Project Roadmap

Project Plan: Algebraic Framework for Skill Generation and Self-Improving LLM Agents

Executive Summary

This project aims to develop a rigorous algebraic framework for modeling skill composition in LLMs and implement an agent capable of autonomously generating new skills to improve performance on given tasks. Building on recent theoretical advances (Arora & Goyal, 2023) and empirical findings (SKILL-MIX, SELF, Transformers Meta-skills), we will formalize skill creation mechanisms and develop a self-improving agent that can introspect, assess skill gaps, and synthesize novel capabilities.

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Phase 1: Literature Review and Conceptual Foundation (Weeks 1-3)

1.1 Core Papers Analysis

Completed deliverables:

• Comprehensive overview of 10 papers in project knowledge
• Cross-referenced comparison of problems, methods, and definitions
• Identification of conceptual tensions and synthesis opportunities

Key findings:

• Three distinct skill definitions requiring unification
• Emergence can be smooth (competence) and abrupt (tasks) simultaneously
• Meta-skills enable compositional generalization
• “Beyond stochastic parrots” has formal mathematical characterization

1.2 Extended Literature Review (Week 2-3)

Additional readings:

• Quantization Model of Neural Scaling (if available)
• Mechanistic interpretability papers (Anthropic circuit analysis)
• Formal ontology and category theory literature for compositional structures
• Recent work on skill extraction and labeling systems

Deliverable: Annotated bibliography with focus on:

• Formal composition operators
• Skill ontology proposals
• Self-improvement mechanisms in RL and meta-learning

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Phase 2: Definitional Framework Development (Weeks 4-6)

2.1 Unified Skill Taxonomy

Objective: Reconcile different skill definitions into coherent hierarchy

Approach:

1. Atomic Skills: Define primitive capabilities that cannot be decomposed

   • Question: Do atomic skills exist, or is decomposition context-dependent?
   • Hypothesis: Atomicity is relative to model architecture and training data

2. Composite Skills: Skills formed through composition operations

   • Operations: Sequential, parallel, conditional, hierarchical
   • Formalize as: $s_{\text{comp}}=\mathcal{O}(s_1,s_2,\dots ,s_k)$ where $\mathcal{O}$ is composition operator

3. Meta-Skills: Skills that operate on skills

   • Self-feedback, self-refinement (SELF paper)
   • Skill identification, skill creation, skill composition planning

Deliverable: Formal taxonomy document with:

• Mathematical definitions for each skill type
• Composition algebra specification
• Worked examples from SKILL-MIX dataset

2.2 Operations Over Skills

Composition Operators to Define:

Operator	Symbol	Description	Example
Sequential	$s_1\circ s_2$	Apply $s_2$ after $s_1$	Summarize $\circ$ Translate
Parallel	$s_1\oplus s_2$	Apply both simultaneously	Metaphor $\oplus$ Modus ponens
Conditional	$s_1◃c▹s_2$	Choose based on condition	IF complex THEN decompose ELSE direct
Hierarchical	$s_1\succ s_2$	$s_1$ governs application of $s_2$	Planning $\succ$ Execution

Key Questions:

• Is composition commutative? When is $s_1\oplus s_2=s_2\oplus s_1$?
• Can all compositions be uniquely decomposed?
• What is the closure property? (composing skills always yields valid skills?)

Deliverable: Technical report defining algebra with:

• Axioms and properties (associativity, commutativity, etc.)
• Proofs of key theorems (closure, uniqueness of decomposition)
• Connection to existing frameworks (Arora & Goyal’s graph structure)

2.3 Performance Metrics for Skill Evaluation

Extend existing metrics:

1. Single-Skill Competence (from Arora & Goyal): $\text{Comp}(s)=\mathbb{P}[\text{success}\mid \text{task requires }s]$

2. Compositional Competence (generalize SKILL-MIX): $\text{Comp}(\mathcal{O}(s_1,\dots ,s_k))=\mathbb{P}[\text{success}\mid \text{task requires }\mathcal{O}(s_1,\dots ,s_k)]$

3. Meta-Skill Competence (from SELF): $\text{MetaComp}(m)={\mathbb{E}}_T[\text{improvement on }T\mid \text{applying meta-skill }m]$

4. Novelty Score (based on SKILL-MIX’s “beyond stochastic parrots”): $\text{Novelty}(s)=1-{\max }_{s^{\prime }\in S_{\text{training}}}\text{Similarity}(s,s^{\prime })$

Deliverable: Evaluation protocol document

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Phase 3: Mechanisms for Skill Creation (Weeks 7-10)

3.1 Synthesis from Literature

Mechanism 1: Compositional Synthesis (from Fan et al.)

• Train on basic skills $\{s_1,\dots ,s_n\}$ with limited compositions
• Emergent ability to compose unseen combinations
• Implementation: Fine-tune on skill pairs, test on triples/higher

Mechanism 2: Context-Enhanced Learning (from Arora transcription)

• Provide skill description in context during training
• Use dropout to force internalization
• Implementation: Two-phase curriculum with progressive dropout

Mechanism 3: Self-Evolution (from SELF)

• Meta-skills enable autonomous data generation
• Iterative refinement with quality filtering
• Implementation: Generate → Critique → Refine → Filter → Train loop

Mechanism 4: Skill Extraction from Tasks (novel contribution)

• Given task $T$ where model fails, extract required skills
• Compare to available skills, identify gaps
• Synthesize new skills to fill gaps

3.2 Addressing “Stochastic Parrot” and “Leaderboard Cramming” Concerns

Stochastic Parrot Prevention:

• Use SKILL-MIX’s probabilistic novelty verification: ${\alpha }_k>\frac{3}{2}\sqrt[k]{p_s^k{p}_{t}L}$
• Create evaluation sets with $\left(\genfrac{}{}{0ex}{}{N}{k}\right)$ combinations where $\left(\genfrac{}{}{0ex}{}{N}{k}\right)\gg L$
• Test on held-out skill categories

Leaderboard Cramming Prevention:

• Release only 10% of skills/topics (SKILL-MIX approach)
• Rotate evaluation sets periodically
• Focus on compositional generalization metrics, not memorization

Deliverable:

• Risk analysis document
• Mitigation strategies for each mechanism
• Verification protocols

3.3 Research Questions: Ontological Novelty

Question 1: Formally, in which sense can skills be composed into “ontologically” novel skills?

Approach:

• Define ontological novelty: $s_{\text{new}}$ is ontologically novel if it cannot be expressed as $\mathcal{O}(s_1,\dots ,s_k)$ for any $s_i\in S_{\text{base}}$ and operator $\mathcal{O}\in {\Omega }_{\text{known}}$
• Investigate whether composition always stays within closure of base skills, or if emergence creates genuinely new primitives
• Connection to Arora & Goyal’s “slingshot generalization”

Question 2: Are there atomic skills?

Approach:

• Empirically: Attempt to decompose SKILL-MIX’s 101 skills into smaller components
• Theoretically: Prove/disprove existence of irreducible skill basis
• Hypothesis: Atomicity is relative to model capacity and training distribution

Question 3: Is compositional decomposition unique?

Approach:

• Test whether $s_{\text{comp}}={\mathcal{O}}_1(s_1,s_2)={\mathcal{O}}_2(s_3,s_4)$ for different decompositions
• If non-unique, characterize equivalence classes
• Practical implication: Multiple paths to achieve same capability

Deliverable: Technical paper addressing these questions with formal proofs/counterexamples

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Phase 4: Algebraic Framework Development (Weeks 11-14)

4.1 Core Framework Structure

Mathematical Objects:

1. Skill Space $\mathcal{S}$:

   • Set of all possible skills
   • Equipped with similarity metric $d:\mathcal{S}\times \mathcal{S}\to {\mathbb{R}}^{+}$
   • Partition into equivalence classes: $\mathcal{S}={\bigcup }_i{S}_i$ where $S_i$ are skill categories

2. Operation Algebra $(\Omega ,\circ )$:

   • $\Omega$: Set of composition operators
   • $\circ$: Operator composition (how operators combine)
   • Properties: closure, associativity, identity element

3. Skill Generation Function $\mathcal{G}:{\mathcal{S}}^k\times \Omega \to \mathcal{S}$: $s_{\text{new}}=\mathcal{G}(s_1,\dots ,s_k;\mathcal{O})$

4. Task-Skill Mapping $\Phi :\mathcal{T}\to 2^{\mathcal{S}}$:

   • Maps task $T$ to set of required skills
   • Extends Arora & Goyal’s bipartite graph
   • Formalization: $\Phi (T)=\{s\in \mathcal{S}\mid s\text{ necessary for }T\}$

Axioms:

A1 (Closure): $\forall s_1,\dots ,s_k\in \mathcal{S},\mathcal{O}\in \Omega :\mathcal{G}(s_1,\dots ,s_k;\mathcal{O})\in \mathcal{S}$

A2 (Identity): $\exists s_{\text{id}}\in \mathcal{S}:\mathcal{G}(s,s_{\text{id}};\oplus )=s$ for all $s$

A3 (Associativity - conditional): $(s_1\oplus s_2)\oplus s_3=s_1\oplus (s_2\oplus s_3)$ for parallel composition

A4 (Monotonicity): If $\text{Comp}(s_i)>\text{Comp}(s_i^{\prime })$ for all $i$, then $\text{Comp}(\mathcal{G}(s_1,\dots ,s_k;\mathcal{O}))>\text{Comp}(\mathcal{G}(s_1^{\prime },\dots ,s_k^{\prime };\mathcal{O}))$

A5 (Decomposition): $\exists \mathcal{D}:\mathcal{S}\to 2^{\mathcal{S}\times \Omega }$ such that $\mathcal{D}(\mathcal{G}(s_1,\dots ,s_k;\mathcal{O}))$ contains $(s_1,\dots ,s_k,\mathcal{O})$

4.2 Connection to Existing Frameworks

Integration with Arora & Goyal:

• Their skill graph $(S,T,E)$ becomes ${\Phi }^{-1}$: for each $s$, find tasks requiring it
• Their random $k$-tuple model corresponds to parallel composition ${⨁}_{i=1}^k{s}_i$
• Their emergence theorem predicts $\text{Comp}({⨁}_{i=1}^{k^{\prime }}{s}_i)$ as function of scaling

Integration with SKILL-MIX:

• Their evaluation SKILL-MIX($k$) tests $\text{Comp}({⨁}_{i=1}^k{s}_i)$ for random $k$-tuples
• Their auto-grading provides empirical estimates of $\text{Comp}$
• Their novelty criterion formalizes when ${⨁}_{i=1}^k{s}_i\notin S_{\text{training}}$

Integration with SELF:

• Meta-skills $m\in \mathcal{M}\subset \mathcal{S}$ satisfy: $m:\mathcal{S}\to \mathcal{S}$ (skills that transform skills)
• Self-feedback: $m_f(s,T)\to$ critique of $s$ on task $T$
• Self-refinement: $m_r(s,m_f(s,T))\to s^{\prime }$ (improved skill)

Deliverable:

• Formal algebraic specification document
• Proof document for key theorems
• Worked examples mapping concrete skills to algebraic operations

4.3 Implementation in Code

Develop Python library:

class Skill:
    def __init__(self, name, description, examples):
        self.name = name
        self.description = description
        self.examples = examples
        self.embedding = self._compute_embedding()
    
    def similarity(self, other_skill):
        """Compute semantic similarity"""
        return cosine_similarity(self.embedding, other_skill.embedding)
 
class CompositionOperator:
    def __init__(self, name, operation_type):
        self.name = name
        self.type = operation_type  # 'sequential', 'parallel', etc.
    
    def apply(self, skills: List[Skill]) -> Skill:
        """Generate composite skill"""
        raise NotImplementedError
 
class SkillAlgebra:
    def __init__(self):
        self.skills = {}
        self.operators = {}
    
    def compose(self, skills, operator):
        """Core composition function"""
        return operator.apply(skills)
    
    def decompose(self, composite_skill):
        """Attempt to find constituent skills"""
        # Implement decomposition algorithm
        pass
    
    def assess_novelty(self, skill, training_corpus):
        """Check if skill is beyond training distribution"""
        # Implement SKILL-MIX criterion
        pass

Deliverable:

• Python package with documentation
• Unit tests for algebraic properties
• Jupyter notebooks with examples

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Phase 5: Agent Design - Self-Improving System (Weeks 15-20)

5.1 Agent Architecture

Core Components:

┌─────────────────────────────────────────────────────────┐
│                    SELF-IMPROVING AGENT                  │
├─────────────────────────────────────────────────────────┤
│                                                           │
│  ┌──────────────┐      ┌────────────────┐              │
│  │   Task       │─────▶│  Performance   │              │
│  │  Assessor    │      │   Evaluator    │              │
│  └──────────────┘      └────────────────┘              │
│         │                       │                        │
│         ▼                       ▼                        │
│  ┌──────────────────────────────────────┐              │
│  │      Skill Gap Analyzer              │              │
│  │  - Required skills: Φ(T)             │              │
│  │  - Available skills: S_current        │              │
│  │  - Gap: Φ(T) \ S_current             │              │
│  └──────────────────────────────────────┘              │
│         │                                                │
│         ▼                                                │
│  ┌──────────────────────────────────────┐              │
│  │    Introspection Module              │              │
│  │  - Meta-skill inventory               │              │
│  │  - Similarity comparison              │              │
│  └──────────────────────────────────────┘              │
│         │                                                │
│         ▼                                                │
│  ┌──────────────────────────────────────┐              │
│  │   Skill Synthesis Planner            │              │
│  │  - Select composition strategy        │              │
│  │  - Design synthesis procedure         │              │
│  └──────────────────────────────────────┘              │
│         │                                                │
│         ▼                                                │
│  ┌──────────────────────────────────────┐              │
│  │   Skill Generator                    │              │
│  │  - Execute G(s₁,...,sₖ; O)          │              │
│  │  - Validate new skill                 │              │
│  └──────────────────────────────────────┘              │
│         │                                                │
│         ▼                                                │
│  ┌──────────────────────────────────────┐              │
│  │   Skill Integration & Testing        │              │
│  │  - Add s_new to S_current            │              │
│  │  - Re-evaluate on T                   │              │
│  └──────────────────────────────────────┘              │
│         │                                                │
│         ▼                                                │
│  ┌──────────────────────────────────────┐              │
│  │   Meta-Learning Module               │              │
│  │  - If improvement → reinforce         │              │
│  │  - If no improvement → modify         │              │
│  └──────────────────────────────────────┘              │
│                                                           │
└─────────────────────────────────────────────────────────┘

5.2 Detailed Component Specifications

5.2.1 Task Assessor & Performance Evaluator

Input: Task $T$, current model state $M_{\theta }$

Process:

1. Generate $n$ samples from model on task $T$
2. Compute performance metrics:
   • Accuracy/F1 for classification
   • BLEU/ROUGE for generation
   • Task-specific metrics
3. Compare to threshold $\tau$: $\text{Perf}(M_{\theta },T)<\tau$?

Output: Performance score $p\in [0,1]$, binary flag (below threshold?)

Implementation:

class TaskAssessor:
    def assess(self, model, task, threshold):
        samples = model.generate(task.prompts, n=100)
        performance = task.evaluate(samples)
        return {
            'performance': performance,
            'below_threshold': performance < threshold,
            'gap': threshold - performance
        }

5.2.2 Skill Gap Analyzer

Input: Task $T$, current skill set $S_{\text{current}}$

Process:

1. Extract required skills: $S_{\text{req}}=\Phi (T)$

   • Method 1: Manual annotation (for evaluation tasks)
   • Method 2: LLM-based extraction (prompt: “What skills are needed for this task?“)
   • Method 3: Error analysis (identify failure modes, map to skills)

2. Compare with available skills: $S_{\text{gap}}=\{s\in S_{\text{req}}\mid {\min }_{s^{\prime }\in S_{\text{current}}}d(s,s^{\prime })>ϵ\}$

3. Rank gaps by importance:

   • Estimate impact: ${\Delta }_{\text{perf}}(s)=$ expected improvement if $s$ acquired
   • Prioritize high-impact skills

Output: Ranked list of skill gaps

Implementation:

class SkillGapAnalyzer:
    def analyze(self, task, current_skills, embedding_model):
        required_skills = self.extract_required_skills(task)
        
        gaps = []
        for req_skill in required_skills:
            # Find closest existing skill
            similarities = [req_skill.similarity(curr_skill) 
                          for curr_skill in current_skills]
            max_similarity = max(similarities)
            
            if max_similarity < self.threshold:
                impact = self.estimate_impact(req_skill, task)
                gaps.append({
                    'skill': req_skill,
                    'similarity_to_closest': max_similarity,
                    'estimated_impact': impact
                })
        
        return sorted(gaps, key=lambda x: x['estimated_impact'], reverse=True)

5.2.3 Introspection Module

Input: Skill gap $s_{\text{gap}}$, current skill set $S_{\text{current}}$, meta-skill set $\mathcal{M}$

Process:

1. Self-assessment:

   • What skills do I have that are similar to $s_{\text{gap}}$?
   • Find $s_1,\dots ,s_k\in S_{\text{current}}$ that are semantically closest

2. Meta-skill inventory:

   • What composition operations can I perform? (${\Omega }_{\text{available}}$)
   • What meta-skills do I have? ($m_{\text{feedback}},m_{\text{refine}},\dots$)

3. Semantic comparison:

   • Decompose $s_{\text{gap}}$ into potential components
   • Compare with available skills
   • Identify bridging concepts

Output: Synthesis plan candidates

Implementation:

class IntrospectionModule:
    def introspect(self, gap_skill, current_skills, metaskills):
        # Find semantically similar skills
        similar_skills = self.find_k_nearest(gap_skill, current_skills, k=5)
        
        # Analyze gap structure
        gap_decomposition = self.attempt_decomposition(gap_skill)
        
        # Match with available operators
        synthesis_candidates = []
        for operator in self.operators:
            # Try different combinations of similar skills
            for skill_combo in itertools.combinations(similar_skills, operator.arity):
                feasibility = self.assess_feasibility(
                    skill_combo, operator, gap_skill
                )
                if feasibility > threshold:
                    synthesis_candidates.append({
                        'base_skills': skill_combo,
                        'operator': operator,
                        'feasibility': feasibility
                    })
        
        return sorted(synthesis_candidates, 
                     key=lambda x: x['feasibility'], 
                     reverse=True)

5.2.4 Skill Synthesis Planner

Input: Synthesis candidates from introspection

Process:

1. Strategy selection:

   • Compositional synthesis: $s_{\text{new}}=\mathcal{O}(s_1,s_2)$
   • Context-enhanced learning: Train with skill description in context
   • Self-evolution: Generate examples, refine, train
   • Hybrid: Combine multiple approaches

2. Procedure design:

   • Define training data requirements
   • Specify training hyperparameters
   • Plan evaluation protocol

3. Risk assessment:

   • Estimate probability of success
   • Identify potential failure modes
   • Plan contingencies

Output: Detailed synthesis procedure

5.2.5 Skill Generator

Input: Synthesis procedure

Process - Example for Compositional Synthesis:

def generate_skill_compositional(base_skills, operator, target_skill, 
                                 num_examples=1000):
    """
    Generate new skill through composition
    """
    # Step 1: Generate training examples
    examples = []
    for i in range(num_examples):
        # Create prompt requiring composed skills
        prompt = create_composite_prompt(base_skills, operator, topic=random_topic())
        
        # Generate with base model
        response = base_model.generate(prompt)
        
        # Grade using meta-skill (self-feedback)
        grade = metaskill_feedback(response, base_skills, operator)
        
        if grade > quality_threshold:
            examples.append((prompt, response))
    
    # Step 2: Fine-tune on examples
    new_model = fine_tune(base_model, examples, 
                         learning_rate=1e-5, 
                         epochs=3)
    
    # Step 3: Validate new skill
    validation_score = evaluate_skill(new_model, target_skill, 
                                     validation_set)
    
    if validation_score > success_threshold:
        return new_model, True
    else:
        return new_model, False

Process - Example for Context-Enhanced Learning:

def generate_skill_context_enhanced(target_skill, base_model, 
                                   num_examples=1000):
    """
    Generate new skill through context-enhanced learning
    """
    # Step 1: Create skill description context
    skill_context = f"""
    Skill: {target_skill.name}
    Description: {target_skill.description}
    Examples: {target_skill.examples}
    """
    
    # Phase 1: Train with random contexts (teach pattern)
    phase1_data = []
    for i in range(num_examples // 2):
        random_context = generate_random_skill_context()
        task = generate_task_requiring_skill(random_context.skill)
        phase1_data.append((random_context + task.prompt, task.answer))
    
    model_phase1 = fine_tune(base_model, phase1_data)
    
    # Phase 2: Train with target context + dropout
    phase2_data = []
    for i in range(num_examples // 2):
        task = generate_task_requiring_skill(target_skill)
        # With probability 0.8, include context
        if random.random() < 0.8:
            input_text = skill_context + task.prompt
        else:
            input_text = task.prompt
        phase2_data.append((input_text, task.answer))
    
    final_model = fine_tune(model_phase1, phase2_data)
    
    # Test with 100% dropout (no context)
    test_score = evaluate_skill(final_model, target_skill,
                               validation_set, context_dropout=1.0)
    
    return final_model, test_score > success_threshold

5.2.6 Skill Integration & Testing

Input: New skill $s_{\text{new}}$, updated model $M_{{\theta }^{\prime }}$

Process:

1. Add skill to inventory: $S_{\text{current}}\leftarrow S_{\text{current}}\cup \{s_{\text{new}}\}$
2. Re-evaluate on original task $T$
3. Compute improvement: $\Delta =\text{Perf}(M_{{\theta }^{\prime }},T)-\text{Perf}(M_{\theta },T)$
4. Side-effect testing: Check performance on other tasks (no catastrophic forgetting?)

Output: Performance delta, updated skill inventory

5.2.7 Meta-Learning Module

Input: Synthesis procedure $P$, outcome (success/failure), performance delta $\Delta$

Process:

1. If success ($\Delta >0$):

   • Reinforce procedure $P$
   • Increase prior probability: $p(P)\leftarrow p(P)\cdot (1+\alpha \Delta )$
   • Store successful pattern for future use

2. If failure ($\Delta \le 0$):

   • Analyze failure mode:
     • Insufficient training data?
     • Wrong composition operator?
     • Base skills too distant from target?
   • Modify procedure:
     • Increase training examples
     • Try different operator
     • Select different base skills
   • Decrease prior: $p(P)\leftarrow p(P)\cdot (1-\beta )$

3. Update strategy distribution:

   • Maintain distribution over synthesis strategies
   • Use multi-armed bandit / Thompson sampling to balance exploration/exploitation

Output: Updated procedure, modified strategy distribution

5.3 Complete Agent Algorithm

class SelfImprovingAgent:
    def __init__(self, base_model, skill_algebra, metaskills):
        self.model = base_model
        self.algebra = skill_algebra
        self.current_skills = skill_algebra.get_base_skills()
        self.metaskills = metaskills
        self.strategy_distribution = initialize_uniform_distribution()
        
    def improve_on_task(self, task, performance_threshold, max_iterations=10):
        """
        Main loop: iteratively improve on task
        """
        for iteration in range(max_iterations):
            # 1. Assess performance
            perf_report = self.task_assessor.assess(
                self.model, task, performance_threshold
            )
            
            if not perf_report['below_threshold']:
                print(f"Success! Performance {perf_report['performance']:.3f} exceeds threshold")
                return True
            
            # 2. Analyze skill gaps
            gaps = self.skill_gap_analyzer.analyze(
                task, self.current_skills, self.embedding_model
            )
            
            if not gaps:
                print("No identifiable skill gaps, but performance insufficient")
                return False
            
            # 3. Introspect and plan
            top_gap = gaps[0]
            synthesis_candidates = self.introspection_module.introspect(
                top_gap['skill'], self.current_skills, self.metaskills
            )
            
            # 4. Select synthesis strategy
            strategy = sample_strategy(self.strategy_distribution)
            procedure = self.synthesis_planner.design_procedure(
                synthesis_candidates[0], strategy
            )
            
            # 5. Generate new skill
            new_model, success = self.skill_generator.execute(
                procedure, self.model
            )
            
            # 6. Integrate and test
            if success:
                self.current_skills.add(procedure.target_skill)
                delta = self.integration_tester.test(
                    new_model, self.model, task
                )
                
                # 7. Meta-learn
                if delta > 0:
                    self.model = new_model
                    self.meta_learner.reinforce(procedure, delta)
                    print(f"Iteration {iteration}: Improvement {delta:.3f}")
                else:
                    self.meta_learner.penalize_and_modify(procedure)
                    print(f"Iteration {iteration}: No improvement, modifying strategy")
            else:
                self.meta_learner.penalize_and_modify(procedure)
                print(f"Iteration {iteration}: Skill generation failed")
        
        return False  # Max iterations reached without success

5.4 Evaluation Protocol

Benchmark Tasks:

1. SKILL-MIX variants:

   • Original: k=3,4,5 with full skill list
   • Novel: k=3,4,5 with unseen skills added to ontology
   • Measure: Can agent acquire new skills to handle unseen combinations?

2. Math problem solving:

   • GSM8K, MATH dataset
   • Introduce novel problem types requiring new skill combinations
   • Measure: Improvement from baseline after self-improvement iterations

3. Code generation:

   • HumanEval, MBPP
   • Add tasks requiring rare programming patterns
   • Measure: Success rate improvement through skill acquisition

4. Multi-hop reasoning:

   • HotpotQA, StrategyQA
   • Increase complexity by requiring longer reasoning chains
   • Measure: Maximum chain length handled after improvements

Metrics:

1. Task Performance:

   • ${\Delta }_{\text{task}}={\text{Perf}}_{\text{after}}-{\text{Perf}}_{\text{before}}$

2. Skill Acquisition Rate:

   • Number of novel skills successfully acquired per iteration

3. Transfer Efficiency:

   • Improvement on held-out tasks after acquiring skills for target task

4. Novelty Score:

   • Fraction of acquired skills that are genuinely new (not in training)

5. Computational Cost:

   • Training tokens used per skill acquired
   • FLOPs per iteration

Baselines:

1. Standard fine-tuning on task examples
2. SELF framework (meta-skill learning only)
3. Prompt engineering without model updates
4. Static model (no improvement mechanism)

Deliverable: Comprehensive evaluation report with ablations

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Phase 6: Extension - Open-Ended Skill Generation (Weeks 21-24) [BOLD GOAL]

6.1 Objective

Move beyond targeted skill acquisition to creative exploration of skill space. Can agent discover ontologically novel skills unprompted?

6.2 Approach

Curiosity-Driven Exploration:

1. Novelty Search:

   • Randomly compose existing skills
   • Evaluate novelty of resulting capability
   • Keep skills that maximize novelty while maintaining usefulness

2. Skill Space Mapping:

   • Use dimensionality reduction (UMAP/t-SNE) on skill embeddings
   • Identify unexplored regions
   • Synthesize skills to fill gaps

3. Counterfactual Reasoning:

   • “What if I combined skills X and Y that are never combined in corpus?”
   • Generate hypothetical tasks that would require such combinations
   • Test if synthesis is possible and useful

Mathematical Formulation:

Novelty Score: $\text{Novelty}(s)=\frac{1}{|S_{\text{current}}|}{\sum }_{s^{\prime }\in S_{\text{current}}}d(s,s^{\prime })$

Usefulness Score: $\text{Useful}(s)={\mathbb{E}}_{T\sim \mathcal{T}}[1(s\in \Phi (T))\cdot \text{Value}(T)]$

Objective: ${\max }_{s\in \mathcal{S}}\lambda \cdot \text{Novelty}(s)+(1-\lambda )\cdot \text{Useful}(s)$

6.3 Implementation

class OpenEndedExplorer:
    def explore_skill_space(self, num_iterations=1000):
        """
        Discover novel skills through guided exploration
        """
        discovered_skills = []
        
        for i in range(num_iterations):
            # 1. Sample composition
            base_skills = random.sample(self.current_skills, k=2)
            operator = random.choice(self.operators)
            
            # 2. Synthesize
            candidate_skill = self.synthesize(base_skills, operator)
            
            # 3. Evaluate novelty and usefulness
            novelty = self.compute_novelty(candidate_skill)
            usefulness = self.estimate_usefulness(candidate_skill)
            
            score = self.lambda_param * novelty + (1 - self.lambda_param) * usefulness
            
            # 4. Accept if above threshold
            if score > self.exploration_threshold:
                self.current_skills.add(candidate_skill)
                discovered_skills.append({
                    'skill': candidate_skill,
                    'novelty': novelty,
                    'usefulness': usefulness,
                    'base_skills': base_skills,
                    'operator': operator
                })
                
                print(f"Discovered: {candidate_skill.name} (novelty={novelty:.3f}, useful={usefulness:.3f})")
        
        return discovered_skills

Deliverable:

• Open-ended exploration system
• Case studies of discovered novel skills
• Analysis of skill space coverage

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Timeline and Milestones

Phase	Weeks	Key Deliverable	Success Criterion
1	1-3	Literature review complete	Comprehensive comparison document
2	4-6	Unified skill taxonomy	Formal definitions for all skill types
2	4-6	Composition algebra	Axioms + theorems + proofs
3	7-10	Mechanism synthesis	4 working skill creation methods
3	7-10	Ontology questions	Technical paper with formal results
4	11-14	Algebraic framework	Mathematical specification + code library
4	11-14	Framework integration	Successful mapping of 3+ papers to framework
5	15-20	Self-improving agent	Working implementation
5	15-20	Evaluation results	Positive Δ on 3+ benchmark tasks
6	21-24	Open-ended exploration	Discovery of 10+ genuinely novel skills

Critical Milestones:

• Week 6: Formal algebra specification (enables implementation)
• Week 14: Code library ready (enables agent development)
• Week 18: Agent demonstrates improvement on at least one task
• Week 20: Full evaluation complete (determines project success)

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Resource Requirements

Computational Resources

• Model training: 8x A100 GPUs for fine-tuning experiments
• Evaluation: 4x A100 GPUs for continuous evaluation
• Storage: 5TB for datasets, model checkpoints, logs

Software/Tools

• Python 3.10+, PyTorch 2.0+
• Transformers library (HuggingFace)
• Weight & Biases for experiment tracking
• Symbolic math libraries (SymPy) for algebraic proofs

Data

• SKILL-MIX dataset (already available)
• GSM8K, MATH, HumanEval (public benchmarks)
• Custom synthetic datasets for skill composition

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Risk Management

Risk	Probability	Impact	Mitigation
Skill decomposition is non-unique (breaks framework)	High	High	Characterize equivalence classes instead of uniqueness
Agent fails to improve on any task	Medium	Critical	Start with simpler tasks, incremental complexity
Computational cost exceeds budget	Medium	High	Use smaller models for prototyping, scale selectively
”Ontological novelty” is ill-defined	High	Medium	Operational definition via novelty metrics, defer philosophy
Skills cannot be reliably extracted from tasks	Medium	High	Use multiple extraction methods, human verification

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Success Criteria

Minimum Viable Success:

1. Formal algebraic framework published
2. Agent demonstrates improvement on ≥1 benchmark task
3. At least one mechanism for skill creation validated

Target Success:

1. Comprehensive framework integrating all papers
2. Agent improves on ≥3 diverse benchmark tasks
3. Discovery of ≥5 novel skills through open-ended exploration
4. Publication-quality theoretical results on skill composition

Stretch Goals:

1. Agent achieves superhuman performance through self-improvement
2. Framework adopted by other researchers (citation impact)
3. Ontologically novel skills with practical applications discovered
4. Connection to Gödel machine for formal self-improvement

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Next Steps (Immediate Actions)

1. Week 1:

   • Finalize literature review (extend beyond current 10 papers)
   • Set up computational infrastructure
   • Begin formal taxonomy development

2. Establish collaborations:

   • Reach out to Sanjeev Arora’s group (mentioned in notes)
   • Connect with SKILL-MIX, SELF paper authors
   • Recruit collaborators with expertise in formal methods

3. Prototype core components:

   • Implement skill similarity metrics
   • Build composition operator framework
   • Test on SKILL-MIX dataset

4. Set up evaluation infrastructure:

   • Prepare benchmark datasets
   • Implement auto-grading systems
   • Establish baseline measurements

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This project plan integrates the theoretical foundations from the literature with an ambitious agentic scheme. The progression from formal foundations → mechanisms → framework → agent → open-ended exploration provides a coherent path toward the ultimate goal of autonomous skill generation and model self-improvement.