U-shaped and Inverted-U Scaling behind Emergent Abilities of Large Language Models

Authors: Tung-Yu Wu, Melody Lo
Primary area: foundation or frontier models, including LLMs
Keywords: large language models, emergent abilities, scaling laws

Abstract:
Large language models (LLMs) have been shown to exhibit *emergent abilities* in some downstream tasks, where model performance stagnates at first and then improves sharply and unpredictably with scale beyond a threshold. In this work, we investigate the phenomenon by grouping questions based on difficulty level and provide a possible explanation for emergent abilities. Specifically, we observe U-shaped scaling for hard questions and inverted-U scaling followed by steady improvement for easy questions. The two scaling patterns initially offset each other, causing stagnant overall performance. The performance starts to soar when the scaling pattern of easy questions reverts from inverse to standard scaling, leading to emergent abilities. Based on this finding, we propose a simple yet effective pipeline, called *Slice-and-Sandwich*, to predict the emergence threshold and model performance beyond the threshold. Our code is publicly available at https://github.com/tony10101105/ExpEmergence.

Decision: accept (poster)
Reviewer scores: 8, 6, 6, 6

Reviews and discussion:
<thread>
creation date:
1729352630754

thread_type:
official_review

<thread_object>
creator: reviewer_k8wa

<content>
summary:
By dividing questions according to difficulty levels, the authors observe U-shaped scaling for hard questions and inverted-U scaling for easy questions, followed by steady improvement. Then the authors propose a simple pipeline, called Slice-and-Sandwich, to predict the model performance.

strengths:
1. The inverted-U vs. U-shape scaling seems interesting.

2. The pipeline seems to be novel. Although the basic principle of the method is fairly easy to understand, it substantially outperforms the baseline methods.

3. The experiments are diverse. The authors uses 56 LLMs to evaluate the difficulty levels, and the effectiveness of Slice-and-Sandwich is demonstrated on 6 benchmarks.

weaknesses:
1. The fitted scaling trend is either underestimated or overestimated the actual scaling trend. Although the authors have given explanations, it should be clearly pointed out that why this  underestimation /  overestimation is acceptable. For example, the prediction error of the score of hard problems on Arithmetic benchmark (figure 7) seems actually comparable to the prediction error of the score of all problems on Arithmetic benchmark  (figure 9). 

2. The authors provide possible explanations for the inverted-U vs. U-shape scaling. They provide examples of easy/hard problems and give very brief descriptions of the performance of different models. The concepts would be easier to understand if the authors can provide some specific outputs of weak/strong LLMs on easy/hard problems in the appendix. 

3. There are some typos in the paper. For example, in all figures, the authors use exactly the same color for the easiest and most difficult problems, making it a bit confusing. In line 484, $F_s^c$ should be $F_e^c$. Also, the layout of the appendix seems weird. For instance, P21 only contains a single huge image.

4. Perhaps the authors should use one sentence in the introduction section (line 53) to explain what is `binary brier score` to make it easier for readers to understand.

questions:
1. I am interested in understanding why averaging the scaling trends of both the easy group and the hard group leads to an accurate prediction of the overall aggregate scaling trend. It's possible that with one prediction being overestimated and the other underestimated, the outcome appears to be coincidentally accurate. The author should provide further (possibly qualitative) explanation to demonstrate that the good result is not a matter of chance. If this question is resolved, I believe the soundness and contribution of the paper can be substantially improved.

2. The authors provide a separate figure for each benchmark. Would it be possible to also provide a figure that includes the results from all benchmarks?
</content>
</thread_object>

</thread>
<thread>
creation date:
1730010019189

thread_type:
official_review

<thread_object>
creator: reviewer_9qww

<content>
summary:
This paper investigates the emergent behavior of large language models as their effective size increases. The authors observe that for certain downstream tasks, the continuous metric exhibits a U-shape for easier questions and an inverted U-shape for more difficult questions below the emergence threshold, with steady improvements beyond this threshold. Additionally, they propose a method for forecasting model performance beyond the emergence threshold.

strengths:
1. The paper connects the scaling trend on easy questions with the double descent phenomenon in deep learning is quite interesting.
2. A group-wise scaling law based on difficulty level can offer more detailed insights compared to a single overall scaling curve.

weaknesses:
1. This paper primarily focuses on the continuous Brier score as a metric, which may not fully capture the validity of the observed scaling trends. It is unclear whether these trends are specific to this single metric. To establish this phenomenon as more general, validation across other metrics is needed.
2. The writing and organization of this paper are difficult to follow. For instance, the scaling plots for various LLMs are presented without clearly specifying the types or model families used, as these details are only provided in the appendix. Additionally, the figure depicting the scaling curves lacks clarity; for example, the label '0_1502_brier' in Figure 3 is ambiguous. It appears to represent the index of data points, but using the difficulty level as a label would be more informative.
3. The proposed task in Section 4 is using data collected before the emergence threshold to forecast the occurrence of emergent abilities and the scaling trend beyond this threshold. However, the paper lacks details on how the threshold is selected and how this choice is validated. Furthermore, more explanation is needed to highlight the significance of this task. Why is predicting the scaling curve beyond the emergence threshold based on early performance trends important?
4. In the experiments, a few multiple-choice datasets are utilized to demonstrate the presence of emergent behavior, but they also note that such behavior is absent in other BIG-Bench datasets, as detailed in the appendix. This raises questions about whether the observed emergence is specific to certain datasets rather than a general phenomenon. A more comprehensive analysis is needed to uncover broader patterns and clarify the characteristics of these emergent behaviors—for instance, identifying which types of tasks exhibit this emergence and which do not. This would help determine the conditions under which such emergence is applicable.

questions:
1. The authors appear to categorize difficulty levels as easy, medium, and hard. How might the granularity of these difficulty levels affect the results?
2. In Section 4, the proposed method Slice-and-Sandwich first fits a function to the binary Brier score and then projects the forecasted scaling trend back to accuracy. I am trying to understand the benefit of this additional step. My interpretation is that fitting the binary Brier score curve is more precise, particularly for extrapolation, compared to directly fitting the accuracy curve. Could the authors elaborate on why this approach is preferred over fitting the accuracy curve directly?
</content>
</thread_object>

</thread>
<thread>
creation date:
1730556138738

thread_type:
official_review

<thread_object>
creator: reviewer_njbx

<content>
summary:
The paper investigates explanations for LLM emergent abilities (sharp transitions from chance-level to high performance as a function of model size) and proposes a method to predict emergent performance. They show that on certain task, when questions are group by difficulty, two distinct scaling trends appear: U-shaped scaling for hard questions and inverted U-shaped followed by improvements for easy questions. This motivated a method for capabilities forecasting: fitting polynomial regression separately for easy and hard questions and then aggregating the results.

strengths:
1. The paper addresses a very timely topic: the science of LLM evals and predictive evals. It is relevant for scientific understanding of frontier LLMs as well as for AI policy (informing capability thresholds and buffers in responsive scaling policies).
2. The paper described an interesting insight that easy and hard questions have different scaling trends and that this could explain sharp transitions of aggregate performance

weaknesses:
1. I found the paper hard to follow. I think the clarity of writing and figures could be substantially improved. Some concrete suggestions:
(a) the name "binary Brier score" is very confusing. What's binary about it? It's continuous for each question.
(b) I think it's very confusing to say Brier score is "noisy due to insufficient confidence calibrations" (line 161). What does it mean to be sufficient? Why is this noise? Why "calibrations" in plural?
(c) what are "available choices' (line 159)? Is it the same as classes (line 139)?
(d) names for difficulty levels (e.g. 0_1502_brier) are confusing. They're okay in Python code, but I'd expect something more readable and abstracting away from distracting details (e.g. I don't really care it's 1502 questions).
(e) The colors on plot could match the semantics of difficulties, e.g. you could use a continuous color map (e.g. red-blue transition to represent easy-hard transition).
(f) plots for different datasets could be next to each other
(g) "Slide-and-sandwich approach allows the data to speak for itself" is misleading. The whole point is that you bake in more assumptions in the method.
2. The name "effective compute" is somewhat misleading if it's just log compute. Ideally, effective compute could be something like g-factor [1] or the first principal component from observational calling laws [2, 3]. It would be great to use that as a metric. Or at the very least, it should be just called "log compute".
3. Authors only evaluate their methods on 3 datasets (plus 3 more in the appendix). I don't think that's enough to justify the claim that they found a general method for capabilities forecasting and a general explanations of emergent features. BIG-Bench includes more than 200 tasks.
4. Relatedly, all of them do show emergent capabilities. I wonder how Slice-and-sandwich methods behaves on tasks that do not show emergent capabilities: does it falsely predict them? I think this is an important point: so far the authors only presented a method for emergent capabilities forecasting conditional on the presence of emergent capabilities. But in practice we don't know that in advance.
5. Overall, I'm somewhat underimpressed by predictive performance and I worry that the authors overfit to a particular set of 6 tasks. If find some patterns in a dataset and bake in this pattern as an inductive biases of your method (polynomial regression), surely you'll improve predictive performance. But will those inductive biases generalize to entirely new tasks you never looked at? The authors provide no evidence for that (unless I'm missing something, happy to be proved wrong).
6. I'm not super convinced this is a general explanation of emergent capabilities. Maybe others have different scaling trends? The whole point that a superposition of U-shaped and inverted-U shaped scaling explains emergent capabilities is also somewhat qualitative. It's not starkly clear from the plots, e.g. MMLU does not seem consistent with this explanation.
7. Consider discussing [4] as related work

[1] Unveiling the general intelligence factor in language models: A psychometric approach

[2] Observational Scaling Laws and the Predictability of Language Model Performance

[3] Safetywashing: Do AI Safety Benchmarks Actually Measure Safety Progress?

[4] The Quantization Model of Neural Scaling

questions:
1. How were the 6 tasks you experiment on selected? What were the criteria and at what point were they selected?
2. What's the functional form of the polynomials you fit (line 416)?
</content>
</thread_object>

</thread>
<thread>
creation date:
1730648629738

thread_type:
official_review

<thread_object>
creator: reviewer_bboc

<content>
summary:
The paper connects two well-known phenomena in deep learning, namely emergent capabilities and inverse scaling. By breaking down the scaling curve to different difficulty levels, the authors show that LLM skill emergence can be the result of an inverted-U-shaped scaling on easy questions, where the easy-questions curve cancels out the scaling on hard questions. The authors propose a method to predict emergent scaling by fitting sub-groups of the dataset separately.

strengths:
- The main result of this paper connects two important topics in the field, inverse scaling and emergence, making it a very relevant research topic, especially since neither phenomenon is well understood.
The results on MMLU and Persian-QA are clear enough, in my opinion, to establish the existence of a connection between U-shaped scaling and emergence. Although Big-bench arithmetic and the appendix results are less clear, they do not diminish from the other results. Even if the easy-hard data split failed to visualize inverse scaling in the latter experiments, a different split might still show similar curves to MMLU.

- The proposed fitting method, slice-and-sandwich, showcases how the documented phenomenon can be used to predict emergence in practice.

- The paper is clearly written and easy to follow.

weaknesses:
- While the fitting method is an overall positive addition to the paper, it is less convincing than the scaling results (which are, for me, the main result of the paper). The fits shown in the main section use knowledge of the point where emergence starts, making them of little value, since predicting that point is the main reason why we care about fitting emergence.
Appendix F2 fixes that issue by showing fits on random sections of the scaling curve before emergence, but these fits are not so convincing as a reliable method to predict the point of emergence.

- Appendix E shows the scaling curves when plotting accuracy, and attributes the difference from the main results to accuracy being a 'crude measure', which is not a satisfactory explanation to me.

questions:
- Minor correction: In the definition of $M$ (line 128), $N$ and $D$ are mixed up.

- Naming $M$ 'effective model size' is confusing for the reader, since model size is $N$ while $M$ is actually the log of compute $C$. Maybe a better name would be 'effective compute' or just 'log compute?
</content>
</thread_object>

</thread>
<thread>
creation date:
1732042465870

thread_type:
official_comment

<thread_object>
creator: authors

<content>
title:
Official Response to Shared Concerns (1)

comment:
This response is for shared concerns about paper writing and experiments.

1. Paper writing.

    **a. Ambiguous name of effective model size**  
        As suggested, we revised the term to "log compute" in all figures, tables, text body, and appendix.  

    **b. Ambiguous name of binary Brier Score**  
       We revised the term to "Target-Conditioned (TC) Brier Score" to make it clear in all figures, tables, text body, and appendix. TC means our Brier Score takes the **target** label **conditioned** on the probability sum of all classes.  

    **c. Ambiguous names for difficulty levels (e.g. 0_1502_brier)**  
       We revised their names to difficulty levels 1, 2, ..., and 10 and have marked 1 as easiest and 10 as hardest in all figures, tables, text body, and appendix.  

    **d. Regarding the color map for different question groups**  
       We revised the color of different difficulties to gradual transitions from red to blue, as suggested by Reviewer nJbX, including all figures in the main paper and appendix.  

    **e. A plot that different datasets are next to each other**  
       We added it as Fig. 4 in the revised manuscript and added proper descriptions for it in Sec.2.3.  

    **f. Self-correction**   
       Fig.1(b) in the original submission was the un-redistributed Brier Score version that didn’t apply Eq. (4). We have fixed it.  

2. Detailed experiment settings  
    **a. How are these six datasets selected?**  
        The 6 datasets are chosen due to their representativeness as MCQ tasks with ability emergence. We aim to choose MCQ datasets with apparent emergence, i.e., model performance is flat across scales and soars at a certain threshold.  
        The MMLU was first chosen since it is a classic LLM multiple-choice question (MCQ) dataset. Arithmetic and Persian-QA (MCQ version) are subsequently selected from Fig.2 of [R1] (Word unscramble also has an apparent ability emergence, while it didn't have MCQ implementation). This is how we select the 3 datasets.  
        We further tested more tasks (datasets) in Big-Bench. [R1] has documented 66 Big-Bench datasets with emergence. However, they only used PaLM, GPT-3, or LaMDA to test the scaling trend. When evaluating with 56 LLMs, we found some tasks were, in fact, non-emergent. They have different scaling trends, such as steady improvement or even U-shaped scaling. Plus, some tasks in Big-Bench have a tiny number of questions, resulting in reduced statistical reliability, especially when they are grouped into subsets.  
        Specifically, we have examined an additional 17 tasks in Big-Bench with 56 LLMs, totaling 20 tasks. 5 of them only have string-matching task implementation. Evaluation order is simply alternating between [R1]’s App.E.2 and App.E.3. The 3 datasets in our App. are the **largest** datasets that are (1) MCQ tasks and (2) still have clear emergence when evaluated with 56 LLMs. Experimental results of the remaining 9 datasets, which are either MCQ or True/False questions, are in https://anonymous.4open.science/r/iclr_rebuttal-583C/. Detailed information of the 17 tasks following our evaluation order is as follows:

### Table 1: Examination of 17 Big-Bench tasks. Tasks marked with strikethrough only have string-matching task implementation; tasks marked with bold are those adopted in our paper. Our experiments stopped at Hindu knowledge. 
| Task name | Clear emergence | Question number > 100 | MCQ | Level of U-shaped vs. inverted-U Scaling | Sec. in [R1] | 
|-------------|-------------|--------------|---------|--------------|--------|
| analytic entailment       | inverted-U    | &#10008;      | &#10008; | Medium | App.E.2 |
| anachronisms       | &#x2714;   | &#x2714;      | &#10008; | Medium | App.E.3 |
| **analogical similarity**      | &#x2714;   | &#x2714;      | &#x2714; | Medium | App.E.3 |
| ~~ascii word selection~~      | N/A  | N/A      | &#10008; | N/A | App.E.3 |
| ~~auto debugging~~      | N/A  | N/A      | &#10008; | N/A | App.E.3 |
| ~~codenames~~      | N/A  | N/A      | &#10008; | N/A | App.E.2 |
| common morpheme     | &#10008;  | &#10008;      | &#x2714; | Low-to-no | App.E.2 |
| causal judgment     | &#10008;  | &#x2714;     | &#10008; | Strong | App.E.3 |
| code line description     | &#x2714;  | &#10008;     | &#x2714; | Medium | App.E.3 |
| **conceptual combinations**     | &#x2714;  | &#x2714;     | &#x2714; | Strong | App.E.3 |
| crass ai     | U-shaped  | &#10008;     | &#x2714; | All U-shaped | App.E.3 |
| ~~cryptonite~~     | N/A  | N/A     | &#10008; | N/A | App.E.3 |
| cs algorithms     | &#x2714;  | &#x2714;     | &#10008; | No | App.E.3 |
| fact checker     | &#x2714;  | &#x2714;     | &#10008; | Inverted | App.E.2 |
| figure of speech detection     | &#10008;  | &#10008;     | &#x2714; | No | App.E.2 |
| ~~gender inclusive sentences german~~     | N/A  | N/A     | &#10008; | N/A | App.E.2 |
| **Hindu knowledge**     | &#x2714;  | &#x2714;     | &#x2714; | Strong | App.E.2 |
</content>
</thread_object>

</thread>


Metareview:
metareview:
The paper connects two well-known phenomena in deep learning, namely emergent capabilities and inverse scaling. And it proposes a simple yet effective pipeline, called Slice-and-Sandwich, to predict both the emergence threshold and model performance beyond the threshold. Although there were some disagreements, the reviewers generally found that the paper was well-written, thoroughly experimented, and presented novel ideas. After the rebuttal, the clarity and solidity of the paper was improved, and it received unanimous recognition and higher scores from the reviewers. AC believes these evaluations were well-considered and credible, therefore recommends accepting this paper.