Language models, once pretrained, are routinely assessed by the holdout method, which estimates their per-token cross-entropy risk---a quantity that may be finite or infinite. We prove possibility and impossibility theorems that characterize the hardness of this estimation problem. The estimand depends on two objects: the unknown data distribution $p$ and the distribution $q$ induced by the language model under study. Our ignorance of $p$ is empirical, whereas our ignorance of $q$, or at least of its tail, is mathematical; so the parameter space {\em should} range over pairs $(p,q)$---a product space---to fully capture our ignorance. {\em Negative results}: no estimator is pointwise consistent, even when the parameter space is restricted to full-support models and to data distributions whose expected text length is below a fixed bound; worse, every estimator fails on a dense set of pairs. {\em Positive results}: the holdout estimator becomes pointwise consistent when the models are further restricted to Transformers or, architecture-neutrally, to full-support models with a finite context window; and, where no such design is in place, it still consistently estimates the risk capped at any fixed threshold, which suffices for screening candidate models in practice. {\em Lesson}: designs such as the Transformer architecture and finite context windows, although adopted for computational or engineering reasons, pay a {\em statistical dividend}: they suffice to make pointwise consistency possible---a dividend visible only in how they turn a negative result into positive ones.

翻译：暂无翻译