Rescoring the same GPT-3 outputs turns a leap into a slope

A phenomenon named after the fact

By 2022, the standard picture of language-model scaling was smooth. Train a bigger model on more compute and its loss falls along a predictable curve, and downstream task performance mostly follows. Wei et al. collected a specific counterexample to that picture: tasks where accuracy sits at or near chance across several orders of magnitude of training compute, then rises sharply once a model crosses a particular scale, in a shape no small-model trend would have predicted.¹ They named this pattern an emergent ability, borrowed a definition of emergence from physicist Philip Anderson's 1972 essay "More Is Different," and set out to catalog how often it happens and on what kind of task.¹ A year and a half later, Schaeffer, Miranda, and Koyejo argued that much of the catalog was an artifact of how the tasks were scored, not of anything happening inside the models.² Their paper went on to receive a NeurIPS 2023 Outstanding Paper award, one of the conference's top distinctions, a sign that the field treated the challenge as substantial rather than as a footnote.³⁴ The two papers do not disagree about what the number on the page does. They disagree about what to conclude from it.

What Wei et al. called emergent

Wei et al. define the term twice, in near-identical words: "an ability is emergent if it is not present in smaller models but is present in larger models."¹ Scale, in their plots, is training compute in floating-point operations (FLOPs), with parameter count given as a secondary view because compute and parameters had scaled roughly together across the model families they studied.¹ The definition is narrower than it first sounds. It says nothing about which scale triggers a given ability. The paper is explicit that it isn't attempting to predict one: "our goal in this paper is not to characterize or claim that a specific scale is required to observe emergent abilities." It aims only to document examples of the pattern already reported in prior work.¹ The claim is an existence claim about the shape of a curve, not a theory of why the curve bends where it does.

The catalog behind that claim draws mostly on BIG-Bench, a crowdsourced collection of 204 tasks spanning linguistics, math, common-sense reasoning, and more, assembled specifically to probe language models beyond the tasks any one lab would think to write.⁵ Wei et al. found eight examples of the emergent pattern in few-shot prompting across five model families, and four more once fine-tuning or an augmented prompting technique enters the picture. Compute thresholds for a sample of them:

GSM8K, the benchmark behind that last row, is a set of 8,500 grade-school math word problems. Its authors reported that even the largest transformers of 2021 failed most of them despite the problems' conceptual simplicity, exactly the kind of task where a threshold effect matters to anyone deciding whether a model is worth deploying.⁶ Read across the table, the pattern Wei et al. describe is real in the sense that the reported numbers do this: near-chance for a long stretch of compute, then sharply above it. The open question is what produces the shape.

The objection already in the paper

Wei et al. do not miss the obvious challenge to a threshold claim built on accuracy scores. Section 5.1 raises it directly: "using exact string match as the evaluation metric for long-sequence targets may disguise compounding incremental improvements as emergence."¹ The intuition: a model generating a long answer can get more of its tokens right as it scales. Exact-match scoring gives no credit for a nearly-correct answer, so a smooth rise in per-token accuracy can look like nothing happens and then everything happens.

Having raised it, Wei et al. argue the objection covers only part of the catalog. It says nothing, they argue, about the tasks scored by classification rather than exact match: multiple-choice questions like TruthfulQA and MMLU, or the two-way judgment in Word in Context, where there's no long sequence to get partially right. Emergent-looking jumps show up on those tasks too, so the metric explanation is, in their own words, "at best an incomplete explanation."¹ That distinction, between long-sequence tasks scored by exact match and short-answer tasks scored by classification, is the exact seam Schaeffer et al. cut into. Their central claim is that classification metrics are not the exception Wei et al. took them to be.

One task, two rulers

Schaeffer et al. open with a formal distinction between two kinds of scoring rule. A nonlinear or discontinuous metric assigns credit in a way that doesn't track the model's underlying per-token error rate smoothly. Multiple Choice Grade scores 1 if the correct option carries the highest probability and 0 otherwise. Exact String Match scores 1 only for a perfect match.² Both are step functions: an answer one token off from correct scores the same zero as an answer with no tokens right. A linear or continuous metric, by contrast, tracks the error rate directly. Token Edit Distance counts how many tokens are wrong, scaling roughly linearly with the per-token error rate. Brier Score, used for classification tasks, is the mean squared error between the model's predicted probability and the correct outcome.² Their argument, stated plainly, is that emergent abilities appear "due [to] the researcher's choice of metric rather than due to fundamental changes in model behavior with scale."²

To test it, they ran GPT-3 and InstructGPT on the exact kind of task Wei et al. had flagged as emergent: 2-shot multiplication of two 2-digit integers, and 2-shot addition of two 4-digit integers.² Scored under Accuracy, "the GPT family displays emergent abilities if the target has 4 or 5 digits": small models sit near zero, larger ones jump.² Then they rescored the identical model outputs, unchanged, using Token Edit Distance instead of Accuracy. Under that metric, "the family's performance smoothly, continuously and predictably improves with increasing scale."² No model was retrained. No output changed. The only thing that moved was the rule for turning a string of digits into a number between zero and one, and that alone converted a cliff into a slope.

A second test in the same paper attacks a different weak point: the small sample sizes typical of a benchmark like BIG-Bench. Collecting more test examples to raise the resolution of the Accuracy measurement itself, Schaeffer et al. found that "all models in the InstructGPT/GPT-3 family achieve above-chance accuracy," even under the original discontinuous metric.² The small models were never truly at zero. They were sampled too thinly at the original test-set size to register their real, nonzero, above-chance performance. Two independent failures, metric choice and sample size, were both quietly doing the work that "emergence" got credit for.

Neither of these demonstrations exists as a table of per-model numbers in the published paper; both are given as figures, and the exact percentage each model scored is not stated in the running text. What is stated in words is the qualitative result: identical outputs, two metrics, two shapes.

The tasks Wei et al. called metric-proof

The arithmetic demonstration only settles the long-sequence half of Wei et al.'s objection, the half about exact string match on tasks with a multi-token target. It leaves their other claim standing: that classification tasks, scored by a rule with no partial credit to withhold in the first place, show the same jumps for a different reason. Schaeffer et al.'s second experiment goes after that claim specifically.

They ran a meta-analysis of BIG-Bench's own 39 preferred metrics. By an automated scoring-discontinuity measure, at most 5 of the 39 show anything that looks like emergence. Hand-annotating the results narrows that further, to 4 of 39.² Of those four, two metrics account for more than 92 percent of all claimed emergent cases in BIG-Bench: Multiple Choice Grade and Exact String Match.² Multiple Choice Grade is a classification metric, exactly the kind Wei et al. treated as beyond the metric objection's reach. Its appearance at the top of Schaeffer et al.'s list is a direct hit on Wei et al.'s stated reason for calling the objection incomplete, not a restatement of the exact-match half they had already conceded ground on.

Schaeffer et al. show a second concrete case of the same swap, independent of the arithmetic task: on LaMDA, an emergent-looking jump visible under Multiple Choice Grade on certain BIG-Bench tasks "disappeared when we changed the metric to the continuous Brier Score."² A separate model family, a separate metric pair, and the same result: the discontinuity travels with the scoring rule, not with the model.

What a smoother curve doesn't buy a deployment

Schaeffer et al. are careful about how far this goes. Their own Discussion draws the line explicitly: "nothing in this paper should be interpreted as claiming that large language models cannot display emergent abilities; rather, our message is that previously claimed emergent abilities... might likely be a mirage induced by researcher analyses."² They also offer a statistical reason some jumps would appear even if metric choice explained nothing: BIG-Bench alone contains more than 220 tasks scored on roughly 40 metrics across roughly 10 model families, on the order of a million task-metric-family combinations, enough that some fraction will look like a jump by chance alone.² The mirage paper's own claim is narrower than "emergence isn't real." It's "much of the specific evidence for it doesn't hold up the way it was read."

Jason Wei responded to the argument directly, and did not concede the practical point. He grants what produces it: "if you plot... log-probability of the target sequence, performance improves smoothly."⁷ But he argues the discontinuous metric is the one a real use case actually needs: "you want the answer to be 38, and nothing else. Maybe 37 is closer to 38 than -2.591, but assigning some partial credit to that answer seems unhelpful."⁷ A calculator that is usually close is not a calculator. If the product a model ships in only pays out for an exact right answer, the smooth curve underneath is real but beside the point. A team shipping that product still faces a genuinely unpredictable question: at what scale does the metric it's stuck with cross the line into usefulness.

Boaz Barak, writing as an outside theorist rather than an author of either paper, accepts the metric-swap effect and then names a second gap Schaeffer et al.'s model doesn't cover. Their mathematical treatment scores a single per-token error rate. Many of the tasks in question, like the scratchpad-assisted 8-digit addition or the chain-of-thought math problems in Wei et al.'s Figure 3, require several correct sub-steps in sequence. Barak's point: "even if we can precisely predict the loss of a model trained using N flops, we may not be able to predict which tasks it would solve beyond those solvable by an N/10 flop model." The reason: the probability that every sub-step in a chain succeeds can still rise sharply even when each sub-step's own probability rises smoothly.⁸ Schaeffer et al.'s own model does not address that compounding case, and neither Wei et al.'s scratchpad example nor their chain-of-thought example is covered by either side's math. It is a live gap, not a resolved one.

The debate did not end with either paper. A year after the mirage paper, Snell, Wallace, Klein, and Levine treated emergence as exactly the kind of thing worth forecasting rather than dissolving. Fitting curves to finetuning results on MMLU, GSM8K, CommonsenseQA, and CoLA, they report being able to predict when a benchmark score will cross a threshold "up to 4x the FLOPs in advance."⁹ That is a research program built on the premise that discontinuous thresholds are a real target worth predicting, not a mirage to explain away. As of mid-2025, the ontological question hadn't settled by consensus either. A philosophy-of-science analysis argued that "current debates over metrics, pre-training loss thresholds, and in-context learning miss the fundamental ontological nature of emergence in DNNs." It proposed reading large language models instead as complex dynamical systems, with properties analogous to emergence in other natural systems.¹⁰ The metric argument reframed the conversation. It did not close it.