Vol. VIII · The Pretraining Run Plate 6 of 10emergent abilities, exact match, metric choice, sharp transitions, scaling, per-token probability, SchaefferLE

Emergence, Measured.

When an answer needs k steps all correct, a smooth gain per step looks like a sudden leap.

In 2022 Wei and colleagues catalogued emergent abilities: tasks on which small models score at chance and, past some scale, larger models suddenly do well — multi-digit arithmetic, some word puzzles, certain reasoning benchmarks. If abilities appear unpredictably, scaling is a gamble; if they follow smooth trends hidden by the way they are measured, it is not.

Schaeffer, Miranda and Koyejo (2023) argued that much of the suddenness is in the metric. Exact match on an answer of k tokens scores 1 only if every token is right; if each token is right with probability p, the score is p^k. As p climbs smoothly from 0.5 to 0.95, p^10 stays near zero and then shoots up. Measured instead by per-token probability or by loss, the same models improve smoothly. The workshop shows the arithmetic and then checks it on micro.

Below: left, the per-step accuracy p rising smoothly with scale, and the exact-match score p^k for an answer of k steps. Right, the same arithmetic on a real model: micro copying sentences it has just read, scored per token and by exact match over k tokens.

What you will learn
Workshop
Left: a smooth per-step accuracy p over eight orders of magnitude of scale (blue) and exact match p^k for the chosen k (red). Right: micro copying 40 English sentences it has just read: share of copied tokens that are its top prediction (grey), share of sentences whose first k copied tokens are all right (red), and p^k (blue, dashed).

Smooth underneath, sudden on top

With per-step accuracy rising smoothly — it reaches 0.75 at scale 10^4.0 on the left — exact match on a 10-step answer does not reach 0.5 until 10^5.6. With one step the two curves coincide; with 30 steps exact match stays near zero for longer still and then climbs more steeply. Nothing about the model changes between these settings, only the scoring rule. At p = 0.9, exact match is 0.35 for 10 steps and 0.042 for 30.

micro shows the same arithmetic on a real task. Copying a sentence it has just read, it gets 0.41 of the copied tokens right one at a time, but only 0.23 of sentences right across the first two tokens, 0.12 across three and 0.00 across four. Its errors are not independent — at three tokens measured exact match (0.125) sits above p^k (0.069), because some sentences are easy throughout — yet within a few tokens it collapses all the same. Read across model sizes instead of k, this is how a task scored all-or-nothing can sit at zero and then jump, while the per-token picture improves steadily. Schaeffer and colleagues showed that many reported emergent abilities become smooth under continuous metrics; some remain sharp, and the debate over how much is real continues.

Even critics of the mirage view agree on the arithmetic; the disagreement is about how much of real emergence it explains. Some tasks remain abrupt under continuous metrics, and a capability that matters only when it is fully right — a working program, a correct proof — is experienced as sudden whatever the metric.

Terms

Emergent ability
A capability absent in smaller models that appears, often sharply, in larger ones.
Exact match
A score of 1 only if the whole answer is right, else 0.
Continuous metric
A score that changes smoothly with the model, such as per-token probability or loss.
Metric artefact
An apparent jump created by the scoring rule rather than the model.
Exact match on k steps scores p^k: smooth progress per step can look like a sudden leap — so measure with a continuous metric too.

Sources

  1. Wei et al., “Emergent Abilities of Large Language Models”, TMLR 2022.
  2. Schaeffer, Miranda and Koyejo, “Are Emergent Abilities of Large Language Models a Mirage?”, NeurIPS 2023.