Random Scaling of Emergent Capabilities

Core findings

• Random seeds can produce either emergent or linear scaling trends despite identical data and hyperparameters, because consecutive points may sample different or similar performance clusters.Figure 1 illustrates this mechanism using the count task and seed-specific scaling trends.Source: Figure 1 · p. 2

Study approach

• Because large-scale training across numerous seeds is expensive, the study uses partially reinitialized LLMs and toy models to characterize full multimodal performance distributions.This approach extends beyond prior work that summarized only a few training runs.Source: §1 · p. 2

Core findings

• Breakthroughs arise from bimodal performance distributions across random seeds, with some runs showing linear scaling and others emergent trends on synthetic, language-modeling, and multiple-choice tasks.The distribution becomes bimodal around breakthrough thresholds, making emergence stochastic rather than deterministic.Source: §1 · pp. 1–2
• Continuous loss metrics can remain visibly bimodally distributed, including on both synthetic and natural tasks, so continuous measurement does not eliminate apparent breakthroughs.The effect is strongest when failed and successful runs are more evenly represented.Source: §1 · pp. 1–2
• Scaling jumps reflect continuously changing success probabilities and successful-run performance, not necessarily a fixed scale at which a concept becomes learnable.Bimodality may appear abruptly at a minimum capacity scale, but that threshold need not match individual runs’ breakthrough scales.Source: §1 · pp. 1–2
• Bimodality can create nonmonotonic overall scaling, with inverse-scaling regions arising from changing success probability even when mean performance among successful runs remains monotonic.This shows that U-shaped mean curves can be artifacts of the mixture of failure and success distributions.Source: §1 · p. 2

2.1. Tasks and Setup

• The experiments train decoder-only Transformers from hundreds of seeds on counting and reverse-order addition, testing length generalization beyond training lengths while varying width or depth.Training lengths are 30 for counting and 35 for addition; evaluation lengths are 60 and 40, respectively, with the main text focusing on addition.Source: §2.1 · p. 3, Figure 2 · p. 3

2.2. Emergence is a Sign of Bimodal Variation

• Bimodal seed distributions explain why individual scaling curves can look either smooth or emergent: staying within a performance cluster is smooth, while switching from low to high performance creates a breakthrough.They also produce variability in breakthrough timing and magnitude, including oscillating scaling curves.Source: §2.2 · p. 3

2.3. Sudden Jumps From Gradual Distribution Shifts

• The mode can spike sharply while the mean, successful-run probability, and successful-run mean improve continuously, showing that apparent jumps arise from gradual distributional changes.Successful runs are defined here as those reaching at least 20% accuracy; the continuous trends have an exception from depth 2 to 3.Source: §2.3 · p. 4, Figure 3 · p. 4

2.4. Is Bimodality a Mirage?

• Bimodality persists across random seeds under a continuous loss-based error metric, confirming that the phenomenon is not caused by exact-match thresholding alone.The metric uses the maximum per-token loss over the output sequence and is continuous when per-token losses are continuous.Source: §2.4 · p. 4

2.5. Bimodality Emerges Abruptly

• The performance distribution shifts abruptly from unimodal to bimodal at sufficient capacity, with depth 3 or width 2 heads marking the emergence of highly successful runs.This transition identifies a minimum capacity for learning the task, whereas single-run analyses can misidentify that threshold.Source: §2.5 · pp. 4–5, Figure 4 · p. 5

3.1. Data and Setup

• The experiments simulate independent runs by reinitializing Qwen2.5’s final attention layer and LM head, then continuing full-parameter training on C4–MMLU mixtures across random seeds.They evaluate MMLU, vary the MMLU proportion, and train Qwen2.5-0.5B and Qwen2.5-1.5B reinitializations under fixed optimization schedules.Source: §3.1 · p. 5

3.2. Experiment Results

• At 10% MMLU data, Qwen2.5-0.5B develops near-0% and ≥25% accuracy clusters, while Qwen2.5-1.5B performs around or above the random baseline across seeds.With >20% MMLU, all 80 seeds perform around or above the random baseline; the larger model recovers similar capabilities at much lower mixture ratios.Source: Figure 5 · p. 6, §3.2 · p. 6

3.3. Is MMLU emergence a mirage?

• NLL loss remains bimodal for 10% MMLU data (p < 0.001), exposing clusters concealed by accuracy metrics, including among low-performing and larger models.Hartigan’s Dip Test establishes statistically significant bimodality, contradicting the claim that continuous metrics eliminate discontinuities.Source: §3.3 · p. 6
• Pretraining loss is smooth, unimodal, and nearly symmetric, contrasting with MMLU’s multimodality and linking clustering to emergent tasks rather than in-distribution training noise.The comparison suggests that multimodality is specific to emergent-task performance distributions.Source: §3.3 · p. 6

4.1. Data and Setup

• Decoder-only Transformers trained from scratch on synthetic question-formation data vary in depth and hierarchical-data proportion across 80 random seeds to study bimodal generalization.Ambiguous examples support either hierarchical or linear rules, and OOD tests distinguish them with 100% versus 0% accuracy.Source: §4 · p. 6, §4.1 · p. 7

4.2. Experiment Results

• At 10% hierarchical data, mean OOD accuracy jumps from near 0% to 80%, while seeds split between 0% and 80-100% and hierarchical data only increases hierarchical-rule probability.Even with 100% hierarchical-inducing data, the competing linear solution persists, possibly because the training set lacks sufficient diversity.Source: §4.2 · p. 7, Figure 6 · p. 7
• With 90% hierarchical data, 3-layer models are unimodal around 70%, whereas 6-layer models become bimodal, with most seeds at 100% and some at 0%.Bimodality remains under a continuous metric and is statistically significant under Hartigan’s Dip Test.Source: §4.2 · p. 7, Figure 7 · p. 7

Random variation and emergence

• Even continuous NLL exhibits multimodal performance distributions across random seeds, unlike the smooth unimodal distribution of training loss.Figure 8 reports this for MMLU and OOD QF, while Figure 9 shows the unimodal training-loss contrast.Source: Figure 8 · p. 8, Figure 9 · p. 8
• Bimodality can emerge before most runs or the distribution’s mode exhibits a breakthrough, indicating that abrupt metric jumps reflect gradual distributional improvement.This transition from unimodal to bimodal variation can itself be sudden, but the breakthrough typically follows it.Source: §5 · p. 8

Related work

• The study connects random clustering in shortcuts, training dynamics, length generalization, and syntax acquisition to emergence at scale.It extends prior evidence of seed-dependent variability and bimodal variation across these behaviors.Source: §5 · p. 8

Emergent abilities of LMs

• Statistical analysis is necessary when evaluating emergence claims because some breakthroughs reflect evaluation metrics, while others remain emergent on specific tasks.The discussion notes that 8% of high-breakthroughness BIG-Bench datasets remain emergent according to Schaeffer et al. (2024).Source: §5 · p. 8

Depth versus Width Scaling

• Increasing width can damage compositional-task performance while increasing depth improves it, revealing an architectural regime where depth–width tradeoffs shape random variation.This regime appears in Appendix C.2 and motivates further study of architectural effects on emergent tasks.Source: §5 · p. 8

A. Experimental details on synthetic tasks

• 250 seeds are trained for count and 200 for reverse-order addition, with models trained to convergence on their respective training distributions.All model scales run on a single 40GB A100 GPU; count runs finish within 2 hours and addition runs within 6 hours.Source: A. Experimental details on synthetic… · p. 12
• Count uses vocabulary size 150, with thresholded evaluation computed as exact-match accuracy across all consecutive subsequences of the test length.Source: A. Experimental details on synthetic… · p. 12
• Count models scale width with depth fixed at 4 layers or scale depth with width fixed at hidden dimension 512.Fixed-depth widths are {64, 128, 256, 384, 512, 640, 768, 1024}, while fixed-width depths are {1, 2, 4, 6, 8}; training uses 1e −3 learning rate, weight decay 0.1, 10000 steps, batch size 128, and context length 256.Source: A. Experimental details on synthetic… · p. 12
• Reverse-order addition with index hints uses thresholded exact-match evaluation across 500 batches of 128 examples each.Source: A. Experimental details on synthetic… · p. 12
• Addition models scale width with depth fixed at 6 layers or scale depth with width fixed at hidden dimension 512.Fixed-depth widths are {64, 128, 256, 384, 512, 640, 768}, while fixed-width depths are {1, 2, 3, 4, 6, 8, 10, 12}; training uses 1e −4 learning rate, weight decay 0, 30000 steps, batch size 64, and context length 512.Source: A. Experimental details on synthetic… · p. 12

B. Breakthroughness and Linearity

• Breakthroughness and linearity metrics quantify whether performance improves suddenly or reliably with scale, using performances y_i at ascending model scales x_i.The paper defines the metrics through the ordered performance values and the function I(y) = sign(arg maxi yi −arg mini yi)(maxi yi −mini yi).Source: B. Breakthroughness and Linearity · p. 12
• Figure 10 samples the five top seeds for breakthroughness and linearity separately for count and addition.Source: B. Breakthroughness and Linearity · p. 12

• Given two numbers in increasing order, the model is trained to generate a sequence counting consecutively from the first number to the second, while limiting sequence length during training.For example, “5, 9 >, 5, 6, 7, 8, 9” maps the numbers to the consecutive counting sequence.Source: C. Count task · p. 12
• Zhou et al. (2024a) showed that counting models can generalize to more than twice the training length.Source: C. Count task · p. 12
• Appx. C.2 provides a more nuanced account of length generalization by examining its distribution across independent model runs.Source: C. Count task · p. 12

Count performance distributions

• At the highest tested widths and depths, the count-task histogram’s failing-run mode disappears, unlike addition, indicating task-dependent differences in random variation.Figure 11 shows count-task accuracy distributions for fixed test length 60; the main-paper addition distributions use fixed test length 40.Source: §C.1 · pp. 12–13
• Figure 10 ranks the top five count and addition seeds by breakthroughness and linearity using metrics defined in Appendix B.Source: Figure 10 · p. 13
• Regardless of width, one-layer count-task models cannot length-generalize while achieving near-perfect in-distribution accuracy.This example motivates the claim that performance distributions become bimodal at the minimum capacity required by the task.Source: §C.1 · p. 13

C.2. Count task as a case of U-shaped random scaling

• The counting task produces a U-shaped mean-accuracy scaling curve, mirrored by the performance distributions’ W2-distance evolution rather than caused by the chosen summary statistic.This establishes counting as a distinct case of random scaling with bimodally distributed performance.Source: §C.2 · p. 13
• The mean of successful runs—those achieving at least 50% accuracy—increases monotonically with width, while the overall U-shape reflects changing success probabilities.Thus, even when inverse scaling appears across the full distribution, successful-run performance can respond conventionally to scale.Source: §C.2 · p. 13

D. Additional Continuous Metric Plots

• At one-layer depth, all count-task seeds achieve near-perfect in-distribution accuracy but fail to length generalize at test length 60.This contrasts test length 30, which is in-distribution, with length 60, which is out-of-distribution.Source: Figure 12 · p. 14
• Count-task distributions change slowly across intermediate scales but suddenly near the nadir of a U-shaped distance curve, while successful-run performance improves continuously.The count-task summary statistics show a U-shaped mean across all runs when scaling width at fixed depth, but the mean of successful runs continues improving.Source: Figure 13 · p. 14, Figure 14 · p. 15
• Bimodality persists in continuous minimum-token probability metrics for addition and count, with stronger accuracy bimodality corresponding to stronger probability bimodality.The metric tracks the average minimum token probability because sequence averaging can obscure rare generalization failures; its lowest-probability token can shift continuously as models improve.Source: D. Additional Continuous Metric Plots · p. 15, Figure 15 · p. 16
• Loss distributions likewise retain the association between stronger accuracy or probability bimodality and stronger bimodality, although count-task failure modes are more diffuse.For addition, the strongest bimodality occurs at widths 4 and 8 or depths 2 and 4; count remains visibly bimodal even when failure probability is about 25%.Source: D. Additional Continuous Metric Plots · p. 15
• In large language-model experiments, high-performance clusters appear in both accuracy and loss, while continuous NLL exposes low-performance clusters concealed by thresholded accuracy.Across data mixes, models trained on 20 random seeds show bimodal accuracy and multimodal continuous NLL distributions.Source: D. Additional Continuous Metric Plots · pp. 15–16

E.1. Task Description

• The question-formation task asks models to transform declarative English sentences into questions by moving an auxiliary verb according to sentence syntax.It tests hierarchical syntactic generalization rather than reliance on a superficial linear heuristic.Source: §E.1 · p. 16
• Training permits either moving the first auxiliary verb, an incorrect linear rule, or moving the syntactically determined main auxiliary verb, the correct hierarchical rule.Examples include an ambiguous case where both rules yield the correct output and an unambiguous case where only the hierarchical rule does.Source: §E.1 · p. 16

• Training data contains question-formation examples that are ambiguous between linear and hierarchical rules, plus declaration-copying examples that may have unambiguous structures.Source: §E.2 · p. 16
• Following Qin et al. (2024), the declaration-copying subset varies the ratio of center-embedded to right-branching sentences.This manipulation controls exposure to different syntactic structures.Source: §E.2 · p. 16
• Center-embedded sentences require recursive clause embedding and encourage hierarchical-rule learning, whereas right-branching sentences support simpler linear n-gram solutions.Source: §E.2 · pp. 16, 18
• Training mixtures include Quest Only, Center Embed, Right Branch, and intermediate ratios within the declaration-copying subset.Source: §E.2 · p. 18

Evaluation

• ∼100% OOD accuracy indicates acquisition of the hierarchical syntactic rule, whereas linear-rule models achieve ∼0% OOD accuracy.Hierarchical-rule models achieve ∼100% accuracy on both in-distribution and OOD tests, while linear-rule models reach ∼100% only in distribution.Source: §E.3 · p. 18
• The in-distribution test contains ambiguous examples solvable by either rule, while the OOD test contains unambiguous examples requiring the hierarchical rule.Question formation requires moving the main auxiliary verb to the front; ambiguous examples permit either linear or hierarchical solutions, whereas unambiguous examples require the hierarchical rule.Source: §E.3 · p. 18, Table 1 · p. 19