Hallucinations are inevitable but can be made statistically negligible

TL;DR

This paper demonstrates that hallucinations in language models can be made statistically negligible with sufficient training data and improved algorithms, despite theoretical inevitability on some inputs.

Contribution

It provides a probabilistic theoretical framework showing hallucinations can be minimized, contrasting with computability-theoretic limitations.

Findings

Hallucinations probability can be reduced with better training data.

Computability theory implies some hallucinations are inevitable, but their likelihood can be minimized.

The probabilistic approach aligns better with practical language model deployment.

Abstract

Hallucinations, a phenomenon where a language model (LM) generates nonfactual content, pose a significant challenge to the practical deployment of LMs. While many empirical methods have been proposed to mitigate hallucinations, recent studies established a computability-theoretic result showing that any LM will inevitably generate hallucinations on an infinite set of inputs, regardless of the quality and quantity of training datasets and the choice of the language model architecture and training and inference algorithms. Although the computability-theoretic result may seem pessimistic, its significance in practical viewpoints has remained unclear. This paper claims that those "innate" inevitability results from computability theory and diagonal argument, in principle, cannot explain practical issues of LLMs. We demonstrate this claim by presenting a positive theoretical result from a probabilistic perspective. Specifically, we prove that hallucinations can be made statistically negligible, provided that the quality and quantity of the training data are sufficient. Interestingly, our positive result coexists with the computability-theoretic result, implying that while hallucinations on an infinite set of inputs cannot be entirely eliminated, their probability can always be reduced by improving algorithms and training data. By evaluating the two seemingly contradictory results through the lens of information theory, we argue that our probability-theoretic positive result better reflects practical considerations than the computability-theoretic negative result.