Research record

The Law Overshoots Long Memories

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. We had found two rules for the fastest learning speed a small model can use: it drops as the memory it needs gets longer, and it rises with bigger batches. We combined them into one formula and used it to forecast four settings we had never tried.

What we found. The forecasts were in the right range, all within a factor of 1.6, but every one was too high, and more so for longer memories. Either the two effects do not simply multiply, or our new runs were too short: we ran them for fewer steps than the experiment the memory rule came from, which can make slow learning look like failure.

Why it matters. The lesson: when testing a forecast, run it under the same conditions as the data the forecast was built on, or you cannot tell a wrong law from a changed setup.

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, 84 training runs of 4000 steps. The fit, the four forecast cells and the kill test were committed (a7f96bf) before any run.

Program v2 Bucket T, item T24. Decisive computation: analysis/critical_rate_law.py. Output: analysis/critical_rate_law.json.

The question

T22 found the critical learning rate (the largest at which the model still learns) falls with the copy lag; T23 found it rises with batch. T24 asks whether they combine as one product law, fitted on those two sweeps alone:

A combined law gets the scale right but overshoots for long memories
A combined law gets the scale right but overshoots for long memories. A law combining two earlier findings (the limit falls with memory span and rises with batch size) forecast the fastest usable learning rate at four settings neither earlier experiment tried. Grey: forecast. Blue: measured. Every forecast is within a factor of 1.6, but all are too high, and more so for longer memories. Part of that may be our own design: these runs were shorter than the ones the memory law came from.

c = 0.0170 lag^-0.725 batch^0.419

and forecast four (lag, batch) cells neither sweep ran, each measured on a grid of rates a factor 1.2 apart spanning the forecast, three seeds, 4000 steps.

Kill test, fixed before execution: the median absolute forecast error exceeds 20% (one grid step).

Result: the kill test fires, narrowly

CellForecastMeasuredError
lag 2, batch 2560.10500.0988-6%
lag 3, batch 320.03280.0273-17%
lag 8, batch 2560.03840.0267-31%
lag 6, batch 160.01480.0097-35%

Median absolute error 23.6%, against a limit of 20%. The law gets the order of magnitude everywhere and the short-lag cells within one grid step, but every forecast is too high, and the error grows with the lag.

Two explanations, not separated here

  • The two effects may interact: the lag exponent could be steeper away from batch 64, where T22 measured it -- the same kind of failure of a product form that T13 found for batch and sequence length.
  • The budgets differ, and that is a fault in this design. T22 fitted the lag effect with 5000-step runs, T23 the batch effect with 3000, and T24 measured with 4000. At long lags learning is slow, so a shorter budget can make a rate that would eventually learn look like a failure, lowering the measured critical rate exactly where the misses are largest. The design should have used T22's budget for the long-lag cells.

What stands

  • Kill test fires: median forecast error 23.6%, all four forecasts too high, the error growing with lag.
  • The product law is right in direction and scale (every cell within a factor 1.6), not within one grid step.
  • Method: a forecast test fitted on sweeps run at different budgets inherits their mismatch; match the budget of the fit before testing a forecast.

Limits

  • Four cells, three seeds each; one seed at lag 8, batch 256 was censored.

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.

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.
learning rate
How big a step training takes each time it updates the model. Too small and nothing happens; too big and it never settles.
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.

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