Research record

On Modular Sum the Mixture Holds Up

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 copy task, our easy-first warm-up stopped looking useful once plain training was given its best learning rate. We ran the same fair check on the second task, adding numbers, where the warm-up had looked strongest.

What we found. Here it held up. At every learning rate we tried, starting with easier sums beat plain training, and it solved the task on six of eight runs where plain training managed at most three. But plain training was still getting better at the smallest rate we tried, so we cannot call the comparison yet.

Why it matters. The next run tries smaller rates, adds a standard warm-up schedule, and doubles the number of runs, because on this task runs mostly either succeed or fail outright.

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, 48 new training runs of 8000 steps plus 16 re-used. The design and the kill test were committed (9c427e5) before any run.

Program v2 Bucket A, item A32. Decisive computation: analysis/sum_mixture_tuned.py. Output: analysis/sum_mixture_tuned.json. Descriptive post-hoc (solved counts): analysis/sum_mixture_tuned_posthoc.py and its output.

The question

A18 found an early random mix of easier sums raises final (2, 7) accuracy by +0.574 at rate 0.005, a rate never swept for this task. On the copy task, the same kind of gain vanished once both arms were tuned (A31). Does it survive tuning here?

On the second task, the mixture stays ahead at every learning rate tried
On the second task, the mixture stays ahead at every learning rate tried. The second task: add two symbols from 2 and 7 steps back. Plain training on it, against training that starts with a random mix of easier sums, each at four learning rates. Runs here mostly either solve the task or stay near chance. Unlike the copy task, the mixture is ahead at every rate and solves the task on six of eight runs at three of them. But plain training was still improving at the smallest rate tried, so the fair comparison is not finished.

Design: A18's task, seeds (J8's receivers), 8000 steps and 2000-step mix. Fixed (2, 7) and mixed at rates 0.002, 0.003, 0.008 (new) and 0.005 (A18). Kill test, fixed before execution: best mixed minus best fixed, paired at each arm's own best rate, includes zero or lies below it; a best rate at a grid edge is reported as not found. Anchor, in code: fixed at 0.005 on the first seed reproduces A18 -- held.

Results

RateFixed (2, 7)Solved (>= 0.9)Mixed early, then (2, 7)Solved (>= 0.9)
0.0020.546 [0.204, 0.889]3 of 80.844 [0.628, 1.058]6 of 8
0.0030.530 [0.220, 0.839]2 of 80.872 [0.677, 1.067]6 of 8
0.005 (A18)0.286 [-0.001, 0.573]1 of 80.860 [0.673, 1.048]6 of 8
0.0080.065 [0.050, 0.080]0 of 80.715 [0.516, 0.913]2 of 8

(Intervals are t-intervals on bimodal data, which is why some run past 1.)

The kill test fires. Best mixed (0.872 at 0.003) minus best fixed (0.546 at 0.002) is +0.325 [-0.072, +0.723]. But the test is undecided rather than negative, for two reasons. The fixed arm's best rate is at the bottom edge of the grid, so its optimum is not found -- it was still improving as the rate fell. And the outcomes are bimodal (a seed solves the task or stays near chance), so eight seeds give an interval nearly 0.8 wide.

What the table does show, without a test: unlike the copy task (A31), the mixed arm is above the fixed arm at every rate, and solves the task on 6 of 8 seeds at every rate from 0.002 to 0.005; the fixed arm never solves it on more than 3. Lowering the rate helps the plain run (from 1 to 3 solved) and has not yet caught up.

What it says

On the copy task, tuning the rate erased the warm-up's gain. On modular sum it has not -- yet. Two things are needed to decide it: the plain run's grid extended below 0.002 until its optimum is found, and a learning-rate warm-up arm, which on the copy task matched every data warm-up (A33). That is A36, with more seeds because the outcomes are bimodal.

What stands

  • A32: kill test fires, +0.325 [-0.072, +0.723], with the plain run's best rate not found (grid edge).
  • The mixed arm is higher at every rate tried and solves the task on 6 of 8 seeds at three of them; the plain run never on more than 3.

Limits

  • Eight seeds on a bimodal outcome; the fixed arm's optimum not found; no learning-rate warm-up arm.

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.
post hoc
Worked out after the fact, rather than decided in advance. We report such checks separately and never let them decide a result, because it is far too easy to find a pattern once you already know the answer.
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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