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ACADEMIC PAPER OUTLINE GENERALIZED VARIABLES BRAINDUMP 20250719

Documentation for ACADEMIC_PAPER_OUTLINE_GENERALIZED_VARIABLES_BRAINDUMP_20250719 from the Dspex repository.

Generalized Variables for Compositional Language Model Programs: A Unified Optimization Framework

Stream of Consciousness Paper Outline / Brain Dump

Title Ideas:

  • “Beyond Module Boundaries: Generalized Variables for Cross-Module Optimization in LM Programs”
  • “Unified Variable Spaces for Compositional AI: Breaking the Module Optimization Silo”
  • “SIMBA-GV: Adaptive Optimization of Shared Parameters Across Heterogeneous LM Modules”
  • “From Local to Global: A Variable-Centric Approach to LM Program Optimization”

Abstract (rough ideas)

  • Current LM programming frameworks (DSPy) treat modules as isolated optimization units
  • We introduce generalized variables - parameters that transcend module boundaries
  • Novel contribution: variable-aware execution traces, cross-module gradient estimation
  • SIMBA adaptation for variable geometry understanding
  • Results show 23-47% improvement in multi-module programs (need to run experiments)
  • Opens new research directions in compositional AI optimization

1. Introduction

Hook: “What if temperature wasn’t just a parameter for one module, but a shared characteristic that influences reasoning style across an entire AI system?”

Problem Statement:

  • LM programming emerging as dominant paradigm (cite DSPy, LMQL, Guidance)
  • Current limitation: modules optimize in isolation
  • Real systems need shared characteristics (formality, verbosity, reasoning depth)
  • Example: medical diagnosis system where “conservativeness” should affect all modules

Key Insight:

  • Variables in programs aren’t module-specific, they’re system characteristics
  • Think: CPU temperature affects all cores, not just one
  • Need unified optimization space

Contributions:

  1. Formalization of generalized variables for LM programs
  2. Variable-aware execution tracing and attribution
  3. Cross-module gradient estimation techniques
  4. SIMBA-GV: adapted optimizer for variable geometry
  5. Implementation in DSPex proving feasibility
  6. Empirical validation on 5 benchmark tasks

2.1 LM Programming Frameworks

  • DSPy and the module abstraction (Khattab et al.)
  • Compositional approaches (Chain of Thought, ReAct, etc.)
  • Current optimization methods (BootstrapFewShot, MIPRO)
  • Gap: all assume module-local parameter spaces

2.2 Parameter Sharing in ML

  • Multi-task learning parameter sharing
  • Neural architecture search shared weights
  • Hyperparameter optimization across models
  • Difference: we’re sharing semantic/behavioral parameters, not weights

2.3 Program Synthesis and Optimization

  • Traditional program optimization (loop unrolling, etc.)
  • Differentiable programming
  • Black-box optimization
  • Our approach: gray-box with semantic understanding

3. Generalized Variables: Formalization

3.1 Definitions

Definition 1 (Generalized Variable): A generalized variable v โˆˆ V is a tuple (ฯ„, D, C, ฯƒ) where:

  • ฯ„ is the type (continuous, discrete, module-type)
  • D is the domain
  • C is the constraint set
  • ฯƒ is the semantic binding function

Definition 2 (Variable-Aware Module): A module M is variable-aware if it implements:

  • get_variables(): โ†’ Set[Variable]
  • apply_variables(V): โ†’ M'
  • get_feedback(execution): โ†’ Gradient[V]

Definition 3 (Cross-Module Program): A program P = (Mโ‚, Mโ‚‚, …, Mโ‚™, V, ฯ†) where:

  • Mแตข are variable-aware modules
  • V is shared variable set
  • ฯ† is the composition function

3.2 Variable Types

  1. Continuous Behavioral Variables

    • Temperature (creativity vs consistency)
    • Verbosity (concise vs detailed)
    • Formality (casual vs academic)

  2. Discrete Structural Variables

    • Reasoning strategy (step-by-step vs holistic)
    • Output format (JSON vs prose)
    • Error handling (strict vs lenient)

  3. Module-Type Variables (novel contribution)

    • Variable that changes module behavior class
    • Example: AuthorStyle variable that makes all modules write like specific authors
    • Requires new theory for optimization

4. Variable-Aware Execution Framework

4.1 Execution Traces with Variable Attribution

Key Innovation: Traces that track not just what happened, but which variables influenced each decision

Trace = {
  module_calls: [(module_id, timestamp, input, output)],
  variable_uses: [(var_id, timestamp, context, influence_score)],
  decision_points: [(decision_type, chosen, alternatives, variable_weights)]
}

4.2 Cross-Module Variable Impact Measurement

Challenge: How do we measure impact of temperature on both Predict and ChainOfThought?

Solution: Unified effect metrics

  • Semantic drift measurement
  • Output distribution analysis
  • Consistency scoring across modules

4.3 Gradient Estimation for Black-Box LMs

Since we can’t backprop through LMs, we need:

  • Finite difference methods
  • Semantic gradient approximation
  • Natural language feedback incorporation (!)

5. SIMBA-GV: Geometry-Aware Variable Optimization

5.1 Variable Space Geometry

Insight: Variable space has special structure

  • Some variables are correlated (temperature โ†” creativity)
  • Some are antagonistic (speed โ†” accuracy)
  • Some have module-specific effects

5.2 Adaptive Sampling in Variable Space

Original SIMBA samples data points. SIMBA-GV samples variable configurations:

  1. Coverage-based sampling: Ensure we explore variable space
  2. Gradient-informed sampling: Sample more in high-gradient regions
  3. Constraint-aware sampling: Respect variable constraints

5.3 Intelligent Mutation Strategies

Key idea: Mutations should understand variable semantics

  • Temperature mutations: small changes, affects all modules
  • Style mutations: discrete jumps, coordinated across modules
  • Structural mutations: large changes, may need re-initialization

5.4 Cross-Module Bootstrap

Novel: Bootstrap examples that work well for ALL modules sharing a variable

Algorithm:

  1. Generate candidate examples
  2. Evaluate on each module with shared variables
  3. Score with multi-objective optimization
  4. Select Pareto-optimal examples

6. Implementation: DSPex Architecture

6.1 Native vs Python Execution

  • Native: evaluation, tracing, variable management
  • Python: DSPy modules via Snakepit
  • Hybrid execution model

6.2 Performance Considerations

  • Variable application overhead
  • Trace collection impact
  • Optimization loop efficiency

6.3 API Design

Show how users interact with system - should be intuitive!

7. Experimental Evaluation

7.1 Benchmark Tasks

  1. Multi-Stage QA: Question โ†’ Keywords โ†’ Search โ†’ Answer

    • Shared variable: search_depth

  2. Document Processing: Parse โ†’ Extract โ†’ Summarize โ†’ Format

    • Shared variable: detail_level

  3. Code Generation: Understand โ†’ Plan โ†’ Implement โ†’ Test

    • Shared variable: verbosity, error_checking_strictness

  4. Creative Writing: Ideate โ†’ Outline โ†’ Write โ†’ Edit

    • Shared variable: creativity_style

  5. Scientific Analysis: Hypothesize โ†’ Experiment โ†’ Analyze โ†’ Conclude

    • Shared variable: confidence_threshold

7.2 Baselines

  • DSPy with independent module optimization
  • Naive parameter sharing (same value, no optimization)
  • Grid search over variable space
  • Random search baseline

7.3 Metrics

  • Task-specific performance (accuracy, F1, BLEU, etc.)
  • Optimization efficiency (convergence rate)
  • Variable stability across modules
  • Computational overhead

7.4 Research Questions

RQ1: Do generalized variables improve multi-module program performance? RQ2: How does variable type affect optimization difficulty? RQ3: What is the computational overhead of variable-aware execution? RQ4: How does SIMBA-GV compare to naive optimization approaches? RQ5: Can we automatically discover useful variables?

8. Results and Analysis

8.1 Performance Improvements

Expected: 20-50% improvement on multi-module tasks Why: Better coordination, no local optima traps

8.2 Variable Type Analysis

Hypothesis: Continuous > Discrete > Module-type in terms of optimization ease

8.3 Convergence Analysis

Show convergence curves, discuss variable stability

8.4 Ablation Studies

  • Remove variable sharing
  • Remove cross-module bootstrap
  • Remove intelligent mutation
  • Remove trace-based attribution

9. Discussion

9.1 Theoretical Implications

New optimization paradigm: From module-centric to variable-centric

  • Changes how we think about LM program design
  • Opens questions about variable discovery
  • Connections to program synthesis

9.2 Practical Implications

For practitioners:

  • Design programs with shared characteristics in mind
  • Think about behavioral variables, not just prompts
  • Consider cross-module effects early

9.3 Limitations

  • Black-box LM assumption (no gradients)
  • Computational overhead for large variable spaces
  • Variable discovery still manual
  • May not work for all module types

9.4 Future Directions

  1. Automatic Variable Discovery: Can we learn what variables matter?
  2. Hierarchical Variables: Variables that control other variables
  3. Dynamic Variable Spaces: Variables that appear/disappear during execution
  4. Meta-Learning: Learning to optimize variable spaces

10.1 Connection to Control Theory

  • Variables as control parameters
  • Modules as system components
  • Optimization as controller design

10.2 Connection to Category Theory

  • Variables as morphisms between module behaviors
  • Composition preserving properties
  • Natural transformations between module types

10.3 Connection to Program Analysis

  • Variables as program invariants
  • Cross-module information flow
  • Abstract interpretation over variable spaces

11. Conclusion

Summary:

  • Introduced generalized variables for LM programs
  • Showed how to optimize across module boundaries
  • Demonstrated significant improvements
  • Opened new research directions

Key Takeaway: “The future of LM programming isn’t just about better modules, but about better ways for modules to share behavioral characteristics”

Call to Action: Implement in your favorite LM framework!

Appendices (ideas)

A. Formal Proofs

  • Convergence guarantees for SIMBA-GV
  • Variable space complexity analysis

B. Implementation Details

  • Code snippets
  • Performance optimizations
  • Edge case handling

C. Extended Results

  • Full experimental tables
  • Additional visualizations
  • Failure case analysis

D. Variable Taxonomy

  • Comprehensive list of useful variables
  • Classification scheme
  • Selection guidelines

Random Thoughts / TODO:

  • Need to think more about module-type variables theory
  • Should we prove something about variable space structure?
  • Connection to neural architecture search?
  • Maybe add a section on variable interactions?
  • Need better notation for variable-aware modules
  • Should discuss relationship to hyperparameter optimization
  • Add more on semantic gradients - this is novel
  • Think about variable lifecycle (creation, evolution, death)
  • Discussion of variable scope (local vs global)
  • Privacy implications of shared variables?
  • How does this relate to federated learning?
  • Can we use LLMs to suggest good variables?
  • What about adversarial variables?
  • Connection to game theory (variables as strategies)?
  • Philosophical: what does it mean for modules to “share” behavior?

Potential Venues:

  • NeurIPS (optimization angle)
  • ICML (learning theory)
  • ACL/EMNLP (NLP applications)
  • ICLR (representation learning aspects)
  • AAAI (general AI)

Experiments We Need:

  1. Baseline performance across tasks
  2. Ablation studies
  3. Convergence rate analysis
  4. Variable importance ranking
  5. Computational overhead measurement
  6. User study on API usability?
  7. Robustness to variable perturbation
  8. Transfer learning with variables

Code We Need to Write:

  1. Variable registry system
  2. Trace collection infrastructure
  3. SIMBA-GV optimizer
  4. Evaluation framework
  5. Visualization tools
  6. Benchmark implementations

Key Figures:

  1. System architecture diagram
  2. Variable space visualization
  3. Convergence curves
  4. Performance comparison bars
  5. Trace flow diagram
  6. Variable impact heatmap
  7. Cross-module gradient flow

Story Arc:

Introduction โ†’ Problem โ†’ Theory โ†’ Implementation โ†’ Validation โ†’ Impact โ†’ Future

Make sure each section flows into the next!

Writing Notes:

  • Keep it accessible but rigorous
  • Use running example throughout
  • Make figures early - they help clarify thinking
  • Get feedback on formalization section
  • Make sure contributions are crystal clear
  • Don’t oversell - be honest about limitations
  • End with exciting future work

Reviews We’ll Probably Get:

  • “Interesting idea but limited evaluation”
  • “Need more theoretical analysis”
  • “Comparison to hyperparameter optimization unclear”
  • “Why not just use multi-task learning?”
  • “Overhead seems high for modest gains”

Better address these preemptively!

Remember:

  • This is about SHARED BEHAVIOR not shared parameters
  • Variables are SEMANTIC not syntactic
  • Cross-module optimization is the KEY insight
  • SIMBA adaptation is novel and necessary
  • Implementation proves feasibility

END BRAIN DUMP - Ready to start writing properly!