Research record

Subtraction Too

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: Is the task we are studying actually hard? – Often it is not. A rule from 1990 with no parameters beats the trained model on the task most of these results were measured on, and what an intervention costs is set by the task's own structure.

In plain English

What we asked. Every result so far used addition as the hard task. Was the speed-up something special about adding? We made the hard task a subtraction instead, with easier subtractions as the warm-up.

What we found. The speed-up held at the same size: the typical run solved the hard subtraction in about 3,300 steps with the easy start, against 6,800 without.

Why it matters. What carries over is the kind of task -- remember two symbols and combine them -- not addition itself. The next test tries a way of combining that is not arithmetic at all.

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, 32 training runs of 16000 steps on sixteen fresh seeds. The design and the kill test were committed (b13d665) before any run.

Program v2 Bucket A, item A57. Decisive computation: analysis/difference_mixture.py. Output: analysis/difference_mixture.json.

The question

Every positive result in the modular-sum thread has addition as the hard task (A44, A53, A58). Does the speed-up hold for another two-read-and-combine target?

With subtraction as the hard task, the easy start still halves the steps
With subtraction as the hard task, the easy start still halves the steps. The hard task here is to subtract the symbol 7 steps back from the one 2 steps back (mod 32). One recipe starts with easier subtractions; the other is ordinary training with a learning-rate warm-up. The typical run solves it in about 3,300 steps with the easy start against 6,800 without: the same size of saving as for addition. The effect is not about one operation.

Design: GRU, width 48, rate 0.003, 16000 steps, sixteen fresh seeds. Hard task (x[t-2] - x[t-7]) mod 32; easy warm-up of differences on (2,3)/(2,4)/(2,5) for 2000 steps; targets computed in the pilot from the sum generator's inputs, for training and held-out alike. Mixed against plain with a 500-step learning-rate warm-up; steps to 0.9 (never = 16000). Kill test, fixed before execution: mixed minus plain includes zero or lies above it. Anchor, in code: with targets left as sums, the loop reproduces A40 -- held.

Results

Arm (sixteen fresh seeds, hard difference)Median steps to 0.9Solved by 16000
easy differences first331515
plain, learning-rate warm-up684012
  • Mixed minus plain: -4649.4 [-6799.0, -2499.7] steps; ratio of mean steps 0.484.

The kill test does not fire. With subtraction as the hard task, the easy warm-up reaches the solution in about half the steps -- the same size of saving as for addition (0.578 at width 48, 0.544 at width 96).

What it says

The speed-up is not a property of addition: it holds for a second two-read-and-combine operation, with the same structure of easier versions sharing the target's first read. With A52 (easy differences help a hard sum) the evidence points to the task's shape -- two reads combined -- rather than to one operation.

What stands

  • A57: kill test does not fire. Hard difference: mixed minus plain -4649.4 [-6799.0, -2499.7] steps; ratio 0.484; solved 15 against 12.

Limits

  • One more operation, a close relative of addition; sixteen seeds; the GRU at width 48; rate not re-tuned for the difference task.

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.

GRU
Gated Recurrent Unit. A compact design for processing sequences one item at a time, with internal switches controlling what it keeps in memory.
held-out
Data the model was never trained on, kept back specifically to test it. Scoring a model on data it has already seen measures memorisation, not learning.
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.
LSTM
Long Short-Term Memory. An older and larger relative of the GRU, also gated, also for sequences.
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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