Cross-Entropy Risk Estimation for Language Models: Inconsistency Must Be Dense, and the Holdout Method Is No Exception
Language models are compared by their held-out per-token cross-entropy risk---the quantity scaling laws are fitted to. We show that it cannot be consistently estimated. Consistency, or convergence to the estimand, is defined relative to a \emph{possible state of the world}: a pair consisting of a data-generating distribution and a model we turn out to train. Quantifying over models as well as data-generating mechanisms is essential, because what decides whether a model's risk is estimable is a tail property of the distribution its weights induce, which no sample reveals. The per-token cross-entropy risk is hard to estimate because of a topological fact: among the possible states, finite r...
| Authors | Hanti Lin |
|---|---|
| Primary category | cs.LG |
| Published | 16 August 2026 |
| Last updated at source | 16 August 2026 |
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