Herrmann, Csordás and Schmidhuber (2025) — ARXIV-2503.13431
Key takeaways
- The diagnosis is sharp and is the reason the paper exists. Next-token loss conflates hard to predict with requiring computation. Uniform random tokens maximize it and demand nothing; reciting a memorized licence minimizes it and demands nothing either. Both endpoints are uninteresting and the metric puts them at opposite extremes.
- The instrument. Split an autoregressive model in two, insert a PHi
layer between the halves: a posterior encoder to a latent, a decoder back
to the hidden state, and — the novel part — a learned autoregressive
prior that predicts the latent from its own past. The metric is
KL(posterior ‖ prior), the nats of novel information at each step. Train jointly with the next-token loss, or bolt onto a frozen pretrained model. - It separates boring from interesting where next-token loss does not. Across four tasks (memorized subsequences, memorized automata, in-context learning of a fresh automaton, uniform noise), on both a Transformer and an LSTM, only the in-context learning task shows high PHi loss. Next-token loss instead orders them memorized < in-context < random.
- It tracks complexity beyond next-token loss, which is the load-bearing control. Binning tokens by next-token loss and stratifying by the analytically computed description length of the generating automaton, more complex automata give higher PHi loss within every bin. Partial correlation with next-token loss controlled is significant.
- It survives transplant into a frozen LLM. A single PHi layer trained into Llama-3.2-3B reproduces the task separation — unseen literature and a private code base score high, trivial tasks, memorized licences and shuffled tokens score low — while next-token loss shows no consistent pattern across the same five.
- Placement matters and the failure mode is named. Inserting the bottleneck in early layers causes posterior collapse: next-token loss degrades badly and PHi loss goes to nearly zero. Layers 18–24 of 3B are where the metric discriminates and the model is undisturbed.
- On self-generated reasoning: given a correct and an incorrect rationale for a GSM-8K problem, picking the higher-PHi-loss one beats chance, and does so on the adversarial subset where the lower next-token-loss answer is wrong. The paper is candid that choosing by lower next-token loss alone already scores 71%, and it does not claim PHi beats that as a standalone selector — the demonstrated value is incremental.
- A tentative and interesting asymmetry: on counterintuitive questions, high PHi loss predicts correctness for hard problems and not for easy ones. The authors suggest hard failures come from rationales that are too simple, easy failures from genuine misconceptions.
Standing in the anthology
The record’s analysis-and-evaluation line is largely about what a
measurement cannot tell you — SOTA-200 on metric artefacts,
SOTA-270 on what an aggregate loss curve hides. This is the same
genre with a constructive answer attached: a named failure of the field’s
default metric, and an instrument that does not share it.
It does not disturb SOTA-101, which uses perplexity to filter training data for quality. That is a different job for the same number — ranking documents by how typical they are — and nothing here argues against it. The claim is bounded to asking what computation a model is doing.