy sampling and more diverse than the nucleus sampling with p=0.5. Furthermore, the predicted asymptotic entropy is also a useful unsupervised signal for hallucination detection tasks.