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Predicting Emergent Abilities with Infinite Resolution Evaluation

Shengding Hu, Xin Liu, Xu Han, Xinrong Zhang, Chaoqun He, Weilin Zhao, Yankai Lin, Ning Ding, Zebin Ou, Guoyang Zeng, Zhiyuan Liu, Maosong Sun

arXiv:2310.03262v3cs.CL

TL;DR

Task performance remains difficult to predict from established loss-scaling laws, in part because it can appear trivial before surging with model size. The paper introduces PASSUNTIL to expose subtle improvements and quantify task scaling; it reports accurate code-generation prediction and identifies accelerated emergence that remains difficult to fit with a standard scaling law.

  • Problem

    Loss scaling laws predict optimization loss, but task performance remains difficult to predict and can appear discontinuous as model size increases.

    Source: §1 · p. 2

  • Method

    PASSUNTIL repeatedly samples generations until one passes the target test, estimating target-task metrics with theoretically infinite resolution when computation is unbounded.

    Source: §4.1 · p. 4

  • Results

    0.05% deviation from actual values was reported for code-generation predictions, alongside accelerated emergence whose concave scaling curve does not fit the typical task-scaling function.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The analysis makes subtle task improvements measurable and predicts some task performance, while accelerated emergence remains hard to predict and may go out of control as models grow.

    Source: §7 · p. 9, ETHICAL STATEMENT · p. 10

    PASSUNTIL is designed for tasks with a zero random baseline and currently targets random sampling; non-zero-baseline tasks and beam-search decoding remain for future study.

    Source: §4.1 · p. 5

Abstract

from arXiv · show

The scientific scale-up of large language models (LLMs) necessitates a comprehensive understanding of their scaling properties. However, the existing literature on the scaling properties only yields an incomplete answer: optimization loss decreases predictably as the model size increases, in line with established scaling law; yet no scaling law for task has been established and the task performances are far from predictable during scaling. Task performances typically show minor gains on small models until they improve dramatically once models exceed a size threshold, exemplifying the ``emergent abilities''. In this study, we discover that small models, although they exhibit minor performance, demonstrate critical and consistent task performance improvements that are not captured by conventional evaluation strategies due to insufficient measurement resolution. To measure such improvements, we introduce PassUntil, an evaluation strategy with theoretically infinite resolution, through massive sampling in the decoding phase. With PassUntil, we conduct a quantitative investigation into the scaling law of task performance. The investigation contains two parts. Firstly, a strict task scaling law that is not conventionally known to exist, is identified, enhancing the predictability of task performances. Remarkably, we are able to predict the performance of the 2.4B model on code generation with merely 0.05\% deviation before training starts, which is the first systematic attempt to verify predictable scaling proposed by GPT-4's report. Secondly, we are able to study emergent abilities quantitatively. We identify a kind of accelerated emergence whose scaling curve cannot be fitted by standard scaling law function and has a increasing speed. We then examine two hypothesis and imply that the ``multiple circuits hypothesis'' might be responsible for the accelerated emergence.

1 INTRODUCTION

Loss scaling laws predict optimization loss, but task performance remains difficult to predict because it often appears discontinuous as models scale. The introduction proposes higher-resolution evaluation to reveal small-model improvements and investigate task scaling and accelerated emergence.

1 INTRODUCTION

  • Established scaling laws predict decreasing training loss as model size and pretraining data increase, but do not predict practical task performance.This leaves a gap in guidance for scaling methodology.Source: §1 · pp. 1–2
  • Task performance can remain trivial below a size threshold and then surge, complicating prediction as models scale.The introduction describes this pattern as emergent abilities and notes it across model families and tasks.Source: §1 · p. 2
  • The authors hypothesize that limited evaluation resolution makes gradual improvements look discontinuous, and extend the increase-resolution approach to target metrics such as exact match.They distinguish this from replacing target metrics with smoother alternatives.Source: §1 · p. 2
  • PASSUNTIL repeatedly samples generations until one passes the target test, offering theoretically infinite resolution when computational resources are unbounded.The introduction also proposes fitting instance-level scaling laws because test instances may improve at different rates.Source: §1 · p. 2
  • A task-scaling law predicts performance from model size, with code-generation predictions deviating just 0.05% from actual values.The reported relationship is nearly linear between log(−log(PU)) and log(N).Source: §1 · p. 3
  • Accelerated emergence follows a concave scaling curve that the typical task-scaling function cannot fit, and the authors propose a potential transformer-circuits hypothesis.They use a quantitative definition to exclude a possible multi-step-reasoning explanation; the circuits hypothesis is described as consistent with the observed curve.Source: §1 · p. 3
  • The work presents itself as the first open-source attempt to investigate task-performance predictability and says it will release all checkpoints.The authors cite GPT-4’s report as initiating the exploration without comprehensive details.Source: §1 · p. 3

2 RELATED WORK

Prior scaling-law work primarily models loss, while task performance can show varied and emergent behavior that loss scaling does not reliably predict. This paper positions its contribution as a practical resolution increase and an open-source quantitative investigation of task scaling.

2 RELATED WORK

  • Predicting task performance before training is framed as an objective pursued by studies using different perspectives.Source: §2 · p. 3
  • Loss-scaling research formalized power-law behavior for model loss, while this work extends that mathematical form to task performance.The cited literature spans architectures and settings, but focuses on training and validation loss metrics.Source: §2 · p. 3
  • Task performance may improve progressively or show breakthrough behavior, and loss scaling offers no assurance of task outcomes or pretraining guidance.Prior reports include GPT-4 performance predictions but do not disclose the methodology and acknowledge abilities beyond prediction.Source: §2 · p. 3
  • The authors align with increasing evaluation resolution but propose doing so without adding test samples, targeting exact match and related metrics.They describe their work as the first open-source quantitative investigation proposing task scaling laws and accelerated emergence.Source: §2 · p. 4

3 PILOT EXPERIMENTS ON INCREASING RANDOM SAMPLE NUMBERS

The pilot tests whether more random decoding samples reveal performance on difficult tasks for small models. The authors report that many instances become passable with sufficient sampling, exposing subtle task-performance improvements.

3 PILOT EXPERIMENTS ON INCREASING RANDOM SAMPLE NUMBERS

  • Four small models were evaluated on Emoji Movie and Date Understanding using beam search and random sampling at 1, 100, and 10,000 samples.An instance counted as passed if any sampled answer was correct; the figure reports passed-instance counts.Source: §3 · p. 4, Figure 2 · p. 4
  • With enough random sampling, most tested instances were passable even for small models on difficult tasks, revealing subtle performance improvements.Source: §3 · p. 4

4 METHODS

The methods use PASSUNTIL to measure task success probabilities at fine resolution, derive a task scaling law from loss scaling, and fit performance at dataset and instance levels.

4.1 INFINITE RESOLUTION WITH PASSUNTIL

  • PASSUNTIL samples until r generations pass, records the sampling count K, and estimates an instance’s success probability as PU; theoretically it can measure infinitesimally small success rates.PU is a maximum-likelihood estimate, and practical evaluation bounds K to avoid endless sampling.Source: §4.1 · pp. 4–5
  • The method is designed for tasks with zero random-baseline success and currently targets random sampling, leaving non-zero baselines and beam-search decoding for future study.With non-zero random baselines, noise can overshadow smaller-model improvements.Source: §4.1 · p. 5

4.2 FROM LOSS-SCALING LAW TO TASK SCALING LAW

  • The task scaling law is derived by assuming next-token test loss follows the loss scaling law, with a dominant correct sequence, near-zero irreducible loss, and uniform token-level scaling speed.The resulting mathematical model is described as similar to forms in the GPT-4 technical report and Schaeffer et al. (2023).Source: §4.2 · p. 5

4.3 FITTING STRATEGY

  • Dataset-level fitting averages each model’s instance-level PU over the test set before regressing log(−log(PU)) against log model size.Instance-level fitting instead models each question separately and aggregates those fits, addressing differing instance difficulty and scaling behavior.Source: §4.3 · p. 5
  • Instance-level fitting accommodates diverse question difficulty by fitting an individual PASSUNTIL score for each question and aggregating the estimates for the dataset.Easy questions can saturate at PU=1 on small models while hard questions retain trivial performance.Source: §4.3 · p. 5

5 PREDICTABLE SCALING EXPERIMENTS

The experiments train two model series from 0.03B to 2.4B and evaluate loss and task-performance scaling. Task scaling fits predict 2.4B performance, with instance-level predictions reaching small deviations on two evaluations.

5.1 SCALING CONFIGURATIONS.

  • The two model series range from 0.03B to 2.4B, use consistent Transformer shapes, and are pretrained on StarCoder or StarCoder mixed with Pile.The maximum pretraining tokens per model are set to 20N, where N is the number of non-embedding parameters.Source: §5 · p. 6, §5.1 · p. 6

5.2 LOSS SCALING LAW VERIFICATION.

  • End-step training loss decreases in line with the scaling law, including eventual convergence for the 1.5B and 2.4B models despite a loss spike.Figure 3 displays reducible loss against model size on a logarithmic scale for the two data mixtures.Source: §5.2 · p. 6, Figure 3 · p. 6

5.3 DATASET-LEVEL FIT

  • Evaluation covers HumanEval in zero-shot settings and Emoji Movie and Date Understanding in four-shot settings, with Chain-of-Thought reasoning additionally used for Emoji Movie.The experiments remove distracting test instances, including common-word movie names for Emoji Movie.Source: §5.3 · p. 6
  • All three tasks show a strong linear relationship between log(−log(PU)) and log(N), and fits from 0.03B to 1.5B predict 2.4B performance with small deviations.The 2.4B model’s real-performance points are close to the fitted task scaling laws.Source: §5.3 · p. 7, Figure 4 · p. 7

5.4 INSTANCE-LEVEL FIT

  • Instance-level curves have different starting points and scaling speeds; some instances deviate from the task scaling law and need further investigation.The study also uses ground-truth test loss to predict PU for hard instances lacking enough non-zero PU values for fitting.Source: §5.4 · p. 7
  • 0.05% deviation for HumanEval series 1 and 1.7% for Date Understanding series 2 were reported for predictions of 2.4B model performance.Figure 6 presents the predictions; the estimates use instance-level PASSUNTIL and test-loss prediction.Source: Figure 6 · p. 7, §5.4 · p. 7

6 QUANTITATIVE ANALYSIS OF EMERGENCE

Across UICL tasks, the section distinguishes predictable linear or convex scaling from concave accelerated emergence and examines explanations for the latter. Five of eight tasks show concavity, and a two-circuit fit aligns more closely with the observed curve.

Quantitative analysis

  • Even with refined evaluation, some task abilities remain difficult to predict, motivating formal definitions and analysis of possible explanations.Source: §6 · p. 8

Categorization of Emergence

  • 5/8 UICL tasks show strictly concave scaling curves, exemplified by the Dates and Identity curves in Figure 7.The study used 20 questions per task, 4-shot examples, exact match, and a 10^5 sampling-time cap.Source: §6 · p. 8, Figure 7 · p. 8
  • Task scaling-law and sub-scaling-law growth are easier to predict and control than accelerated emergence, which may become harder to control as models grow.Figure 8 visualizes the three curve types: linear, convex, and concave.Source: Figure 8 · p. 8, §6 · p. 9

Cause of Shape of Scaling Curve

  • When each reasoning step follows scaling-law growth, the combined multi-step success rate follows sub-scaling-law growth.The paper attributes this to growth speed that is initially boosted by easier steps and eventually bounded by the hardest steps.Source: §6 · p. 9
  • The multiple-circuits hypothesis, inspired by Varma et al. (2023), yields concavity when task success follows the best circuit.Source: §6 · p. 9
  • A soft two-circuit fit to UICL performance aligns more accurately with the observed scaling curve.The fit uses a weighted combination of two circuits.Source: §6 · p. 9

7 CONCLUSION.

The paper introduces an evaluation strategy for detecting small performance improvements and uses it to study task scaling and emergent abilities.

Conclusion

  • The evaluation strategy detects minimal performance improvements, enabling quantitative measurement of task scaling laws and emergent abilities.Source: §7 · p. 9
  • The strategy has enabled successful prediction of task performance for larger models.Source: §7 · p. 9
  • The paper’s quantitative analysis provides clearer insight into the nature and origins of emergent abilities.Source: §7 · p. 9

ETHICAL STATEMENT

The paper reports that accelerated emergence remains difficult to predict and connects this challenge to unresolved mechanisms and safety concerns.

Ethical Statement

  • Although some emergent abilities are predictable, accelerated emergence remains difficult to predict.Source: ETHICAL STATEMENT · p. 10
  • Accurate predictions for accelerated emergence require a better understanding of the mechanisms producing it.Source: ETHICAL STATEMENT · p. 10
  • An unknown stronger generalization circuit may govern performance at larger scales, and the paper links accelerated emergence to safety concerns.Source: ETHICAL STATEMENT · p. 10

A.1 LIMITATIONS

The supplementary discussion illustrates how easy and hard instances scale differently, while the limitations qualify the models, tasks, and explanations covered by the study.

A.1 LIMITATIONS

  • Predictions do not extend to 10B or larger models, task coverage remains incomplete, and the proposed circuit explanation is not mechanistically established.Source: §A.1 · p. 13

A.2 DISCUSS OF THE USE OF LOSS AS AN ASSISTANCE METRIC

  • Loss alone cannot determine accuracy, but its statistical relationship with PASSUNTIL can improve prediction under computational constraints; sufficiently resolved PASSUNTIL does not require loss.Source: §A.2 · p. 13

B.1 INSTANCE-LEVEL PASSUNTIL INTUITION. ¯

  • Index 24’s easy HumanEval instance improves with model size, while index 20 remains trivial; averaging can hide hard-instance improvements after easy instances saturate.Table 2 contrasts the easy and challenging instances.Source: §B.1 · pp. 13–14, Table 2 · p. 14

C.1 THEORETICAL ANALYSIS OF HYPOTHESIS ¯

The section formalizes two mechanisms for task-level scaling behavior: composing reasoning steps can yield sub-scaling, while majority voting across circuits yields concavity. It also describes the models, training choices, and test sets used in the experiments.

C.1 THEORETICAL ANALYSIS OF HYPOTHESIS ¯

  • When reasoning steps scale at the same speed, their multi-step success rate exhibits sub-scaling law growth.The theorem states that scaling at each reasoning step implies sub-scaling for the multi-step success rate.Source: §C.1 · p. 14
  • When each circuit scales individually, majority voting across circuits makes F(N) concave in log N.The proof notes that the minimum operator preserves concavity.Source: §C.1 · pp. 14–15

D.1 MODEL CONFIGURATION ¯

  • The model series use a Llama 2-like architecture with tied input-output embeddings and gated-GeLU, while configurations keep a uniform shape across model sizes.Table 3 reports non-embedding parameters, batch size, training steps, and total training tokens.Source: §D.1 · p. 15, Table 3 · p. 15

D.2 PRE-TRAINING CORPORA ¯

  • Two model series use different pre-training data mixtures to test whether the experiments generalize across corpora.Tables 4 and 5 list the respective mixture proportions.Source: §D.2 · p. 15, Table 4 · p. 16

D.3 HYPER-PARAMETERS STUDY ¯

  • Training uses a cosine learning-rate schedule with a maximum rate of 0.01; matching its cycle length to the end step is reported as optimal.Longer or shorter cycle lengths result in sub-optimal performance; batch sizes are adjusted slightly from estimated optima to improve GPU utility.Source: §D.3 · pp. 15–16

D.4 TEST SET CONFIGURATIONS ¯

  • The evaluation setup uses zero-shot HumanEval with a 10^4 sampling cap and 4-shot Emoji Movie and Date Understanding with 10^5 caps.HumanEval requires passing all unit tests; the two BigBench tasks use extract string match.Source: §D.4.1 · p. 16, §D.4.2 · p. 16, §D.4.3 · p. 16

D.4.4 UNNATURAL IN-CONTEXT LEARNING TASKS ¯

The appendix specifies the unnatural in-context learning evaluation and discusses removing distracting instances. It also reports additional HumanEval estimates and task-specific relationships between test loss and PASSUNTIL.

D.4.4 UNNATURAL IN-CONTEXT LEARNING TASKS ¯

  • Unnatural in-context learning evaluation samples 20 instances per task, uses 4-shot contexts and extract string match, and caps PASSUNTIL sampling at 10^5.Four examples are randomly selected from the remaining dataset for context.Source: §D.4.4 · p. 16

D.5 REMOVING DISTRACTING FACTOR IS IMPORTANT WHEN MEASURING TINY PERFORMANCE.

  • The authors remove distracting instances because common-word movie names can be guessed and date questions may require extraction rather than date reasoning.The reported movie-name pass rates show no significant association with model scale; GPT-4 report instead bins HumanEval examples by difficulty.Source: §D.5 · p. 17, Table 7 · p. 17, Figure 9 · p. 18

E.1 ADDITIONAL DATASET LEVEL PASSUNTIL RESULT.

  • For Series 2 HumanEval, dataset-level prediction is less accurate than Series 1, but instance-level PASSUNTIL improves prediction precision.Figure 10 presents the Series 2 performances.Source: Figure 10 · p. 18, §E.1 · p. 18

E.2 ESTIMATING PASSUNTIL FROM TEST LOSS ¯

  • For ground-truth answer test loss, PU scales as 0.22L for Series 1 HumanEval, 0.23L for Series 2 HumanEval, 0.96L for Series 2 Date Understanding, and 0.43L for Series 2 Emoji Movie.The authors propose using these relationships to assist PASSUNTIL prediction for hard samples.Source: §E.2 · p. 19, Figure 11 · p. 19

E.3 MORE RESULTS OF THE UNNATURAL IN-CONTEXT LEARNING TASKS ¯

Additional unnatural in-context learning results show concave scaling curves for some subtasks, while other tasks are too difficult for the smallest models to fit reliably. Further instance-level HumanEval curves are also presented.

E.3 MORE RESULTS OF THE UNNATURAL IN-CONTEXT LEARNING TASKS ¯

  • For two subtasks, 0.03B and 0.1B models have zero pass rates, making fit analysis less meaningful; Reverse to Natural Content shows slight sub-scaling, possibly due to its multi-step nature.Source: §E.3 · pp. 19–20
  • Three remaining unnatural in-context subtasks show concave curves; the 2-digits task instead shows an inverse-scaling trend needing further investigation.Figure 12 presents the curves.Source: §E.3 · p. 19, Figure 12 · p. 20

E.4 RESULT OF INDIVIDUAL PASSUNTIL ON MORE SAMPLES ¯

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