No Free Lunch: Fundamental Limits of Learning Non-Hallucinating Generative Models

Generative models have shown impressive capabilities in synthesizing high-quality outputs across various domains. However, a persistent challenge is the occurrence of "hallucinations," where the model produces outputs that are not grounded in the underlying facts. While empirical strategies have been explored to mitigate this issue, a rigorous theoretical understanding remains elusive. In this paper, we develop a theoretical framework to analyze the learnability of non-hallucinating generative models from a learning-theoretic perspective. Our results reveal that non-hallucinating learning is statistically impossible when relying solely on the training dataset, even for a hypothesis class of size two and when the entire training set is truthful. To overcome these limitations, we show that incorporating inductive biases aligned with the actual facts into the learning process is essential. We provide a systematic approach to achieve this by restricting the fact set to a concept class of finite VC-dimension and demonstrate its effectiveness under various learning paradigms. Although our findings are primarily conceptual, they represent a first step towards a principled approach to addressing hallucinations in learning generative models.