Research record

The Sum Crosses Earlier

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. We had found that the easy-start recipe costs time on easy tasks and saves time on hard ones, switching over at a certain difficulty. We tested a second task, adding two symbols, with versions easy enough to reach the switch.

What we found. It switches too, smoothly, from a small cost to saving more than half the steps on the hardest version. But it switches at a lower difficulty than the first task did.

Why it matters. So the pattern carries over, and the exact switching point does not. Where it falls depends on the task, so it has to be measured rather than assumed.

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, a disclosed calibration and the kill test were committed (c99a38b) before any run.

Program v2 Bucket A, item A76. Decisive computation: analysis/sum_ladder_small.py. Output: analysis/sum_ladder_small.json.

The question

On the maximum the easy start crossed from cost to saving where plain training needs about 6680 mean steps (A65). At 32 symbols every rung of the sum was already above that point (A71). With a smaller modulus that reaches the easy region, does the sum cross near the same place?

The sum crosses too, at a lower difficulty than the maximum
The sum crosses too, at a lower difficulty than the maximum. Two ladders of tasks made harder by moving one remembered symbol further back: the maximum of two symbols, and their sum with 12 symbols. Below zero means the easy start was faster. Both switch from a small cost to a growing saving as the task gets harder. The sum switches at about 2,800 steps and the maximum at about 6,700, so the switching point depends on the task.

Design: the sum mod 12 with a 12-symbol model (the model shrinks with the vocabulary; both arms share it), far lags 6, 8, 10, 12 (sequence length 9 + far), chosen by a calibration disclosed before running (plain lag 6 about 1600 steps, lag 10 about 6845). 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 rate warm-up. Kill test, fixed before execution: the interpolated crossing lies outside [3340, 13360] plain mean steps, or there is none. Anchor, in code: A40's first mixed run reproduces -- held.

Results

Far lagPlain mean steps to 0.9Mixed minus plain (steps)Solved (mixed / plain)
62182+501.9 [-14.9, +1018.6]16 / 16
86582-3185.6 [-5839.7, -531.6]16 / 15
1015336-9867.5 [-13936.0, -5799.0]16 / 10
1221886-12907.5 [-16625.2, -9189.8]14 / 5
  • Spearman(plain mean, difference) -1.0: perfectly ordered. Crossing at plain mean 2781 steps.

The kill test fires. The sum crosses, with the same shape as the maximum, but at about 2800 plain steps -- below the band around the maximum's 6680.

What it says

The two-sided pattern of A65 holds on a second operation: on easy versions the easy start costs a little (here +502 steps, an interval touching zero); as the task gets harder for plain training the saving grows, to -12908 steps at lag 12, where plain solves only 5 of 16 runs. But the crossing is not a constant of the rig: the sum's is about 2.4 times lower than the maximum's in plain steps. So "skip the easy start below about 6700 steps" does not carry from one operation to another; where the line falls depends on the operation (and here also on the smaller model).

What stands

  • A76: kill test fires. Sum mod 12: perfectly ordered (Spearman -1.0), crossing near plain mean 2781 steps, outside the predicted band around the maximum's 6680.

Limits

  • Mod 12 changes the model's input and output size as well as the task; the crossing difference may be partly that.
  • Rate 0.003 for both arms throughout. Plain failures at the hard rungs are counted at 24000.

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.

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.
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.
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.
vocabulary
The set of distinct symbols a model can read and produce.
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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