Under Decay the Lead Widens
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. Real training usually shrinks the learning rate towards zero by the end. A recent paper found that this erases most of the benefit of data curricula. Every one of our tests had kept the rate constant after the warm-up.
What we found. With the rate decaying, the easy-sum mix did about as well as before, learning the hard sum on 25 of 32 runs. Ordinary training did much worse than before, on 5. A decaying rate takes away the late steps a slow learner depends on.
Why it matters. The result survives the more realistic schedule. One fair caveat: ordinary training's settings were chosen for a constant rate, so the next test re-tunes them for decay.
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,64training runs of8000steps on thirty-two fresh seeds. The design and the kill test were committed (69db55a) before any run.
Program v2 Bucket A, item A47. Decisive computation: . Output: analysis/sum_mixture_decay.py.analysis/sum_mixture_decay.json
The question
arXiv 2511.18903 finds data curricula help under a constant learning rate and much less once the rate decays, as it does in most training recipes. Every modular-sum comparison here held the rate constant after warm-up. Does the mixture's advantage survive a decaying rate?
Design: A40's two arms on (2, 7), 8000 steps, thirty-two fresh seeds 15001-15032, with a cosine decay to zero at step 8000 added to both: the mixture (no rate warm-up, as A40) decays from 0.003 from step 1; the plain arm warms up linearly over 500 steps, then decays from 0.003. Kill test, fixed before execution: mixed minus the rate-warm-up-and-decay arm includes zero or lies below it. Anchor, in code: with the decay switched off, the loop reproduces A40 on its first seed in both arms -- held.
Results
| Arm (thirty-two fresh seeds, cosine decay) | Final (2, 7) accuracy | Solved (>= 0.9) | Same arm at a constant rate (A40) |
|---|---|---|---|
| easier sums mixed first | 0.892 [0.820, 0.965] | 25 | 0.905 |
| plain, rate warm-up then decay | 0.281 [0.156, 0.405] | 5 | 0.647 |
| mixed minus plain, paired | +0.612 [+0.481, +0.742] | higher on 31 of 32 | +0.258 |
The kill test does not fire. Under a decaying rate the mixture beats the plain recipe on thirty-one of thirty-two fresh seeds, by more than at a constant rate.
What it says, and one thing it does not
The mixture is almost unaffected by the decay (0.892 against 0.905); the plain run loses a great deal (0.281 against 0.647). That fits A42's picture of the mixture as a speed-up: a decaying schedule shrinks the steps that count late in training, which costs a slow learner far more than a fast one. It is the opposite of 2511.18903's finding for its (late-phase, quality-ordered) curricula; ours is an early phase, which a decay does not reach.
What it does not show is that the mixture beats the best plain recipe under decay: the plain arm's peak rate and warm-up were tuned at a constant rate (A36), and a decaying schedule usually wants a higher peak. The gap here is partly that the baseline was not re-tuned for the new schedule -- the programme's standing rule. A51 re-tunes it.
What stands
- A47: kill test does not fire. Under cosine decay, mixed minus the plain rate warm-up:
+0.612[+0.481, +0.742]; solved25against5. - The decay barely affects the mixture (
0.892against0.905at a constant rate) and costs the plain run most of its accuracy (0.281against0.647). The plain decay schedule was not re-tuned (A51).
Limits
- One schedule (cosine to zero), one peak; the plain arm untuned for decay.
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.
- baseline
- The thing you compare against. A result without one is not a result.
- 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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