Research record

The Sum Never Crosses

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. On one task, the easy-start recipe went from costing time to saving it as the task got harder, switching over at a particular difficulty. We asked whether a different task, adding two symbols, switches at the same place.

What we found. We could not tell. Even the easiest version of the sum we could make was harder than that switching point, and the easy start was ahead on every version, by more on the harder ones.

Why it matters. That agrees with a shared switching point but does not test it. We should have checked how easy the easiest version was before running it. The next try uses fewer symbols to make the sum easier.

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, 128 training runs of 24000 steps on sixteen fresh seeds per rung. The design and the kill test were committed (36b9088) before any run.

Program v2 Bucket A, item A71. Decisive computation: analysis/sum_ladder.py. Output: analysis/sum_ladder.json.

The question

On a ladder of maxima (A65) the easy start moved from a cost to a saving as the task got harder for plain training, crossing where plain needs about 6680 mean steps. Does the modular sum cross near the same plain time?

The sum never crosses, because every version of it was already hard
The sum never crosses, because every version of it was already hard. Two ladders of tasks, each made harder by moving one remembered symbol further back: the maximum of two symbols and their sum. Below zero means the easy start was faster. The maximum crosses from cost to saving where ordinary training needs about 6,700 steps. Every version of the sum was harder than that, and the easy start was ahead on all of them, so the sum agrees with the same crossing but cannot test it.

Design: A65's ladder with modular-sum targets (mod 32), far lags 6, 7, 8, 10, all new; sequence length 9 + far; GRU, width 48, rate 0.003, 24000 steps, sixteen fresh seeds per rung; easy sums on (2,3)/(2,4)/(2,5) for 2000 steps; plain with a 500-step learning-rate warm-up. Kill test, fixed before execution: the sum's interpolated crossing lies outside [3340, 13360] plain mean steps, or it never crosses. Anchor, in code: A40's first mixed run reproduces -- held.

Results

Far lagPlain mean steps to 0.9Mixed minus plain (steps)Solved (mixed / plain)
610488-4205.6 [-8693.3, +282.1]15 / 13
714410-6068.8 [-11464.5, -673.0]15 / 10
814982-5303.1 [-11155.4, +549.1]13 / 11
1023566-11858.8 [-16445.9, -7271.6]11 / 1
  • No crossing: the mean difference is negative at every rung. Spearman(plain mean, difference) -0.8.

The kill test fires, on its "never crosses" clause.

What it says, and what it cannot

The test could not do what it was designed to do. Every rung of the sum was harder for plain training than the maximum's crossing point -- the easiest, lag 6, already needs 10488 mean steps against the crossing's 6680 -- so a sum that crossed at the same place as the maximum would show exactly this: the easy start ahead at every rung, by more as the rungs get harder. The data agree with a shared crossing and do not test it. The design fault is that the sum's lowest reachable rung was not calibrated before the kill test was fixed; the easy set uses lags up to 5, so the hard lag cannot go below 6, and at 32 symbols that is already hard.

What does stand is the direction: on the sum, across four difficulties, the easy start is ahead at every one, by 4000 to 12000 steps, and plain solves fewer runs (35 of 64 against 54) within the budget.

A76 reaches the low-difficulty region with a smaller modulus.

What stands

  • A71: kill test fires (no crossing), uninformatively. Every rung (plain 10488-23566 steps) was above the maximum's crossing (6680); the easy start is ahead at all four, as a shared crossing would predict.

Limits

  • The lowest rung was not calibrated against the crossing before the run. Plain failures are counted at 24000, so the means at the harder rungs are partly how often plain solves at all.
  • Rate 0.003 for both arms throughout.

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.

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.
GRU
Gated Recurrent Unit. A compact design for processing sequences one item at a time, with internal switches controlling what it keeps in memory.
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.
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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