Predictive Latents for Autoregressive Language Models: Possibility, Gauge, and Obstruction
Organizations: Zhejiang Sci-Tech University
Abstract
Autoregressive language models are trained by next-token cross-entropy, yet several recent systems also iterate a hidden state without emitting a token: continuous thoughts, pause steps, and recurrent depth. This note separates three risks on one latent process. The token risk scores a readout. The teacher-forced latent risk scores a one-step map that sees the true hidden state. The open-loop risk scores the same map after it has been iterated on its own outputs. If the Bayes next-token posterior is a function of a -dimensional statistic with full-rank covariance, a width- latent, a linear least-squares transition, and a refit unembedding match the Bayes token predictor. The statistic is identified only up to an invertible gauge: whitening the residual stream does not change the token law once the unembedding is refit. The matching obstructions are sharp. A constant latent drives to zero while token cross-entropy falls back to the marginal entropy. A perfect token predictor can still hide a null-space walk whose open-loop error grows linearly in the horizon. A small next-latent loss does not certificate next-token prediction, and a small next-token loss does not certificate free latent rollout, unless a rank or injectivity condition is stated with the number.
Figures & tables
| Small number | Conclusion | Side condition |
|---|---|---|
| near | sufficiency, Thm. 5 | |
| token Bayes | fails under collapse, Thm. 12 | |
| small | fails in a null direction, Thm. 14 | |
| small for all | , Thm. 16 | |
| isotropy of | a new token law | none; it is a gauge, Thm. 11 |
Autoregressive Language Modeling
Autoregressive language modeling predicts sequences one step at a time, using prior tokens to generate text, speech, code, and other data.
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