The Order Comes Right
Ongoing research. This is an experimental result from active work, not a settled conclusion. The numbers are what we measured and the method is described so you can judge it, but the programme is still running and later experiments may revise what it means.
A note on the language in these records. This is a working laboratory notebook for research into training AI models more cheaply and efficiently, so you will read that an approach did not work, that a result did not hold up, or that one method was worse than another. That is the research doing its job, not a verdict on the engineering we deliver to clients. Ruling an approach out is how the search narrows, and these are the pages that teach us the most: nearly every technique we now rely on came from understanding why something else fell short. Testing our own ideas at least as hard as anyone else's is the point of publishing them. More about this programme and why we run it.
Part of a bigger question: What actually makes training cheaper? – One thing has worked: stopping part of the training early saved about 7% with no loss of quality. Everything else tested has been matched by a simpler or cheaper method -- and in two cases the clever method was only winning because it was quietly being given more.
In plain English
What we asked. In an earlier experiment we found a small thinking model spending its thinking in exactly the wrong place: longest on the questions it never got wrong, shortest on the ones where all its mistakes were. A natural fix is to let it stop as soon as its answer looks certain, on the theory that easy questions become certain quickly and hard ones do not. We tried that on the same models, without retraining them.
What we found. The thinking moved to the right place: the model now thought longest about the hardest questions. But its accuracy collapsed, from about 96% to between 50% and 74% depending on how certain it had to be. The reason is that its early answers were confidently wrong. It had been trained in a way that barely taught those early answers anything, so they sounded sure without being right.
Why it matters. The lesson for anyone building a model that decides how long to think: the stopping signal has to get two things right at once. It has to spend more on harder questions, and its confidence has to be trustworthy at every point where it might stop. The model's own learned signal had the second and not the first; confidence had the first and not the second.
The rest of this page is the technical record: the design, every number, and the limits. It is written for a reviewer, and you do not need it to have understood the result above.
New here? How to read a research record
- Start at the verdict. Every record states, before the experiment was run, what result would have made us abandon the idea. That is the "kill test". Then it says whether the test fired. Nothing gets reinterpreted after the fact.
- Numbers in square brackets are uncertainty.
23.4 [18.1, 28.7]means our best estimate is 23.4 and the true value is probably somewhere in that range. If a range includes zero, we cannot claim an effect. - Results that rule an idea out are kept. Roughly half of what is published here says an approach did not work, including plenty of our own. Those pages are the output, not a shortfall: knowing which direction is a dead end is what lets the next experiment go somewhere better, and most of what we now rely on came out of understanding why something else fell short. Work that only publishes what worked is not measuring anything.
- Read the Limits section. Every record ends with what it does not show. It is the most honest part of any experiment and usually the shortest.
- Pro tip: the figures near the top are designed to carry the result on their own. If you read nothing else, read the caption under each one, which says what it shows and what to take from it.
EXPLORATORY. Not a preregistered study. Local CPU, S3's six seeds, no new training beyond reproducing S3's models. The design and kill test were committed (bce0717) before any run.
Program v2 Bucket S, item S9. Decisive computation: . Output: analysis/confidence_halting.py. Reproduce with analysis/confidence_halting.jsonpython analysis/confidence_halting.py (a few minutes); --reuse re-derives every endpoint.
The question
S3 found that a thinking-step penalty cuts 38% of a small model's thinking for at most a point of accuracy, and that a fixed budget of three steps beats every penalised model anyway. The mechanism was the useful part: the learned halting unit is inverted -- eight steps on the inputs it never gets wrong, about three on the ones holding every error -- and a penalty, which charges every step the same, shortens every column without re-ordering them.
The one change aimed at that mechanism: stop when the answer is confident, rather than when a separate halting unit says so. At every thinking step the answer head already gives a probability for its answer; halt at the first step where it clears a threshold. If confidence tracks difficulty, the allocation turns the right way round. No training: the rule is applied at inference to S3's own models, at eight thresholds from 0.55 to 0.999.
Kill test, fixed before execution: at matched mean thinking steps, confidence halting does not beat S3's fixed-budget frontier, paired over S3's six seeds.
Prior recorded in advance: moderate.
Anchor, in code, passed: S3 did not save its models, so they were retrained from S3's committed code and seeds. All six reproduce S3's committed accuracy and mean steps exactly.
Result: the kill test fires, by a wide margin
| Threshold | Accuracy | Mean steps | Against the fixed-budget frontier |
|---|---|---|---|
0.55 | 0.502 | 1.21 | -0.157 [-0.269, -0.043] |
0.8 | 0.531 | 2.25 | -0.327 [-0.481, -0.173] |
0.9 | 0.584 | 3.12 | -0.324 [-0.432, -0.216] |
0.99 | 0.699 | 5.55 | -0.300 [-0.453, -0.148] |
0.999 | 0.742 | 6.72 | -0.257 [-0.430, -0.084] |
(Four more thresholds in the output, all the same shape.) Every threshold is far below the fixed budget. At 0.999 -- a model that stops only when its answer is essentially certain -- the same network that scored 0.96 under its learned halting scores 0.742. The strict envelope gives the same verdict. For reference, S3's penalised models were about 0.015 below the frontier; confidence halting is ten to twenty times further.
And yet the order comes right
Mean thinking steps by number of non-zero entries, S3's learned halting against confidence at 0.99:
| Non-zero entries | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| learned halting | 8.00 | 7.96 | 7.77 | 7.33 | 6.86 | 6.26 | 5.48 | 4.47 | 3.97 | 3.27 |
| confidence | 5.07 | 5.05 | 4.95 | 5.00 | 5.08 | 5.41 | 5.75 | 6.17 | 6.53 | 6.71 |
Confidence halting spends more thinking on the harder inputs -- the allocation S3 found missing is exactly what this rule produces. So confidence does track difficulty. What it does not do is mean the answer is right.
Why: the early answers are confidently wrong
The overthinker was trained with loss sum_n p_n CE_n -- each step's answer weighted by the probability of stopping there. Its halting distribution puts most of its mass late, so its early answers were barely trained at all, and nothing in the loss asked them to be calibrated. They come out confident and wrong: at a 0.999 threshold, a quarter of the answers the network commits to are errors.
A halting signal needs two properties, and S3 and S9 each found one. S3's learned halting unit was calibrated in the sense that stopping where it said gave 0.96, but it was ordered backwards against difficulty. Answer confidence is ordered the right way, but it is miscalibrated at every step except the ones the network was trained to answer at. Adaptive computation needs a signal that is both, and neither falls out of sum_n p_n CE_n for free.
What stands
- The kill test fires. Confidence halting is
0.16to0.34below the fixed-budget frontier at every threshold tested. - Confidence orders the thinking correctly --
5.1steps on the easiest inputs,6.7on the hardest -- the reverse of the learned halting unit. - The early answers are confidently wrong, because the training objective weights them by a stopping probability that was small. That is the mechanism, and it is the general caution: a stop-when-confident rule is only as good as the calibration of the answers it stops on.
- For the report's reasoning-budget idea: removing thinking is easy (S3), allocating it well needs a signal that is both ordered and calibrated, and the obvious candidates here each have only one.
Limits
- No retraining. A model trained so every step's answer is supervised -- deep supervision, or the uniform-weight loss PonderNet's authors discuss -- would have calibrated early answers, and confidence halting on that model is the natural next test. This record tests the rule on the model S3 built, as S9 was written to.
- Parity, one width, S3's scale. As in S3, a task where a fixed budget saturates at three steps is close to the worst case for any adaptive rule.
- Max-probability confidence is the simplest signal. Entropy or a margin would order the same way and be miscalibrated for the same reason, which is a claim this record makes from the mechanism rather than from measuring them.
Terms on this page
Every piece of vocabulary this record uses, in plain language. Generated from the text above, so it cannot drift out of step with it.
- accuracy
- The fraction of answers a model gets right on questions it was not trained on.
- calibrated
- A model is calibrated when its confidence matches how often it is right: answers it gives with 90% confidence should be right about 90% of the time.
- calibration
- Working out an instrument's settings from runs whose answer you already know, so it can be used on a run whose answer you do not.
- deep supervision
- Training a model to give a correct answer at every intermediate stage, not only at the end, so an early answer can be trusted.
- kill test
- A condition written down before running the experiment that says what result would make us abandon the idea. Fixing it in advance is what stops a disappointing result being reinterpreted as an encouraging one.
- parity
- Whether a count is odd or even. A classic test puzzle for models, because the answer flips with every single item and cannot be guessed from a rough impression.
- seed
- The number that fixes all the randomness in a training run. Same seed, same run. Running several seeds is how you tell a real effect from a lucky one.
- width
- How many internal numbers a model uses at each layer. The usual way we vary model size in these experiments.
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