Research record

Late, Not Delayed

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. Grokking, when it happens, can take a very long time. So we gave our model three times longer, with just under the amount of data it had needed before, to see whether the famous delayed insight would finally show up.

What we found. With enough time, the model learned the rule from fewer examples than before, but always slowly and all at once: it got better at its training examples and at new problems together. It never memorised first and understood later. We have now looked for grokking three ways and not found it on this setup.

Why it matters. Useful to know: 'not enough data' can partly mean 'not enough time'. Giving a small model longer let it learn from a third of the data.

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, 9 training runs of 30,000 steps. The design and kill test were committed (e59c3e8, docstring rewrapped in b86e8d1 with no logic change) before any run.

Program v2 Bucket E, item E11. Decisive computation: analysis/grokking_long.py (E10's code with pools and budget changed). Output: analysis/grokking_long.json.

The question

E10 found that with weight decay 1.0 the modular-sum model generalises from 1024 sequences without any delay, and at 256 does nothing in 10,000 steps. Grokking delays can run far longer than 10,000 steps and sit just below the data a network needs. E11 runs E10's pilot at pools 384, 512, 768 for 30,000 steps, with E5's criterion (memorise to 0.99 while held-out is below 0.5, then held-out 0.9 at least three times later).

Given long enough, the model learns the rule late, but never after memorising
Given long enough, the model learns the rule late, but never after memorising. The addition task with only 384 training sequences and weight decay, run for 30,000 steps. Each colour is one training run; solid lines are its training set, dashed lines are problems it has never seen. Two of the three runs eventually solve new problems perfectly, thousands of steps in, but the solid and dashed lines rise together. The model is learning slowly, not memorising first and understanding later, so this is not grokking.

Kill test, fixed before execution: no cell has two of three seeds that grok.

Result: the kill test fires

PoolSeedHeld-out reaches 0.5Training reaches 0.9Final training / held-out
384810110,85010,9001.00 / 1.00
38481117,2507,3001.00 / 1.00
38481177,550never0.68 / 0.49
51281012,2007,8000.86 / 0.77
51281112,05020,6500.81 / 0.74
51281173,3003,5501.00 / 1.00
76881016,500never0.56 / 0.49
76881111,0004,0500.91 / 0.86
7688117nevernever0.32 / 0.26

No run groks. Wherever held-out accuracy rises, it rises with or before training accuracy: the model never fits its training set first and generalises later.

What the longer runs add

  • The edge moves down with the budget. At 384 sequences -- below E10's 1024 -- two of three seeds generalise completely, at steps 7,300 and 10,900. E10's edge was partly its 10,000-step budget. Late learning, but not delayed generalisation: training and held-out accuracy climb together.
  • Weight decay 1.0 makes the middle pools unstable. At 512 and 768 several runs learn partway and then fall back or stall (final held-out 0.26-0.86); only three of six end fully solved.

What stands

  • Kill test fires. Across E5, E10 and E11 -- pools of 8 to 16384 sequences, up to 30,000 steps, with and without weight decay -- no run on this rig memorises first and generalises later.
  • With a longer budget the model generalises from fewer examples (384 at 30,000 steps against 1024 at 10,000), always learning the rule and fitting the data together.
  • The grokking bridge is not built on this rig. The programme's transition and grokking's delayed generalisation remain separate phenomena on present evidence; the E thread closes here.

Limits

  • One task, one width, one rate, one decay strength. Grokking in the literature is usually shown with transformers on two-token arithmetic; a recurrent model on sequences may simply not show it.

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.
grokking
When a network that has memorised its training examples suddenly, much later, learns the general rule and starts getting new examples right.
held-out
Data the model was never trained on, kept back specifically to test it. Scoring a model on data it has already seen measures memorisation, not learning.
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.
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.
transformer
The architecture behind most modern large language models. It uses attention to look at every part of the input at once, rather than reading in order.
weight decay
A common training setting that gently pulls a model's internal numbers toward zero, used to stop it over-fitting.
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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