Research record

The Rate Policy Sets the Exponent

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 actually makes training cheaper? – One thing has worked: stopping part of the training early saved about 7% with no loss of quality. Everything else tested has been matched by a simpler or cheaper method -- and in two cases the clever method was only winning because it was quietly being given more.

In plain English

What we asked. We have been testing a published rule for how much faster a model learns when it sees more examples per step. Each time we changed how the learning speed was chosen, the rule came out different. This time we used the fastest learning speed at each batch size.

What we found. That gave yet another answer. Across four ways of choosing the learning speed, the same small model on the same task produced a spread of results wider than the difference between it and the large language models the rule came from. The fastest settings also gave up some accuracy when batches were small.

Why it matters. In practice: a rule about batch size only means something alongside how the learning speed was set. When you see one quoted, ask which.

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, 40 training runs. The multipliers were fixed from T12's committed tuning data, and the design and kill test committed (69fa22c), before any run.

Program v2 Bucket T, item T15. Decisive computation: analysis/batch_law_speed_optimal.py. Output: analysis/batch_law_speed_optimal.json. Reproduce with python analysis/batch_law_speed_optimal.py (about twenty minutes on a throttled laptop CPU); --reuse re-derives every endpoint from the saved series.

The question

T12's tuning grid holds an interior speed optimum at every batch: 3x the square-root rule at batches 16-128 and 2x at 256. T15 measures the batch exponent there, on J8's eight receivers, with T8's endpoint (first step at held-out accuracy 0.5) and per-receiver slope.

How the learning rate is set decides the batch-size law
How the learning rate is set decides the batch-size law. One small model, one task. Each bar is how fast the training steps needed fall as the batch size doubles, under a different policy for the learning rate. The grey bar is a published value for large language models trained at a fixed rate. The same model gives anything from 0.31 to 0.57 depending only on the rate policy, a wider range than separates it from the published language models. A batch-size law is a statement about a rate policy too.
The fastest learning rate costs accuracy when batches are small
The fastest learning rate costs accuracy when batches are small. For each batch size, the learning rate that reached 50% accuracy soonest, and how accurate the model then was at the end of a fixed-length run. At large batches the fastest rate also finishes accurate. At small batches it gives up several points of accuracy: the fastest setting is one end of a trade-off, not a free choice.

Kill test, fixed before execution: the exponent's interval overlaps the square-root rule's ([-0.585, -0.556]). Anchor: batch 256 at 2x reproduces T10's cell on every receiver. It holds.

Result: the kill test does not fire

BatchMultiplierStep to 0.5Final accuracy
163117.0 [106.4, 127.6]0.909
32379.3 [72.3, 86.4]0.934
64359.5 [50.9, 68.0]0.941
128342.6 [36.1, 49.1]0.958
256230.4 [27.8, 33.1]0.993
Exponent-0.479 [-0.488, -0.471]

At the speed-optimal rates the batch exponent is -0.479, clear of the square-root rule's interval.

The batch exponent across the thread

How the rate was setExponentRecord
Fixed at 0.006 (as the paper did)-0.315 [-0.329, -0.301]T8
Speed-optimal at each batch-0.479 [-0.488, -0.471]T15
Square-root rule doubled-0.514 [-0.533, -0.495]T10
Square-root rule-0.571 [-0.585, -0.556]T8
Published law (transformers, fixed rate)-0.372511.16893

On one model and one task, how the learning rate is set moves the batch exponent from -0.31 to -0.57, a range wider than the gap between this 17K-parameter recurrent network and the paper's transformer language models. A batch-size exponent is a property of a rate policy, not of batch size alone.

The speed-optimal rates also cost accuracy at small batches (0.909 at batch 16, against 0.975 at the lowest rate in T12's grid): they are the fast end of T12's frontier, not a free choice.

What stands

  • Kill test does not fire. Speed-optimal tuning gives -0.479, not the square-root rule's -0.571.
  • Four rate policies, four exponents from -0.31 to -0.57. Quote a batch exponent with its rate policy.
  • Speed-optimal is not accuracy-neutral: final accuracy at batch 16 falls to 0.909.

Limits

  • Speed optimum on three throwaway seeds and a grid of multipliers; the true optimum lies between grid points.
  • One substrate, 800 steps.

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.
batch size
How many examples the model looks at before updating itself once. Bigger batches give a steadier but more expensive update.
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.
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.
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.
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.
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.

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