Research record

Two Easy Sums Are Traps

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 the adding task, a shuffled mix of three easier sums helped a small model learn a hard sum, but the same sums in the wrong order stopped it learning. Is one of the three doing the work, or the harm?

What we found. Two of them, on their own, stopped learning completely: after warming up on either, none of eight runs learned the hard sum, worse than no warm-up. The third, closest to the hard sum, helped somewhat. The shuffled mix of all three still did best.

Why it matters. Easy practice is not automatically helpful. Some of it can lock a model into a way of working that does not carry over, and mixing it with closer material seems to prevent that.

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, 24 new training runs of 8000 steps plus 8 re-run as anchors. The design and the kill test were committed (ef8f7b4) before any run.

Program v2 Bucket A, item A39. Decisive computation: analysis/sum_single_pair.py. Output: analysis/sum_single_pair.json.

The question

On the copy task one easier version did as well as the mix (A25, A27). On modular sum, the order of the three easy pairs decided between help and harm (A22). Is it the mixing, or one of the pairs?

Two of the three easy sums, on their own, stop the model learning
Two of the three easy sums, on their own, stop the model learning. Every model ends on the hard sum after the same 8,000 steps. The first 2,000 steps are spent on one easier sum, or on all three shuffled together. Plain training with no warm-up reaches 53% here. Error bars are 95% confidence intervals. Warming up on the two easiest sums alone leaves the model near chance on every run, worse than no warm-up. The sum nearest the target helps somewhat; the shuffled mix of all three does best, even though two thirds of it is the harmful material.

Design: A18's task and receivers, 8000 steps, rate 0.003. 2000-step warm-ups of (2,3) only, (2,4) only, (2,5) only, or the shuffled mix (A32's arm), then (2,7). Kill test, fixed before execution: the best single pair minus the mix includes zero or lies above it. Anchor, in code: the mix reproduces A32's mixed runs at 0.003 -- held on all eight seeds.

Results

Warm-up (2000 steps)Final (2, 7) accuracySolved (>= 0.9)Minus the plain run at 0.003
shuffled mix0.872 [0.677, 1.067]6 of 8+0.342
(2,5) only0.708 [0.500, 0.915]3 of 8+0.178
(2,4) only0.127 [0.021, 0.233]0 of 8-0.403
(2,3) only0.092 [0.055, 0.129]0 of 8-0.438
(plain run, A32)0.5302 of 8

The kill test fires. The best single pair, (2,5), minus the mix is -0.164 [-0.362, +0.034]: the mix leads on the point estimate and the interval reaches zero. One pair is not shown to be enough, nor shown to fall short.

The other two pairs are traps. A warm-up of (2,3) or (2,4) alone leaves the model worse than no warm-up at all (-0.438, -0.403), solving the task on no seed. The same two pairs make up two thirds of the mix, which does best.

What it says

With A22 this makes a consistent picture on modular sum. Training for long on an easy pair far from the target -- whether alone or as the last phase of an ordered warm-up -- stops the model learning the target in the budget; mixing it with the nearer pair does not. Two readings, which predict different things: the far pairs put the model into a solution it cannot leave (a trap), or they only delay it (a slower start the budget does not cover). A41 runs (2,3) and the plain run to 20000 steps to tell them apart.

This is also the sharpest difference from the copy task. There, easier versions only ever helped or did nothing, and their benefit was the generic one of a learning-rate warm-up (A33). Here, easy material can actively prevent learning, which a learning-rate warm-up cannot reproduce or explain.

What stands

  • A39: kill test fires. The best single pair (2,5) minus the mix: -0.164 [-0.362, +0.034].
  • (2,3) and (2,4) alone are harmful (0 of 8 solved; -0.44 and -0.40 against no warm-up).

Limits

  • Eight seeds on a bimodal outcome, one rate, one budget.
FOLLOW-UP 2026-09-28 by A41. Run to step 20000, the (2,3) warm-up arm solves 4 of 8 against the plain run's 5 (its kill test does not fire, by one seed), but ends at the same mean accuracy and reaches 0.9 on 6 of 8 seeds at some point. "Trap" above is mostly a delay of several thousand steps. Text and numbers above unchanged.

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.

accuracy
The fraction of answers a model gets right on questions it was not trained on.
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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