Longer Memory, Lower Limit
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: What decides when a model learns, and can you change it? – Settings dominate, data barely matters, and there is a brief window before the jump in which interrupting the model is unusually costly. Timing can be delayed but not brought forward.
In plain English
What we asked. Our last test of a 2025 prediction -- that the fastest usable learning speed drops as a model has to remember further back -- was too coarse to tell. So we repeated it with finely spaced speeds and memory spans from 2 to 8 steps.
What we found. This time the pattern is clear: the limit falls steadily, from 0.072 at 2 steps to 0.020 at 8, close to the power law the paper describes, and it holds even leaving out the shortest span. Above the limit the model does not stop learning entirely; it just gets only partway.
Why it matters. In practice: if your model must hold information over long stretches, expect to need a lower learning rate, by roughly the square root of how much longer the memory is.
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,231training runs of5000steps. The design and kill test (computed without the smallest lag, the lesson of T21) were committed (7835e35) before any run.
Program v2 Bucket T, item T22. Decisive computation: (T21's code on a finer grid). Output: analysis/ghost_critical_rate_fine.py.analysis/ghost_critical_rate_fine.json
The question
arXiv 2501.02378 (the ghost mechanism) predicts that the critical learning rate for abrupt learning in RNNs falls as an inverse power of the timescale being learned. T21 found a fall with the copy lag (-0.437), but its lag-2 point sat at the top of the rate grid and, without it, the grid's factor-of-1.5 spacing could not resolve a slope. T22 reruns T21 on a finer grid with room above: rates a factor 1.2 apart from 0.02 to 0.124, lags 2-8, three seeds, 5000 steps.
Kill test, fixed before execution: the exponent's interval over lags 3-8 includes zero or lies above it (excluding the smallest lag so one step cannot carry the result).
Result: the kill test does not fire
| Lag | Critical rate (three seeds) |
|---|---|
2 | 0.072, 0.072, 0.072 |
3 | 0.041, 0.041, 0.035 |
4 | 0.035, 0.035, 0.035 |
5 | 0.029, 0.029, 0.029 |
6 | 0.035, 0.035, 0.035 |
7 | 0.029, 0.024, 0.024 |
8 | 0.020, none, 0.020 |
| Exponent of critical rate on lag | Value |
|---|---|
Lags 3-8 (the kill test) | -0.529 [-0.621, -0.437] |
All lags 2-8 | -0.746 [-0.760, -0.732] |
The critical learning rate falls as a power of the lag, beyond the first step: from 0.072 at lag 2 to 0.020 at lag 8, an exponent near -0.5 over lags 3-8. Lag 2 is no longer censored (0.072 against a grid top of 0.124), and the finer spacing resolves the slope T21 could not. One point is out of line (lag 6 above lag 5), and at lag 8 one seed learned at no rate in the grid, so the long-lag end is censored from below; the fall there may be steeper, not shallower.
How runs fail above the critical rate
All 140 runs above their critical rate learned partway -- above chance, below 0.5 -- none stayed at chance and none learned and then collapsed. The paper's "no-learning zone", where gradients vanish and the network locks into confident wrong answers, was not seen on this rig; the ceiling on the learning rate here is a ceiling on how far learning gets, not on whether it starts.
What stands
- Kill test does not fire. Over lags
3-8the critical rate falls aslag^-0.53[-0.62, -0.44]; over all lags,lag^-0.75. The ghost mechanism's inverse power law holds on this GRU. T21 carries a RESOLVED banner. - Practical reading: the longer the memory a recurrent model must learn, the lower the learning rate it can tolerate -- here about a factor of
3.5from a memory of2steps to8. - Failure above the limit is partial learning, not a no-learning zone.
Limits
- One width, one task family,
5000steps; lag8partly censored at the bottom of the grid. - The endpoint is reaching accuracy
0.5, not the paper's measure of abrupt learning.
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.
- exponent
- The number in a power law that says how strongly one quantity responds to another. A bigger exponent means a steeper response to the same doubling.
- gradient
- The direction and amount by which each of a model's internal numbers should change to do slightly better. Training is repeatedly following it.
- 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.
- power law
- A relationship where one quantity changes by a fixed percentage whenever another one doubles, rather than by a fixed amount. Most scaling results in AI are stated this way.
- recurrent
- A design that reads a sequence one item at a time, carrying memory forward. The main alternative is attention, which looks at everything at once.
- 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.
- slope
- How steeply one quantity changes as another does. A slope of one between a warning and the event it predicts means the warning shifts exactly in step with the event.
- width
- How many internal numbers a model uses at each layer. The usual way we vary model size in these experiments.
Get new results as we publish them
Roughly monthly, one finding per email, in plain English first. Including the approaches that turned out not to work, which are usually the useful ones. No sales email.
Double opt-in: we send a confirmation link and add nobody who does not click it. One-click unsubscribe on every email.