Literature Overview and Meta-Analysis

This document synthesizes research on skill emergence, composition, and evaluation in large language models. The papers collectively address how LLMs acquire, represent, compose, and apply skills—from theoretical frameworks through empirical characterization to practical applications.

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1. Paper Overviews

1.1 Arora & Goyal (2023) — A Theory for Emergence of Complex Skills in Language Models

Problem: How do complex skills emerge in LLMs when parameters and training corpora are scaled up? Mechanistic explanations via gradient analysis are difficult.

Approach: A statistical framework leveraging empirical Scaling Laws to analyze skill emergence without requiring mechanistic insight into training dynamics.

Core Concepts and Definitions

Skill Graph: A bipartite graph $G=(V_1,V_2,E)$ where $V_1$ is the set of skills $S$, $V_2$ is the set of text-pieces $T$, and an edge $(s,t)\in E$ means that comprehending text-piece $t$ requires applying skill $s$.

Text-piece Distribution: Text-pieces are generated by sampling random $k$-tuples of skills and converting them into text whose comprehension requires those skills, under distributions ${\mu }_1$ (over skills) and ${\mu }_2$ (over text-pieces).

Competence: For a skill $s\in S$, competence is defined as $\text{Competence}(s)\in [0,1]$ representing the model’s success rate on cloze questions from randomly selected text-pieces adjacent to $s$.

Competence on $k^{\prime }$-tuples: The ability to answer cloze questions in randomly selected text-pieces connected to all skills in a $k^{\prime }$-tuple.

Scaling Law (Chinchilla): $L(N,D)=A+\frac{B}{N^{0.34}}+\frac{C}{D^{0.28}}$

Main Results

1. Theorem 14 (Emergence of $k^{\prime }$-tuples): Competence in skill $k^{\prime }$-tuples improves almost as fast as competence on individual skills with scaling. Let $Y$ be text pieces with total measure $\theta$ (error fraction). For $\alpha ,\beta >0$ satisfying: $H(\theta )+k\theta \left(H(\beta \alpha )-\beta \alpha \log \frac{1}{\alpha }-(1-\beta \alpha )\log \frac{1}{1-\alpha }\right)<0$ at least $(1-\alpha )$ fraction of $k^{\prime }$-tuples have $\le \beta \theta$ fraction of edges to $Y$.

2. Corollary 13 (Scaling Effect): When model scales such that loss drops from $\delta$ to $\delta /k^{\prime }$, performance on $k^{\prime }$-tuples equals previous performance on individual skills.

3. Slingshot Generalization: The Scaling Laws imply a strong inductive-bias allowing pre-trained models to learn efficiently—competence levels appear to “violate” usual generalization theory.

4. Poverty of Stimulus: If the model displays competency on even 10% of $k^{\prime }$-tuples, it must have acquired competence in combinations not seen during training, since $|S{|}^{k^{\prime }}\gg D$ (training corpus size).

5. Key Insight: 10× scaling ≈ 2× increase in number of skills that can be composed.

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1.2 Wei et al. (2022) — Emergent Abilities of Large Language Models

Problem: Can we characterize abilities that appear unpredictably with scale—present in larger models but absent in smaller ones?

Approach: Empirical documentation and classification of emergent abilities across prompting paradigms (few-shot, chain-of-thought, etc.).

Core Concepts and Definitions

Emergent Ability: An ability is emergent if it is not present in smaller models but is present in larger models. Formally: “cannot be predicted simply by extrapolating the performance of smaller models.”

Phase Transition: Sharp performance increase at critical scale. Distinction between slow emergence (gradual on linear scale, appears sharp on log scale) and truly discontinuous transitions.

Prompting Paradigms:

• Few-shot prompting: In-context learning with exemplars
• Chain-of-thought (CoT): Intermediate reasoning steps before final answer
• Instruction-following: Zero-shot task completion from natural language instructions

Main Results

1. Documentation of emergent abilities across benchmarks (BIG-Bench, arithmetic, word problems, etc.)
2. Observation that emergence is task-dependent: some tasks exhibit smooth scaling, others show sharp transitions
3. Documents emergence across: few-shot learning, chain-of-thought reasoning, instruction following, task composition
4. The existence of emergent abilities raises questions about future capabilities with continued scaling

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1.3 Michaud et al. (2024) — The Quantization Model of Neural Scaling

Problem: Explain both (i) the power law decrease of loss with scale and (ii) sudden emergence of new capabilities.

Approach: Propose the Quantization Hypothesis—that network knowledge/skills are “quantized” into discrete chunks (quanta) learned in order of decreasing use frequency.

Core Concepts and Definitions

Quantization Hypothesis: Network knowledge and skills are quantized into discrete modules (quanta). Models learn these quanta in order of decreasing “use frequency” in the training distribution.

Quantum (pl. Quanta): A discrete unit of knowledge/skill. Analogous to Minsky’s “Society of Mind” agents.

Monogenic Sample: A prediction problem whose performance is determined by a single quantum; exhibits sharp phase transition at learning threshold.

Polygenic Sample: A prediction problem where multiple quanta influence performance; exhibits gradual improvement with scale.

Q-Sequence: The ordering of quanta by use frequency, determining learning priority.

Key Formal Results

If quanta use frequencies follow a power law $p_k\propto k^{-(\alpha +1)}$ (Zipfian), then:

• Parameter Scaling: $L(N)-L_{\infty }\propto N^{-\alpha }$ where ${\alpha }_N=\alpha$
• Data Scaling (multi-epoch): $L(D)-L_{\infty }\propto D^{-\alpha /(\alpha +1)}$
• Data Scaling (single-epoch): ${\alpha }_D=\alpha /(\alpha +1)$

Validation: Toy dataset “multitask sparse parity” confirms power law scaling and emergence.

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1.4 Didolkar et al. (2024) — Metacognitive Capabilities of LLMs: An Exploration in Mathematical Problem Solving

Problem: Do LLMs possess metacognitive knowledge (knowledge about their own reasoning processes)? Can this be leveraged to improve performance?

Approach: Develop a prompt-guided procedure to elicit LLM-identified skill labels, create a skill exemplar repository, and use skill-based in-context learning.

Core Concepts and Definitions

Metacognitive Knowledge: The learner’s accumulated knowledge about their own cognitive processes and learning-relevant properties of data.

Skill Exemplar Repository: Formally: $\text{Repository}=(s_0,q_0^T,a_0^T),(s_1,q_1^T,a_1^T),\dots ,(s_n,q_n^T,a_n^T)$ where $s_i$ is a skill label, $(q_i^T,a_i^T)$ is a question-answer pair.

Two-Stage Skill Discovery:

• Stage 1: LLM assigns fine-grained skill labels to examples (~5000 for MATH dataset)
• Stage 2: LLM performs semantic clustering to obtain coarse skill families (~117 for MATH)

Main Results

1. Skill-based in-context exemplar selection improves accuracy on GSM8K and MATH for multiple LLMs
2. Skills discovered by strong LLMs (GPT-4) improve performance of weaker LLMs
3. The skill exemplar repository transfers across datasets

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1.5 Yu et al. (2023) — SKILL-MIX: A Flexible and Expandable Family of Evaluations for AI Models

Problem: Traditional LLM evaluations suffer from training-set contamination and saturation at high performance levels. How to evaluate LLM ability to flexibly combine learned skills—a key indicator of general-purpose AI capability?

Approach: SKILL-MIX evaluation: randomly pick $k$ skills from $N$ available, ask LLM to produce text combining all $k$ skills in context of a random topic.

Core Concepts and Definitions

Skills: 101 language skills from Wikipedia entries (e.g., metaphor, modus ponens, self-serving bias), each with definition and example.

Topics: 100 topics with low corpus probability.

SKILL-MIX$(k)$: Given $N$ skills and $T$ topics, sample random subset of $k$ skills and one topic, then prompt model to produce ~3 sentences demonstrating all $k$ skills in context of the topic.

Auto-grading: Uses GPT-4 and LLaMA-2-70B to grade responses on presence of skills, topic relevance, sentence count, and text sensibility.

Beyond Stochastic Parrot Criterion: Model surpasses “stochastic parrot” behavior if: ${\alpha }_k>\frac{3}{2}\sqrt{p_s^k\cdot p_t\cdot L}$ where ${\alpha }_k$ is “Ratio of Full Marks” on SKILL-MIX$(k)$, $p_s$ is skill frequency, $p_t$ is topic frequency, $L$ is training corpus size.

Main Results

1. GPT-4 achieves reasonable performance at $k=5$ with ${\alpha }_5\approx 0.12$
2. For $k\ge 5$, probability calculations show GPT-4 generates combinations not seen in training: $p_s^k{p}_{t}L\le 0.001\text{ for }k=6$
3. Performance follows approximate $Y=X^2$ relationship where $X$ is single-skill performance, $Y$ is composite performance
4. Most models saturate by $k=3$ or $k=4$; only GPT-4 performs well at $k=5$
5. Evidence of “cramming for leaderboards”—models ranking high on standard benchmarks underperform on SKILL-MIX
6. Filtering common skills (frequency >5%) makes evaluation significantly harder

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1.6 Fan et al. (2024) — Transformers Can Learn Meta-skills for Task Generalization in In-Context Learning

Problem: Can Transformers learn “meta-skills” that enable composition of basic skills to generalize to unseen task combinations? Prior work (Kirsch et al.) showed Transformers trained on $A,B$ fail on $A+B$ compositions.

Approach: Train on in-context learning of function classes and their compositions, test on held-out compositions.

Core Concepts and Definitions

Basic Skill: The ability to perform in-context learning (ICL) on a function class $\mathcal{A}$ (e.g., linear, quadratic, sine, sqrt, heaviside).

Composite Skill: ICL on composite function class, e.g., $\mathcal{A}+\mathcal{B}:=f\mid f=f_1+f_2,f_1\in \mathcal{A},f_2\in \mathcal{B}$.

Meta-skill: The high-level skill required for skill composition:

1. Identifying if in-context samples come from a composite function
2. Identifying the needed combination of basic ICL skills
3. Applying a composite ICL skill on-the-fly

Function Composition Operations:

• Addition: $f(x)=f_1(x)+f_2(x)$
• Maximum: $f(x)=\max (f_1(x),f_2(x))$
• Multiplexing: $f(x)=\{\begin{array}{llc}{f}_1(x)& x\in [0,0.5)f_2(x)& x\in [0.5,1]\end{array}$

ICL Loss Function: ${\mathbb{E}}_P\left[{\sum }_{i=0}^{H}l(M_{\theta }(P^i),f(x_{i+1}))\right]$ where $P^i=(x_1,f(x_1),\dots ,x_{i+1})$ and $l$ is $L_2$ loss.

Main Results

1. Partial Composition: Training on $\text{sine}+\text{sqrt}$ and $\text{sine}+\text{linear}$ enables ICL on $\text{sine}+\text{quad}$
2. Cross Composition: Training on $A+B,C+D$ enables generalization to $A+C$
3. Weak-to-strong generalization: Training on 2-function compositions enables performance on 3-5 function compositions
4. Orthogonal Basis Requirement: Training on orthogonal function bases (e.g., Fourier, Legendre polynomials) crucial for generalization
5. Unsupervised meta-skill learning: Models can identify input-output associations without explicit labels

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1.7 Lu et al. (2024) — SELF: Self-Evolution with Language Feedback

Problem: How can LLMs continuously self-improve without external rewards or human intervention? Self-refinement capability exists in top-tier models but is absent in smaller ones.

Approach: Two-phase framework: (1) Meta-skill learning teaches self-feedback and self-refinement, (2) Iterative self-evolution where model generates responses, refines them, filters high-quality data, and self-trains.

Core Concepts and Definitions

Meta-Skills (in SELF context):

1. Self-Feedback Ability: Evaluate own responses using natural language feedback: ${\tau }_{\varphi }(f|p,r)$
2. Self-Refinement Ability: Optimize responses based on self-feedback: ${\tau }_{\varphi }(\hat{r}|p,r,f)$

Meta-Skill Training Corpus: ${\mathcal{D}}_{\text{meta}}=(p,r,f,\hat{r})$ where $p$ is prompt, $r$ is initial response, $f$ is feedback, $\hat{r}$ is refined response.

Self-Refinement Distribution: $\Psi (\hat{r}|p):={\sum }_{r,f}{\tau }_{\varphi }(r|p)\cdot {\tau }_{\varphi }(f|p,r)\cdot {\tau }_{\varphi }(\hat{r}|p,r,f)$

Meta-Skill Learning Objective: ${\mathcal{L}}_{\text{meta}}(\varphi )=-{\mathbb{E}}_{(p,r,f,\hat{r})\sim {\mathcal{D}}_{\text{meta}}}\left[\log {\tau }_{\varphi }(f|p,r)+\log {\tau }_{\varphi }(\hat{r}|p,r,f)+\beta \log {\tau }_{\varphi }(\hat{r}|p)\right]$

Self-Evolution Training (iteration $t$): ${\mathcal{L}}_{\text{evol}}^t(\varphi )=-{\mathbb{E}}_{p_{\text{evol}}}{\mathbb{E}}_{{\hat{r}}_{\text{evol}}\sim {\Psi }^{t-1}}\left[\log {\tau }_{\varphi }^t({\hat{r}}_{\text{evol}}|p_{\text{evol}})\right]$

Total Objective: ${\mathcal{L}}_{self}^t(\varphi )={\mathcal{L}}_{evol}^t(\varphi )+{\mathcal{L}}_{meta}^t(\varphi )$

Main Results

1. Vicuna-7B improved from 14.09% → 29.64% accuracy on GSM8K after 3 self-evolution iterations
2. Progressive improvement (+6.82% on GSM8K, +4.9% on SVAMP)
3. Meta-skill learning alone provides +6.82% boost (from baseline 14.09% to 20.91%)
4. Self-refinement during inference adds +2.58% on GSM8K
5. Self-refinement capability transfers to smaller models (previously emergent only in large models)
6. Meta-skill training implicitly improves direct response generation

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1.8 Lu et al. (2025) — Automated Capability Discovery (ACD)

Problem: How to systematically discover the full spectrum of capabilities and failure modes in foundation models?

Approach: Designate one FM as “scientist” to propose open-ended tasks for a “subject” model (possibly itself).

Core Concepts and Definitions

ACD Framework: Foundation model self-exploration where:

• Scientist model: Proposes new task families
• Subject model: Attempts tasks
• Scoring via programmatic checks or LLM judge

Task Family: Structured set of tasks including:

1. Specific task instances with unique data
2. Instruction provision for subject model
3. Scoring mechanism

Open-ended Archive: Maintains discovered tasks; at iteration $t$, samples artifact $a_t$ from model $M$ conditioned on context $C_{t-1}$.

Interestingness Filter: Uses embedding-based similarity to determine if proposed task is “interestingly new.”

Main Results

1. 5000 generations → 1330 “interestingly new” tasks → 25 distinct capability clusters
2. Human evaluation confirms high validity of auto-generated tasks
3. Self-assessment reasonably aligns with human judgments
4. Automatically generates “Capability Reports” summarizing discoveries

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1.9 Ganguli et al. (2022) — Predictability and Surprise in Large Generative Models

Problem: Reconcile the paradox that large generative models are highly predictable (via scaling-laws) yet unpredictable in specific capabilities and outputs.

Approach: Analyze the combination of predictability and unpredictability features and their policy implications.

Core Concepts and Definitions

Smooth General Capability Scaling: Model performance improves as power law in compute, data, parameters:

• $L\propto C^{-{\alpha }_C}$ (compute)
• $L\propto D^{-{\alpha }_D}$ (data)
• $L\propto N^{-{\alpha }_N}$ (parameters)

Abrupt Specific Capability Scaling: Specific capabilities can suddenly emerge at particular scales, unpredictable from smaller models. Examples:

• GPT-3 3-digit addition: <1% accuracy (N<6B) → 80% accuracy (N=175B)
• Gopher MMLU: ~30% accuracy (N<6B) → 60% accuracy (N=280B)

Open-Endedness: Models can produce outputs for essentially any input, making comprehensive testing impossible.

Distinguishing Features Identified:

1. Smooth general capability scaling
2. Abrupt specific capability scaling
3. Unknown specific capabilities until tested
4. Open-ended outputs

Main Results

1. Scaling laws enable prediction of general performance but not specific capabilities
2. Specific capability emergence can be abrupt even when general loss improves smoothly
3. Analogy: daily weather (specific, volatile) vs. seasonal averages (general, predictable)
4. Economic value analysis shows language models increasingly function as recommendation systems with scale
5. Recommendations for policy: continuous monitoring, staged deployment, capability-discovery protocols

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1.10 Darlow et al. (2025) — Continuous Thought Machines

Problem: Modern NNs abstract away temporal neural dynamics. Can we build architectures that leverage neural timing and synchronization as core computational principles?

Approach: Introduce Continuous Thought Machine (CTM) with neuron-level temporal processing and neural synchronization as latent representation.

Core Concepts and Definitions

Continuous Thought Machine (CTM): Architecture with:

1. Internal tick dimension $t\in 1,\dots ,T$, decoupled from data dimensions
2. Neuron-level models (NLMs): Each neuron has private weights processing activation histories
3. Neural synchronization: Correlation structure across neurons as latent representation

Synapse Model: Generates pre-activations: ${\mathbf{a}}^t=f_{{\theta }_{\text{syn}}}(\text{concat}({\mathbf{z}}^t,{\mathbf{o}}^t))\in {\mathbb{R}}^D$

Pre-activation History: ${\mathbf{A}}^t=[{\mathbf{a}}^{t-M+1},{\mathbf{a}}^{t-M+2},\dots ,{\mathbf{a}}^t]\in {\mathbb{R}}^{D\times M}$

Synchronization Matrix: ${\mathbf{S}}^t=\text{sync}({\mathbf{Z}}^t)$ computed from post-activation history, used as latent representation.

Main Results

1. Solves 2D mazes via internal map formation without positional encodings
2. Learns to “look around” images before classifying (emergent adaptive attention)
3. Native adaptive computation time as emergent property
4. Generalizes to longer sequences in parity computation

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1.11 Chen et al. (2024) — Schema-Guided Scene-Graph Reasoning (SG²)

Problem: LLMs struggle with spatial reasoning over scene graphs due to distraction by redundant information.

Approach: Multi-agent “Reason-while-Retrieve” framework with schema-guided graph query generation.

Core Concepts and Definitions

SG² Framework: Two-module architecture:

1. Reasoner: Decomposes task, generates natural language information queries
2. Retriever: Translates queries into executable graph programs

Scene Graph Schema: Abstract structure that:

• Guides schema-aligned query generation
• Enables structure-aware reasoning
• Provides API for graph database operations

Reason-while-Retrieve Strategy: Iterative retrieval and reasoning, avoiding full graph prompting.

Main Results

1. Outperforms single-agent tool-based approaches (12% improvement over ReAct baseline)
2. Reduces hallucination by filtering irrelevant graph data
3. Effective on numerical Q&A and planning tasks

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1.12 Arora Talk Transcript — LLM Skills and Metacognition

Key Methods Described:

1. Skill Labeling Approach: Four-word underscore-separated format (e.g., “circle_properties_area_calculation”)

2. Skill Extraction from Wikipedia: Named language skills with Wikipedia entries as baseline

3. Direct Elicitation: Prompt for “broad skill with no existing name” → e.g., “linguistic exorcism”

4. Context-Enhanced Learning: Training with contextual information (phrasebooks) that is dropped at test time with curriculum:

   • Phase 1: Train with random contexts to learn context-usage pattern
   • Phase 2: Train with target context + dropout to internalize
   • Test: Full dropout → measure internalization (random → target with 20% dropout)

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2. Meta-Analysis

2.1 Problems Addressed and Their Relations

Problem Taxonomy

Paper	Core Problem	Problem Category
Arora & Goyal (2023)	How complex skills emerge with scale	Emergence Theory
Wei et al. (2022)	Documenting emergent abilities	Emergence Characterization
Michaud et al. (2024)	Explaining scaling laws + emergence	Emergence Mechanism
Didolkar et al. (2024)	Can metacognition improve LLM performance?	Metacognition Application
Yu et al. (2023)	Evaluating skill composition	Skill Evaluation
Fan et al. (2024)	Learning meta-skills for task generalization	Meta-skill Learning
Lu (SELF, 2024)	Autonomous self-improvement	Self-Evolution
Lu (ACD, 2025)	Automated capability discovery	Capability Discovery
Ganguli et al. (2022)	Predictability-unpredictability paradox	Safety/Policy
Darlow et al. (2025)	Temporal dynamics in neural computation	Architecture
Chen et al. (2024)	Spatial reasoning with scene graphs	Reasoning Application

Problem Clustering

Cluster A: Capability Evaluation & Measurement

• SKILL-MIX: How to evaluate skill composition ability resistant to contamination
• Emergent Abilities: What capabilities emerge and when
• Predictability & Surprise: How to predict specific vs. general capability scaling

Relationships: SKILL-MIX operationalizes evaluation of emergent composition abilities identified by Wei et al., while Ganguli et al. explain why specific capabilities (like those tested in SKILL-MIX) emerge unpredictably despite smooth general scaling.

Cluster B: Theoretical Understanding of Emergence

• Arora & Goyal Theory: Why do skills and skill combinations emerge with scaling?
• Emergent Abilities: Empirical catalog of what emerges
• Predictability & Surprise: Smooth general vs. abrupt specific scaling patterns

Relationships: Arora & Goyal provide mathematical explanation for emergence phenomena documented empirically by Wei et al. and Ganguli et al. All three address the “stochastic parrots” debate about genuine understanding vs. pattern matching.

Cluster C: Meta-learning & Composition

• Transformers Meta-skills: Can models learn to compose skills via ICL?
• SKILL-MIX: Can models apply multiple skills simultaneously?
• SELF: Can models develop meta-skills for self-improvement?

Relationships: All three investigate meta-cognitive capabilities. Fan et al. show meta-skills can be learned for function composition, SKILL-MIX tests whether pre-trained models have acquired these naturally, and SELF demonstrates explicit meta-skill training enables self-evolution.

Cluster D: Training & Optimization

• SELF: Autonomous improvement via self-generated data
• Arora & Goyal Theory: How scaling drives skill acquisition
• Transformers Meta-skills: Task distribution design for meta-skill learning

Cross-cluster Insight: The problems form a progression: Arora & Goyal explain why skills emerge → Wei et al. / Ganguli et al. document what emerges → SKILL-MIX / Fan et al. test composition abilities → SELF leverages these for autonomous improvement.

Structural Relations

• Emergence Cluster: Arora & Goyal ↔ Wei et al. ↔ Michaud et al. form a theoretical progression: Wei documents the phenomenon, Arora provides a skill-based framework, Michaud offers a quantization mechanism.

• Skill/Metacognition Cluster: Arora & Goyal → Didolkar et al. → Yu et al. → Fan et al. → Lu (SELF) form a methodological chain from theory to evaluation to learning to self-improvement.

• Practical Application: ACD (Lu 2025) and SG² (Chen 2024) apply skill/capability concepts to automated-evaluation and structured reasoning respectively.

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2.2 Methods and Their Relations

Method Taxonomy

Statistical/Theoretical Methods

Method	Paper(s)	Purpose
Random Graph Theory	Arora & Goyal	Models skill-text relationships as bipartite graph $(S,T,E)$; uses concentration inequalities for emergence analysis
Scaling Law Analysis	Arora & Goyal, Ganguli et al.	Chinchilla law: $L(N,D)=A+B{N}^{-0.34}+C{D}^{-0.28}$; power law relationships between scale and capability

Comparison: Arora & Goyal use scaling laws as input to prove emergence theorems, while Ganguli et al. use them as descriptive tools to explain predictability patterns.

Skill Representation Methods

Method	Paper(s)	Representation
Bipartite Skill Graph	Arora & Goyal	$G=(V_{\text{skills}},V_{\text{text}},E)$
Skill Exemplar Repository	Didolkar et al.	${(s_i,q_i,a_i)}_i$
Function Class Composition	Fan et al.	$\mathcal{A}+\mathcal{B}=f_1+f_2$
Quanta	Michaud et al.	Discrete modules with use frequency

Synthetic Evaluation Methods

Method	Paper(s)	Description
SKILL-MIX Prompting	Yu et al.	Generate text on topic demonstrating $k$ skills; auto-grade with GPT-4/LLaMA-2-70B
ICL Function Composition	Fan et al.	Training on $(x_i,f(x_i))$ pairs → predict $f(x_{H+1})$; test on held-out compositions

Comparison: Both use synthetic tasks to isolate specific capabilities. SKILL-MIX tests natural language skill composition with semantic evaluation, while Fan et al. test mathematical function composition with exact loss metrics. SKILL-MIX prioritizes ecological validity (text generation), Fan et al. prioritize theoretical clarity (function spaces).

Evaluation Frameworks

Framework	Paper(s)	Core Metric
SKILL-MIX$(k)$	Yu et al.	Skill Fraction, Full Marks Ratio
Competence on $k^{\prime }$-tuples	Arora & Goyal	Success rate on cloze questions
ICL Test Error	Fan et al.	$L_2$ prediction error on held-out compositions
ACD Task Families	Lu (2025)	Success/failure rates on generated tasks

Self-Evolution Methods

Method	Paper(s)	Mechanism
Context-Enhanced Learning	Arora (talk)	Train with contextual info, test without
Self-Evolution Training	Lu (SELF)	Iterative refinement via self-feedback
Meta-skill ICL	Fan et al.	Learn composition operators from examples

Key Methodological Innovation: Arora & Goyal’s use of Scaling Laws as an assumption rather than something to derive represents a paradigm shift—accepting empirical regularities to prove theorems about emergence without needing mechanistic gradient descent analysis.

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2.3 Concepts and Definitions: Cross-Paper Comparison

Definition 1: “Skill” Definitions

Paper	Definition	Granularity
Arora & Goyal	Node in skill graph; comprehension requirement for text-pieces	Abstract (set $S$)
Didolkar et al.	LLM-assigned label from hierarchical clustering (e.g., “circle_properties_area_calculation”)	Fine-grained → Coarse
Yu et al.	Wikipedia-documented language/reasoning skill	Named (101 skills)
Fan et al.	ICL capability on a function class	Functional ($\mathcal{A},\mathcal{B},\dots$)
Michaud et al.	Quantum (discrete knowledge/skill chunk)	Discrete module
Lu (SELF)	Meta-skill: Self-feedback + Self-refinement	Procedural

Analysis of Differences:

Aspect	SKILL-MIX	Arora & Goyal	SELF	Fan et al.
Granularity	Named linguistic skills	Abstract, unspecified	Meta-cognitive processes	Mathematical functions
Observability	Human-identifiable	Latent graph structure	Explicit in training	Precisely defined
Compositionality	Simultaneous application	Random co-occurrence	Sequential refinement	Algebraic operations
Domain	Natural language	General (agnostic)	Any task domain	Function spaces

Key Distinction: Arora & Goyal and Michaud et al. treat skills as abstract units in a theoretical framework, while Didolkar et al. and Yu et al. treat them as named, human-interpretable categories. Fan et al. operationalizes skills as function classes for controlled experiments.

Reconciliation: These definitions operate at different levels of abstraction:

• Fan et al.: Most concrete (mathematical functions as skills)
• SKILL-MIX: Intermediate (named linguistic capabilities)
• Arora & Goyal: Most abstract (allows any definition satisfying graph structure)
• SELF: Meta-level (skills for managing skills)

The hierarchy suggests: Mathematical functions ⊂ Named linguistic skills ⊂ Any capabilities forming graph structure ⊃ Meta-skills operating on skills

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Definition 2: “Emergence” Definitions

Paper	Definition	Characterization
Wei et al.	Ability present in larger models, absent in smaller	Binary threshold
Arora & Goyal	Improvement in competence on skills and skill-tuples with scaling	Continuous (random graph analysis)
Michaud et al.	Sharp transition when quantum is learned (monogenic) vs. gradual improvement (polygenic)	Threshold vs. gradual
Fan et al.	Ability to perform ICL on unseen compositions after training on subset	Operational (generalization metrics)

Formal Characterizations:

Wei et al. Definition:

• Emergent Ability: Capability not present in smaller models but appearing in larger ones
• Criteria: Ability absent below threshold, present above it
• Measurement: Task performance vs. model scale

Arora & Goyal Definition: $\text{Emergence of skill }s:\text{competence on }s\text{ increases as }\theta \downarrow$ where $\theta$ is the fraction of text pieces with errors.

Performance Curve $C(k,\theta )$: Boundary of $(\alpha ,\beta )$ pairs satisfying: $H(\theta )+k\theta (H(\beta \alpha )-\beta \alpha \log \frac{1}{\alpha }-(1-\beta \alpha )\log \frac{1}{1-\alpha })<0$

Ganguli et al. Distinction:

• Smooth general emergence: Predictable improvement on broad distributions
• Abrupt specific emergence: Discontinuous transitions on individual tasks
• Mathematical form: General follows power law, specific shows phase transitions

Critical Difference: Wei et al. and Ganguli et al. define emergence empirically (observational), while Arora & Goyal define it theoretically (via mathematical conditions). Fan et al. define it operationally (via generalization metrics).

Reconciliation: Wei et al.’s “emergence” is a special case of Michaud’s monogenic scaling. Arora & Goyal’s framework explains emergence as improvement in $k^{\prime }$-tuple competence—a model trained on $D$ tokens can display competence on skill combinations numbering $\gg D$, creating apparent “sudden” abilities when tested.

Apparent Contradiction — Resolution:

Ganguli et al. observe abrupt specific emergence while Arora & Goyal predict smooth emergence. Resolution:

1. Arora & Goyal’s smoothness applies to competence on skill sets, not individual tasks
2. Individual tasks (Ganguli’s “specific”) may combine multiple skills non-linearly
3. A task requiring perfect execution of $k$ skills shows threshold behavior even if individual skill competence improves smoothly
4. Mathematical formulation: If task success $\propto {\prod }_{i=1}^k{p}_i$ (all skills needed), then:
   • Individual $p_i$ increases smoothly
   • Product $\prod p_i$ can show sharp threshold

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Definition 3: Competence/Performance

Paper	Definition	Aggregation Level	Probabilistic?
Arora & Goyal	$\mathbb{P}[\text{correct cloze}\mid \text{text piece adjacent to }s]$	Per skill	Yes (graph probability)
SKILL-MIX	Full Marks Ratio ${\alpha }_k$: Fraction of $\left(\genfrac{}{}{0ex}{}{N}{k}\right)T$ combinations receiving perfect scores	Per $k$-tuple	Yes (over random samples)
Fan et al.	${\mathbb{E}}_{f\sim \mathcal{F}}\left[{\mathbb{E}}_x\left[\ell (M_{\theta }(P),f(x))\right]\right]$	Per function class	Yes (over functions & inputs)
SELF	Direct Response Accuracy / Self-Refinement Accuracy	Per problem	No (deterministic eval)

Insight: All definitions measure success rate but differ in:

1. Unit of evaluation: Skills (Arora & Goyal), skill tuples (SKILL-MIX), function classes (Fan et al.), problems (SELF)
2. Success criterion: Cloze correctness, multi-aspect scoring, loss threshold, exact match

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Definition 4: “Meta-skill” Definitions

Paper	Definition	Mechanism
Fan et al.	High-level skill for composing basic ICL skills	Learned operator: (identify composite → decompose → apply)
Lu (SELF)	Self-feedback + Self-refinement abilities	Procedural: generate → evaluate → refine
Arora (talk)	LLM’s ability to recognize and classify its own skill usage	Introspective: naming + organizing skill taxonomies

Common Thread: All three treat meta-skills as second-order capabilities—operating on skills rather than tasks directly. Fan et al. focuses on composition operators, Lu (SELF) on evaluation/refinement operators, and Arora on introspection/classification operators.

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Definition 5: “Beyond Stochastic Parrots”

SKILL-MIX Criterion: A model is “beyond stochastic parrots” if: ${\alpha }_k>\frac{3}{2}\sqrt[k]{p_s^k{p}_{t}L}$ where:

• ${\alpha }_k$: Full marks ratio on SKILL-MIX$(k)$
• $p_s$: Maximum skill frequency in corpus
• $p_t$: Maximum topic frequency
• $L$: Training corpus size

Interpretation: Model generates more successful $k$-combinations than expected if memorizing corpus combinations.

Arora & Goyal Implicit Criterion: Model shows “slingshot generalization” when: $\text{Competence on }{k}^{\prime }\text{-tuples}>\text{Expected from }|S{|}^{k^{\prime }}\text{ independent samples}$

Given training on $D$ tokens, competence on $k^{\prime }$-tuples where $D{c}^{k^{\prime }}\ll |S{|}^{k^{\prime }}$ indicates going beyond memorization (paucity of stimulus).

Comparison:

Aspect	SKILL-MIX	Arora & Goyal
Method	Probabilistic counting	Combinatorial argument
Threshold	Quantitative inequality	Asymptotic comparison
Mechanism	Novel combinations generated	Competence despite paucity
Evidence Type	Direct (generation counts)	Indirect (competence on rare tuples)

Philosophical Difference:

• SKILL-MIX: Operational definition (model does something novel)
• Arora & Goyal: Capacity definition (model can do more than training allows)

Both converge on: Genuine understanding requires handling combinations not explicitly trained.

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Definition 6: “Scaling” Models

Paper	Scaling Relation	Key Exponent
Ganguli et al.	$L\propto C^{-{\alpha }_C}\propto D^{-{\alpha }_D}\propto N^{-{\alpha }_N}$	Power law (empirical)
Michaud et al.	$L(N)-L_{\infty }\propto N^{-\alpha }$ from Zipfian quanta frequencies	$\alpha$ from $p_k\propto k^{-(\alpha +1)}$
Arora & Goyal	Scaling model with 10x → 2x skill-tuple complexity	$k^{\prime }\approx 2k$ when model scales 10x

Unification: Michaud provides a mechanism (quanta + Zipf) underlying the empirical scaling-laws that Ganguli et al. document. Arora & Goyal connect scaling to skill composition capacity, showing that the same scaling improvement on individual skills propagates to $k^{\prime }$-tuples.

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Compositional Generalization

Paper	Setting	Key Finding
Arora & Goyal	Skill $k^{\prime }$-tuples	Competence emerges despite $\Vert S{\Vert }^{k^{\prime }}\gg D$
Yu et al. (SKILL-MIX)	$k$ skills + topic	GPT-4 generates novel combinations at $k=5$
Fan et al.	Function class composition	Training on $A+B,C+D$ → generalize to $A+C$

Synthesis: All three demonstrate that models can handle novel combinations of learned components. The key insight is that compositional generalization is implicit in successful pretraining—models don’t memorize compositions but learn underlying composition operators.

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2.4 Key Cross-Paper Synthesis

Unified Framework Emerging from Papers

1. Skills exist (various definitions but consistent concept)
2. Skills can be composed (Arora & Goyal: random tuples; SKILL-MIX: prompted combinations; Fan et al.: function operations)
3. Composition ability emerges with scale (All papers agree)
4. Emergence follows mathematical regularities (Arora & Goyal: theorems; Ganguli et al.: power laws; SKILL-MIX: probability calculations)
5. Meta-skills enable higher-order composition (SELF, Fan et al.)
6. True capability exceeds memorization (SKILL-MIX criterion, Arora & Goyal paucity of stimulus)

Remaining Tensions

1. Smooth vs. Abrupt Emergence: Resolved by distinguishing aggregate skill competence (smooth) from individual task performance (can be abrupt)

2. Skill Definition: Unresolved across papers—ranges from abstract graph nodes to specific named capabilities. This flexibility may be feature not bug: allows framework to apply across domains.

3. Measurement Challenges: Each paper uses different metrics, making quantitative cross-comparison difficult. Need unified benchmark incorporating all approaches.

Future Research Directions Implied

1. Unified skill taxonomy spanning mathematical functions → linguistic skills → meta-skills → algebraic frameworks
2. Formal connection between smooth competence emergence (Arora & Goyal) and abrupt task emergence (Ganguli et al.)
3. Scaling laws for meta-skill acquisition (combining SELF + Arora & Goyal frameworks)
4. Evaluation suite combining SKILL-MIX’s naturalness with Fan et al.’s precision
5. Ontological frameworks for skill discovery and expansion

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3. References

1. Arora, S., & Goyal, A. (2023). A Theory for Emergence of Complex Skills in Language Models.
2. Wei, J., et al. (2022). Emergent Abilities of Large Language Models.
3. Michaud, E. J., et al. (2024). The Quantization Model of Neural Scaling.
4. Didolkar, A., et al. (2024). Metacognitive Capabilities of LLMs: An Exploration in Mathematical Problem Solving.
5. Yu, D., et al. (2023). SKILL-MIX: A Flexible and Expandable Family of Evaluations for AI Models.
6. Fan, Y., et al. (2024). Transformers Can Learn Meta-skills for Task Generalization in In-Context Learning.
7. Lu, X., et al. (2024). SELF: Self-Evolution with Language Feedback.
8. Lu, X., et al. (2025). Automated Capability Discovery via Foundation Model Self-Exploration.
9. Ganguli, D., et al. (2022). Predictability and Surprise in Large Generative Models.
10. Darlow, L. N., et al. (2025). Continuous Thought Machines.
11. Chen, Z., et al. (2024). Schema-Guided Scene-Graph Reasoning based on Multi-Agent Large Language Models.

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This analysis reveals a remarkably coherent research program across papers, despite different formalism and terminology. The field is converging on a compositional view of LLM capabilities with mathematical foundations.