IC-43256 LLMs from 19 families exhibit u-shaped scaling on hard questions and inverted-U scaling on easy questions, with the opposing trends explaining emergent ability stagnation

Tung-Yu Wu, Melody Lo

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

The authors evaluate 56 released LLMs across 19 model families on 6 multiple-choice benchmarks (MMLU, Arithmetic, Persian-QA, Hindu Knowledge, Conceptual Combinations, Analogical Similarity), grouping questions by difficulty level using a target-conditioned Brier score. Performance on the hardest question group shows a U-shaped curve: it first worsens with scale (models are distracted by surface patterns such as negations) before improving past the emergence threshold. Performance on the easiest group shows an inverted-U followed by steady improvement, consistent with deep double descent. Because the two trends offset each other when aggregated, overall accuracy stagnates until the easy-question curve reverts to standard scaling, at which point both groups improve and overall performance soars. The pattern is absent on non-emergent tasks (ARC, HellaSwag, Abstract Narrative Understanding), where performance improves monotonically.

Evidence
correlational
Key metric
56 LLMs, 6 datasets, g=10 difficulty groups; emergence thresholds t = 1.5 (MMLU), 1.8 (Arithmetic), 2.3 (Persian-QA); TC Brier score vs accuracy Pearson correlation: 0.99 (MMLU), 1.00 (Arithmetic), 0.88 (Persian-QA)
Caveat
The authors note that emergent phenomena are common across LLM benchmarks and it might be hard to claim all of them show clear u-shaped vs. inverted-u scaling. The focus is on multiple-choice tasks; applying the method to string-matching tasks requires identifying a suitable continuous metric.
Model
Gemma, Llama 2 / Llama 2 base, RedPajama-INCITE, Yi, StableLM, MPT, Falcon, Pythia, Qwen, Qwen1.5, BLOOM, DeepSeekMoE, OPT, GPT-Neo, CodeGen, XGLM, OpenLLaMA
Concepts
Scale-dependent behaviour
Datasets
MMLU / MMLU-Math [eval], Big-Bench / Big-Bench verb tense [eval], ARC [eval], HellaSwag [eval]
Methods
LM Evaluation Harness [eval], Polynomial Regression [supporting]
Extraction
automatic-extraction