# Cross-Entropy Risk Estimation for Language Models: Inconsistency Must Be Dense, and the Holdout Method Is No Exception

arXiv: https://arxiv.org/abs/2608.15798 · HTML: https://mlsanity.com/paper/2608.15798/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 26 references that are in the map.
- First posted on arXiv 16 Aug 2026.

## Builds on (most cited first, 6 of 6)

- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [ForecastBench: A Dynamic Benchmark of AI Forecasting Capabilities](https://mlsanity.com/paper/2409.19839/) (2024)
- [Local permutation tests for conditional independence](https://mlsanity.com/paper/2112.11666/) (2021)
- [On the Impossibility of Learning the Missing Mass](https://mlsanity.com/paper/1503.03613/) (2015)
- [The Topology of Statistical Verifiability](https://mlsanity.com/paper/1707.09378/) (2017)

## Abstract (arXiv)

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 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 risk and infinite risk each lie arbitrarily close to every instance of the other. Consequently no estimator---not merely the holdout average---is consistent at every state at which the risk is defined. Worse, inconsistent estimation persists under both bounding the expected sequence length and restricting to full-support models; and in that restricted setting the states at which inconsistency occurs are even dense. Two interesting ways out are identified, and neither is free. Way out 1: using a bounded context window, we can floor a model's next-token probabilities, making its risk finite exactly when the data-generating distribution has finite expected sequence length---a new, statistical rationale for a choice that was made on computational grounds, though the assumption it substitutes is itself beyond the reach of any test. Way out 2: reporting the risk only when it falls below a threshold fixed in advance restores consistency, at no cost to what model selection actually requires---but we need to recognize that the goal of estimation is revised.