Research record

One Number Forecasts the Grid

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. If the amount of material per training step really is what sets learning time, a rule fitted on one experiment should predict another it never saw. We fitted a simple rule on five settings and used it to forecast six others from an earlier experiment.

What we found. Five of the six forecasts landed within 10 percent. The sixth, with far less material per step than anything the rule was fitted on, took nearly half as long again as forecast: below some point, learning slows down faster than the rule says.

Why it matters. A forecast that works inside its range and fails outside it is still useful; the failure tells you where the range ends. We are now measuring that edge.

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. No training: a fit to committed outputs, run seconds after its design and kill test were committed (b17a93e).

Program v2 Bucket T, item T18. Decisive computation: analysis/content_law_forecast.py. Output: analysis/content_law_forecast.json. Reproduce with python analysis/content_law_forecast.py.

The question

T16 found that on the random-stream copy task, scored content per step -- batch times scored positions per sequence -- sets the pace. T18 fits one law on T16's five cells alone, log step = 6.592 - 0.240 log(content per step), and forecasts T13's six cells with length >= 16, none of which it saw.

One number forecasts learning time, until there is too little per step
One number forecasts learning time, until there is too little per step. Each point is one combination of batch size and sequence length. The grey line is a simple law fitted on the blue points alone. The orange points come from a different experiment the law never saw. Five of six blind forecasts land within 10%. The far-left point, with eight times less material per step than anything fitted, is much slower than the law predicts: below some amount of material per step, learning slows down faster.

Kill test, fixed before execution: the median absolute forecast error exceeds 15%.

Result: the kill test does not fire

CellContent per stepObservedForecastError
16 x 16176386.9210.4-45.6%
64 x 16704166.3150.8-9.3%
16 x 64944148.6140.5-5.4%
256 x 162816102.3108.0+5.6%
64 x 64377698.5100.7+2.2%
256 x 641510480.272.1-10.0%

Median absolute error 7.5%. Five of six cells land within 10%, including one (256 x 64) nearly three times beyond the fitted range's top. A law in one number, fitted on five cells, forecasts a grid that the two-exponent product law could not describe (T13).

Where it fails

The one cell far below the fitted range, 16 x 16 at 176 per step -- eight times less content than any fitted cell -- takes 46% longer than forecast. Below some content per step, learning slows faster than the power law says. That is a knee at the small end, the mirror of the critical batch size at the large end (T8's bending curve), and one cell cannot say where it is. T19 locates it.

A disclosure

The pilot's docstring listed a reference comparison against the product law k B^b C^c. It was not coded, and it has not been added after the run. The comparison the record can make is T13's own: the interaction there is +0.118 with intervals far from zero, so a product law does not describe this grid.

What stands

  • Kill test does not fire. Median forecast error 7.5% on six unseen cells.
  • Content per step forecasts learning time from 704 to 15104 within 10%, from a fit on 1408-5632.
  • Below that it breaks: at 176 per step the model is 46% slower than forecast.

Limits

  • One task, one rate, one model; six targets, one of them the failure.
  • The fit uses cell means; no interval on the forecast errors.

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.

batch size
How many examples the model looks at before updating itself once. Bigger batches give a steadier but more expensive update.
critical batch size
The batch size beyond which making batches bigger barely reduces the number of training steps needed. Below it, bigger batches help a lot; above it, they mostly cost more compute.
exponent
The number in a power law that says how strongly one quantity responds to another. A bigger exponent means a steeper response to the same doubling.
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.
power law
A relationship where one quantity changes by a fixed percentage whenever another one doubles, rather than by a fixed amount. Most scaling results in AI are stated this way.

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