Are Emergent Abilities of Large Language Models a Mirage? — part 2/3

Cleaned-text code points [15285, 31001). Bibliography omitted here; source.md includes it.

If emergent abilities are real, one should expect task-model family pairs to show emergence for all reasonable metrics. However, if our alternative explanation is correct, we should expect emergent abilities to appear only under certain metrics. To test this, we analyzed on which metrics emergent abilities appear. To determine whether a task-metric-model family triplet exhibits a possible emergent ability, we used a metric from previous work [ 28 ] . Letting y i ∈ ℝ y_{i}\in\mathbb{R} denote model performance at model scales x i ∈ ℝ x_{i}\in\mathbb{R} , sorted such that x i < x i + 1 x_{i}<x_{i+1} , the emergence score is: Emergence Score ​ ( { ( x n , y n ) } n = 1 N ) = def sign ​ ( arg ⁡ max i ​ y i − arg ⁡ min i ​ y i ) ​ ( max i ⁡ y i − min i ⁡ y i ) Median ​ ( { ( y i − y i − 1 ) 2 } i ) \text{Emergence Score}\Big(\Big\{(x_{n},y_{n})\Big\}_{n=1}^{N}\Big)\quad\defeq\quad\frac{\text{sign}(\arg\max_{i}y_{i}-\arg\min_{i}y_{i})(\max_{i}y_{i}-\min_{i}y_{i})}{\sqrt{\text{Median}(\{(y_{i}-y_{i-1})^{2}\}_{i})}} (1)

We found that most metrics used in BIG-Bench have zero task-model family pairs that exhibit emergent abilities: of the 39 preferred metrics in BIG-Bench, at most 5 display emergence (Fig. 5 A). Many of the 5 are nonlinear and/or discontinuous, e.g., Exact String Match, Multiple Choice Grade, ROUGE-L-Sum (App. A.4 ). Notably, because BIG-Bench often scores models on tasks using multiple metrics, the lack of emergent abilities under other metrics suggests that emergent abilities do not appear when model outputs are scored using other metrics.

Because emergence score only suggests emergence, we also analyzed hand-annotated task-metric-model family triplets [ 32 ] , which revealed emergent abilities appear with 4 / 39 4/39 metrics (Fig. 5 B), and 2 metrics account for > 92 % >92\% of claimed emergent abilities (Fig. 5 C): Multiple Choice Grade and Exact String Match. Multiple Choice Grade is discontinuous, and Exact String Match is nonlinear.

#### Prediction: Changing Metric Removes Emergent Abilities

To test our second prediction, we focused on the LaMDA family [ 30 ] because its outputs are available through BIG-Bench. For our analysis, we identified tasks on which LaMDA displays emergent abilities with Multiple Choice Grade, then asked whether LaMDA still displays emergent abilities on the same tasks with a different BIG-Bench metric: Brier Score [ 2 ] . Brier Score is a strictly proper scoring rule for predictions of mutually exclusive outcomes; for a binary outcome, the Brier Score simplifies to the mean squared error between the outcome and its predicted probability mass. LaMDA’s emergent abilities on the discontinuous Multiple Choice Grade disappeared when we changed the metric to the continuous Brier Score (Fig. 6 ). These results support our alternative explanation that emergent abilities are induced by the chosen metric.

## 5 Inducing Emergent Abilities in Networks on Vision Tasks

To demonstrate how emergent abilities can be induced by the researcher’s choice of metric, we show how to produce emergent abilities in deep networks of various architectures: fully connected, convolutional, self-attentional. We focus on vision tasks because abrupt transitions in vision models’ capabilities have not been observed to the best of our knowledge; this is one reason why emergence in large language models is considered so interesting. For the convolutional example, see App. B .

#### Emergent Reconstruction of CIFAR100 Natural Images by Nonlinear Autoencoders

We first induce an emergent ability to reconstruct images in shallow (i.e., single hidden layer) nonlinear autoencoders trained on CIFAR100 natural images [ 19 ] . To emphasize that the sharpness of the metric is responsible for emergent abilities, and to show that sharpness extends to metrics beyond Accuracy, we intentionally define a discontinuous metric that measures a network’s ability to reconstruct a dataset as the average number of test data with squared reconstruction error below threshold c c : Reconstruction c ( { x n } n = 1 N ) = def 1 N ∑ n 𝕀 [ | | x n − x ^ n | | 2 < c ] \text{Reconstruction}_{c}\Big(\{x_{n}\}_{n=1}^{N}\Big)\;\defeq\;\frac{1}{N}\sum_{n}\mathbb{I}\Big[||x_{n}-\hat{x}_{n}||^{2}<c\Big] (2) where 𝕀 ⁡ ( ⋅ ) \mathbb{I}(\cdot) denotes an indicator variable and x ^ n \hat{x}_{n} is the autoencoder’s reconstruction of x n x_{n} . The autoencoder family displays smoothly decreasing squared reconstruction error as the number of bottleneck units increases (Fig. 7 B). Under our newly defined Reconstruction c metric and for particular choices of c c , the autoencoder family exhibits a sharp and seemingly unpredictable image reconstruction ability (Fig. 7 C) that qualitatively matches published emergent abilities (Fig. 7 A).

#### Emergent Classification of Omniglot Characters by Autoregressive Transformers

We next induce emergent abilities in Transformers [ 31 ] trained to autoregressively classify Omniglot handwritten characters [ 20 ] , in a setup inspired by recent work [ 5 ] : Omniglot images are embedded by convolutional layers, then sequences of embedded image-image class label pairs are fed into decoder-only transformers. We measure image classification performance on sequences of length L ∈ [ 1 , 5 ] L\in[1,5] , again via subset accuracy : 1 1 if all L L images are classified correctly (Fig. 8 B), 0 otherwise. Causal transformers display a seemingly emergent ability to correctly classify Omniglot handwritten characters (Fig. 8 C) that qualitatively matches published emergent abilities (Fig. 8 A).

## 6 Related Work

Srivastava et al. [ 28 ] observed that while accuracy at a particular task can empirically appear sharp and unpredictable, cross entropy does not; the authors then hypothesized that emergent abilities may be partially attributed to the metric. Our paper converts their discussion into precise predictions, then quantitatively tests the predictions to reveal that: metric choice is likely wholly responsible for emergent abilities; well-known and widely-used metrics (including ones already used by [ 28 ] ) capture graded improvements; emergent abilities do not appear only for tasks involving multiple steps, and indeed appear most commonly on the discontinuous Multiple Choice Grade; metric choice can be used to induce emergent abilities in a novel domain (vision) in diverse architectures and tasks.

Caballero et al. [ 4 ] explain emergence by assuming a piece-wise power law functional form; under this view, emergent abilities are real, caused by a change in the governing power law. In contrast, our work suggests that emergent abilities are induced by the researcher, even under a single power law. Michaud et al. [ 25 ] posit that emergent abilities may be real under strong data assumptions.

## 7 Discussion

Our paper presents an alternative explanation for claimed emergent abilities of large language models. For a fixed task and a fixed model family, the researcher can choose a metric to create an emergent ability or choose a metric to ablate an emergent ability. Ergo, emergent abilities may be creations of the researcher’s choices, not a fundamental property of the model family on the specific task. We emphasize that 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 in [ 3 , 8 , 28 , 33 ] might likely be a mirage induced by researcher analyses.

Our paper has several implications. Firstly, a task and a metric are distinct and meaningful choices when constructing a benchmark. Secondly, when choosing metric(s), one should consider the metric’s effect on the per-token error rate and adapt their measuring process accordingly, e.g., if one chooses accuracy, one should make sure to have sufficient data to accurately measure accuracy to avoid the risk of drawing invalid scientific conclusions. Thirdly, when making claims about capabilities of large models, including proper controls is critical. In this particular setting, emergent abilities claims are possibly infected by a failure to control for multiple comparisons. In BIG-Bench alone, there are ≥ \geq 220 tasks, ∼ 40 \sim 40 metrics per task, ∼ 10 \sim 10 model families, for a total of ∼ 10 6 \sim 10^{6} task-metric-model family triplets, meaning probability that no task-metric-model family triplet exhibits an emergent ability by random chance might be small. Fourthly, scientific progress can be hampered when models and their outputs are not made public for independent scientific investigation.

## Appendix A Approximate Behavior of Metrics on Sequential Data

How do different metrics behave when used to measure autoregressive model outputs? Precisely answering this question is tricky and possibly analytically unsolvable, so we provide an approximate answer here.

Notationally, we consider N N test data of length L L (here, length is measured in tokens) with targets denoted t n ​ = def ( t n1 , t n2 , … ​ t nL ) t_{n}\defeq(t_{n1},t_{n2},...t_{nL}) , the autoregressive model has a true-but-unknown per-token error probability of ϵ ∈ [ 0 , 1 ] \epsilon\in[0,1] and the model outputs prediction t ^ n ​ = def ( t ^ n1 , t ^ n2 , … ​ t ^ nL ) \hat{t}_{n}\defeq(\hat{t}_{n1},\hat{t}_{n2},...\hat{t}_{nL}) . This assumes that the model’s per-token error probability is constant, which is empirically false, but modeling the complex dependencies of errors is beyond our scope.

### A.1 Per-Token Error Probability is Resolution-Limited

Note that because we have N N test data, each of length L L , our resolution for viewing the per-token error probability ϵ \epsilon is limited by 1 / N ​ L 1/NL . Here, resolution refers to “the smallest interval measurable by a scientific instrument; the resolving power.” To explain what resolution means via an example, suppose one wants to measure a coin’s probability of yielding heads. After a single coin flip, only two outcomes are possible (H, T), so the resolution-limited probability of heads is either 0 0 or 1 1 . After two coin flips, four outcomes are possible (HH, HT, TH, TT), so the resolution-limited probability of heads is now one of 0 , 0.5 , 1 0,0.5,1 . After F F coin flips, we can only resolve the coin’s probability of yielding heads up to 1 / F 1/F . Consequently, we introduce a resolution-limited notation: ⌊ a ⌉ b ​ = def a rounded to the nearest integer multiple of 1 / b \lfloor a\rceil_{b}\defeq\text{$a$ rounded to the nearest integer multiple of $1/b$} (3)

### A.2 Token Edit Distance

We first consider an adaptation of the Levenshtein (string edit) distance for models that function on tokens rather than characters, an adaptation we term the token edit distance . The token edit distance between two token sequences t n , t n ^ t_{n},\hat{t_{n}} is defined as the integer number of additions, deletions or substitutions necessary to transform t n t_{n} into t ^ n \hat{t}_{n} (or vice versa).

Token Edit Distance ​ ( t n , t ^ n ) \displaystyle\text{Token Edit Distance}(t_{n},\hat{t}_{n}) = def Num Substitutions + Num. Additions + Num. Deletions \displaystyle\defeq\text{Num Substitutions}+\text{Num. Additions}+\text{Num. Deletions} (4) = ∑ ℓ = 1 L 𝕀 [ t n ​ ℓ ≠ t ^ n ​ ℓ ] + Num. Additions + Num. Deletions \displaystyle=\sum_{\ell=1}^{L}\mathbb{I}[t_{n\ell}\neq\hat{t}_{n\ell}]+\text{Num. Additions}+\text{Num. Deletions} (5) ≥ ∑ ℓ = 1 L 𝕀 [ t n ​ ℓ ≠ t ^ n ​ ℓ ] \displaystyle\geq\sum_{\ell=1}^{L}\mathbb{I}[t_{n\ell}\neq\hat{t}_{n\ell}] (6)

The expected token edit distance is therefore:

𝔼 ⁡ [ Token Edit Distance ​ ( t n , t ^ n ) ] \displaystyle\mathbb{E}[\text{Token Edit Distance}(t_{n},\hat{t}_{n})] ≥ 𝔼 [ ∑ ℓ = 1 L 𝕀 [ t n ​ ℓ ≠ t ^ n ​ ℓ ] ] \displaystyle\geq\mathbb{E}[\sum_{\ell=1}^{L}\mathbb{I}[t_{n\ell}\neq\hat{t}_{n\ell}]] (7) = ∑ ℓ = 1 L p ⁡ ( t n ​ ℓ ≠ t ^ n ​ ℓ ) \displaystyle=\sum_{\ell=1}^{L}p(t_{n\ell}\neq\hat{t}_{n\ell}) (8) ≈ L ⁡ ( 1 − ϵ ) \displaystyle\approx L(1-\epsilon) (9)

The resolution-limited expected token edit distance is therefore:

⌊ 𝔼 ⁡ [ Token Edit Distance ​ ( t n , t ^ n ) ] ⌉ N ​ L ≥ L ⁡ ( 1 − ⌊ ϵ ⌉ N ​ L ) \lfloor\mathbb{E}[\text{Token Edit Distance}(t_{n},\hat{t}_{n})]\rceil_{NL}\geq L\Big(1-\lfloor\epsilon\rceil_{NL}\Big) (10)

From this, we see that the expected token edit distance scales approximately linearly with the resolution-limited per-token probability. The real rate is slightly higher than linear because additions and deletions contribute an additional non-negative cost, but modeling this requires a model of how likely the model is to overproduce or underproduce tokens, which is something we do not currently possess.

### A.3 Accuracy

Accuracy ​ ( t n , t ^ n ) \displaystyle\text{Accuracy}(t_{n},\hat{t}_{n}) = def 𝕀 [ No additions ] 𝕀 [ No deletions ] ∏ l = 1 L 𝕀 [ t nl = t ^ nl ] \displaystyle\defeq\mathbb{I}[\text{No additions}]\,\mathbb{I}[\text{No deletions}]\,\prod_{l=1}^{L}\mathbb{I}[t_{nl}=\hat{t}_{nl}] (11) ≈ ∏ l = 1 L 𝕀 [ t n ​ l = t ^ n ​ l ] \displaystyle\approx\prod_{l=1}^{L}\mathbb{I}[t_{nl}=\hat{t}_{nl}] (12)

As with the Token Edit Distance (App. A.3 ), we ignore how likely the language model is to overproduce or underproduce tokens because we do not have a good model of this process. Continuing along,

𝔼 ⁡ [ log ⁡ Accuracy ] \displaystyle\mathbb{E}[\log\text{Accuracy}] = ∑ l 𝔼 [ log 𝕀 [ t n ​ l = t ^ n ​ l ] ] \displaystyle=\sum_{l}\mathbb{E}[\log\mathbb{I}[t_{nl}=\hat{t}_{nl}]] (13) ≤ ∑ l log 𝔼 [ 𝕀 [ t n ​ l = t ^ n ​ l ] ] \displaystyle\leq\sum_{l}\log\mathbb{E}[\mathbb{I}[t_{nl}=\hat{t}_{nl}]] (14) ≈ L ​ log ⁡ ( 1 − ϵ ) \displaystyle\approx L\log(1-\epsilon) (15)

Taking an approximation that would make most mathematicians cry:

𝔼 ⁡ [ Accuracy ] \displaystyle\mathbb{E}[\text{Accuracy}] ≈ exp ⁡ ( 𝔼 ⁡ [ log ⁡ Accuracy ] ) \displaystyle\approx\exp(\mathbb{E}[\log\text{Accuracy}]) (16) = ( 1 − ϵ ) L \displaystyle=(1-\epsilon)^{L} (17)

This reveals that accuracy approximately falls geometrically with target token length. The resolution-limited expected accuracy is therefore:

⌊ 𝔼 ⁡ [ Accuracy ] ⌉ N ​ L = ⌊ ( 1 − ϵ ) L ⌉ N ​ L \lfloor\mathbb{E}[\text{Accuracy}]\rceil_{NL}=\lfloor(1-\epsilon)^{L}\rceil_{NL} (19)

From this we can see that choosing a nonlinear metric like Accuracy is affected significantly more by limited resolution because Accuracy forces one to distinguish quantities that decay rapidly.

### A.4 ROUGE-L-Sum

Another BIG-Bench metric [ 28 ] is ROUGE-L-Sum [ 23 ] , a metric based on the longest common subsequence (LCS) between two sequences. Section 3.2 of [ 23 ] gives the exact definition, but the key property is that ROUGE-L-Sum measures the “union” LCS, which means “stitching” together LCSs across the candidate and multiple references. As explained in the original paper: if the candidate sequence is c = w 1 ​ w 2 ​ w 3 ​ w 4 ​ w 5 c=w_{1}w_{2}w_{3}w_{4}w_{5} , and if there are two reference sequences r 1 = w 1 ​ w 2 ​ w 6 ​ w 7 ​ w 8 r_{1}=w_{1}w_{2}w_{6}w_{7}w_{8} and r 2 = w 1 ​ w 3 ​ w 8 ​ w 9 ​ w 5 r_{2}=w_{1}w_{3}w_{8}w_{9}w_{5} , then L ​ C ​ S ​ ( r 1 , c ) = w 1 ​ w 2 LCS(r_{1},c)=w_{1}w_{2} and L ​ C ​ S ​ ( r 2 , c ) = w 1 ​ w 3 ​ w 5 LCS(r_{2},c)=w_{1}w_{3}w_{5} , then the union -LCS of c , r 1 , r 2 c,r_{1},r_{2} is w 1 ​ w 2 ​ w 3 ​ w 5 w_{1}w_{2}w_{3}w_{5} , with length 4. Intuitively, this disproportionately benefits models with smaller error rates because their mistakes can be “stitched” across multiple references; this is confirmed in simulation (Fig. 9 ).

## Appendix B Inducing Emergent Abilities in Networks on Vision Tasks

### B.1 Emergent Classification of MNIST Handwritten Digits by Convolutional Networks