$ attention.sh

knowledge base · entry 10

Most big models are undertrained

A read-through of the paper that showed the era's giant models were built wrong — too big for the data they saw — and gave the rule for fixing it.

paper Training Compute-Optimal Large Language Models
authors Hoffmann, Borgeaud, Mensch, et al. — DeepMind (2022)
source arXiv:2203.15556
why we read it right-sizing a model is the difference between paying for quality and paying for waste

every figure zooms — click it, scroll to magnify, drag to pan

part one

The idea, before any jargon

figure 1 · the whole idea

Most of the giants were trained wrong

For a few years the recipe for a better language model was simple: make it bigger. GPT-3 had 175 billion parameters; Gopher had 280 billion. This paper asked whether all those parameters were actually being used well — and found that they weren't. The giants had been fed far too little data for their size.

To prove it, DeepMind took the same compute budget that trained the 280B Gopher and instead trained a 70B model — a quarter the size — on more than four times the data. They called it Chinchilla, and it beat Gopher across the board.

click any figure to zoom · scroll to magnify · drag to pan

Note the exact claim, because it's precise: not "smaller is better" in general, but "for a fixed compute budget, the field had been choosing the wrong point on the size-versus-data trade-off." Spend the same money differently and you get a better model.

figure 2 · the trade-off

A fixed budget is a pie you have to split

The reason a trade-off exists at all is a piece of arithmetic. Training compute is roughly 6 × parameters × tokens. So for a fixed budget, the product of model size and data is fixed: every parameter you add is a token you don't get to train on, and vice versa.

That means there's a sweet spot. Too small, and the model can't absorb what the data has to teach. Too big, and it's starved — it never sees enough tokens to fill its capacity. The whole paper is a careful search for where that peak sits.

part two · the method

How they found the peak

figure 3 · isoFLOP curves

One budget at a time

The method is more convincing than any single model. For a given compute budget, they trained many models of different sizes and recorded each one's final loss. Plot loss against model size and you get a U — too small underfits, too big starves, and the bottom of the U is the best size for that budget.

Do it for several budgets and the bottoms of the U's line up. Joining them gives a clean rule: as you scale compute, the optimal model size and the optimal amount of data grow together, at roughly the same rate. Three independent approaches in the paper all agree on it.

part three · the number

The rule you can carry around

figure 4 · ~20 tokens per parameter

About twenty tokens per parameter

Distilled to a rule of thumb: a compute-optimal model should see roughly 20 training tokens for every parameter. A 70B model wants about 1.4 trillion tokens — which is exactly what Chinchilla got. Now apply the same rule to the models of the day.

GPT-3 and Gopher wanted trillions of tokens and got 300 billion — undertrained by more than ten times. The field had been pouring its budget into parameters and leaving the models hungry. Model and data are meant to grow in lockstep, and the ratio had been badly backwards.

part four · the payoff

Smaller, better, and cheaper

figure 5 · the result

Right-sizing pays twice

Chinchilla didn't just edge out Gopher — it beat it across language modelling, reading, reasoning, and the MMLU knowledge test, at a quarter of the size.

And there's a bonus that keeps paying. Because the compute-optimal model is smaller, it's also cheaper to run — about a quarter the inference cost, every single time you serve it, forever. Better on the day it ships and cheaper on every day after: the case for right-sizing is hard to argue with.

the math · in the paper's notation

The law, and where 20:1 comes from

Two equations govern the whole paper: the fitted loss law and the constrained optimum it implies. Both decoded here, with Chinchilla's own numbers pushed through them — after this, section 3 of the PDF is review.

the equations · worked

The loss law, then the optimum under a budget

§3.3 · the parametric loss law (approach 3)

L(N, D)  =  E  +  ANα  +  BDβ

read aloud: the loss a trained model lands at is an irreducible floor, plus a penalty for being too small, plus a penalty for seeing too little data. Each penalty melts away as a power law in its own variable. The paper fits the five constants to hundreds of training runs: E = 1.69, A = 406.4, B = 410.7, α = 0.34, β = 0.28.

  • Nparameters
  • Dtraining tokens
  • Ethe entropy of language itself — the loss no model of any size escapes
  • A, αhow hard, and how fast, finite model size hurts
  • B, βhow hard, and how fast, finite data hurts
plug in → Chinchilla itself, N = 70e9, D = 1.4e12: A/Nα = 406.4/70e90.34 ≈ 0.083, B/Dβ = 410.7/1.4e120.28 ≈ 0.163 → L ≈ 1.69 + 0.083 + 0.163 ≈ 1.94. note the split: the data term is twice the size term — even Chinchilla is more data-starved than size-starved.

§3 · the budget constraint and the optimum

C  ≈  6·N·D      Nopt ∝ Ca,   Dopt ∝ Cb,    a = βα+β ≈ 0.46,   b = αα+β ≈ 0.54

read aloud: minimize L(N, D) while 6ND stays at your budget, and the optimal size and data both grow as roughly the square root of compute. a ≈ b ≈ 0.5 is the headline "grow them together"; the ratio D/N that falls out is ≈ 20 tokens per parameter. (GPT-3-era laws had a ≈ 0.73 — nearly all new compute into parameters — which is exactly the error the paper corrects.)

  • Ctotal training FLOPs — 6 per parameter per token (2 forward, 4 backward)
  • a, bthe scaling exponents: how the optimum splits new compute between size and data
plug in → Gopher's budget: C = 6 · 280e9 · 300e9 ≈ 5.0×1023. spend it at N = 70B instead: D = C/6N ≈ 1.2T tokens — a quarter the size, four times the data, same bill. that model is Chinchilla, and it wins. check the rule: 1.4e12 / 70e9 = 20 tokens per parameter.

part five · what we take from it

Budgeting a specialist

applied · our reading

The compass, and its fine print

For a domain model the lesson is to resist the reflex to reach for more parameters. Balance size against data; treat ~20 tokens per parameter as a starting point. But two pieces of fine print matter as much as the rule.

the fine print First, Chinchilla optimizes training compute only — it says nothing about the cost of serving the model afterwards. If you'll run a specialist millions of times, the optimum shifts smaller still, which is exactly what the next entry works out. Second, recall phi-1: better data can move this whole curve, so 20:1 is a compass, not a law.

next in the arc

Where this goes

Chinchilla found the compute-optimal balance for training. Beyond Chinchilla asks the obvious next question: once you also pay to serve the model — on every query, for its whole lifetime — the sums change, and the answer is to train an even smaller model on even more data.

source

Hoffmann, Borgeaud, Mensch, et al. — DeepMind (2022). Training Compute-Optimal Large Language Models. arXiv:2203.15556. The Chinchilla/Gopher numbers and the ~20:1 rule are the paper's; the curves, the isoFLOP sketch, and the worked ratios are our own illustration.

← back to the knowledge base  ·  ⚙ the illustrated decoder  ·  ▶ run the forward pass

attention.sh — a division of Jinacode Systems. Transformers, fine-tuning, and sovereign LLMs.
we're hiring the curious — hello@attention.sh