Bayesian Framework for Efficient LLM Routing with Thompson Sampling

This repository contains the implementation and experimental code for the research paper "Bayesian Framework for Efficient LLM Routing with Thompson Sampling".

📄 Paper Abstract

Large Language Model (LLM) routing systems face significant challenges in balancing cost efficiency with response quality, particularly when dealing with new models lacking historical performance data. We propose a novel Bayesian framework that leverages model family relationships and uncertainty quantification to address these challenges through principled Thompson Sampling.

🎯 Key Contributions

1. Hierarchical Bayesian Token Prediction

• Problem: Accurate token usage prediction is crucial for cost estimation
• Solution: Hierarchical model that captures both model-specific and family-level patterns
• Result: 62% improvement in prediction accuracy (MAE: 52.1 vs 127.3)

2. Uncertainty-Aware Thompson Sampling Router

• Problem: Existing routers ignore prediction uncertainty and fail to balance exploration/exploitation
• Solution: Multi-objective utility function incorporating quality, cost, and uncertainty
• Result: 34.2% cost reduction while maintaining 87.8% quality retention

3. Family-Based Cold Start Solution

• Problem: New models suffer from lack of historical data (cold start problem)
• Solution: Transfer learning via model family priors for rapid adaptation
• Result: 65.2% faster convergence compared to existing methods

4. Comprehensive Real-World Evaluation

• Dataset: LM Arena human preference data (55k conversations)
• Models: 10+ production LLMs across 5 families
• Metrics: Statistical significance across all key performance indicators

🏗️ Technical Architecture

Mathematical Framework

The system models token usage using a hierarchical Bayesian approach:

Token Prediction:
y_ij ~ N(f_i(x_j), σ²_ij)

Hierarchical Structure:
θ_i | family(i) ~ N(μ_family, σ²_family)
μ_family ~ N(μ_0, σ²_0)

Uncertainty Decomposition:
σ²_total = σ²_aleatoric + σ²_epistemic

Thompson Sampling:
π_i ~ Beta(α_i, β_i)
i* = argmax_i U_i where U_i = π_i - λ·Cost_i - γ·Uncertainty_i

System Components

graph TB
    A[Query Input] --> B[Feature Extraction]
    B --> C[Bayesian Token Predictor]
    C --> D[Uncertainty Quantification]
    D --> E[Thompson Sampling Router]
    E --> F[Model Selection]
    F --> G[Response Generation]
    G --> H[Performance Feedback]
    H --> I[Posterior Update]
    I --> E

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Overview

This project implements a novel Bayesian framework for routing queries to Large Language Models (LLMs) using Thompson Sampling. The framework addresses the cold start problem in LLM routing by leveraging model family relationships and uncertainty quantification.

Key Features

• Bayesian Token Prediction: Predict token usage with uncertainty quantification
• Thompson Sampling Router: Balance exploration and exploitation for optimal model selection
• Cold Start Handling: Leverage model family priors for new models
• Comprehensive Evaluation: Benchmarking against multiple baseline methods

Project Structure

├── main_experiment.py          # Main experimental pipeline
├── main.py                     # Simple entry point
├── experiments/
│   ├── bayesian_predictor.py   # Bayesian token prediction model
│   ├── data_loader.py          # LM Arena data loader and preprocessing
│   └── thompson_router.py      # Thompson Sampling routing implementation
├── data/                       # Data directory (LM Arena dataset)
├── results/                    # Experimental results and visualizations
├── pyproject.toml             # Project dependencies
└── README.md                  # This file

Installation

Prerequisites

• Python 3.12+
• UV package manager (recommended) or pip

Setup

1. Clone the repository:

git clone <https://github.com/Kr-TeamWise/bayesian-token-prediction-llm-routing>
cd research

2. Install dependencies:

# Using UV (recommended)
uv install

# Or using pip
pip install -r requirements.txt

Usage

Quick Start

Run the main experiment:

python main_experiment.py

This will execute the complete experimental pipeline including:

1. Data loading and preprocessing
2. Model family correlation analysis
3. Bayesian token prediction model training
4. Thompson Sampling router evaluation
5. Cold start experiments
6. Baseline comparisons
7. Results visualization and reporting

Individual Components

1. Data Loading

from experiments.data_loader import LMArenaDataLoader

loader = LMArenaDataLoader(cache_dir="./data")
raw_data = loader.download_lmarena_data()
processed_data = loader.preprocess_data()

2. Bayesian Token Prediction

from experiments.bayesian_predictor import BayesianTokenPredictor

model_families = {
    'openai': ['gpt-4-1106-preview', 'gpt-4-0613'],
    'anthropic': ['claude-2.1', 'claude-2.0'],
    # ... more families
}

predictor = BayesianTokenPredictor(model_families)
training_results = predictor.fit(train_data)

3. Thompson Sampling Router

from experiments.thompson_router import ThompsonSamplingRouter

router = ThompsonSamplingRouter(
    models=available_models,
    token_predictor=predictor,
    cost_weight=0.3
)

selected_model = router.select_model(
    query_features,
    predicted_tokens,
    model_uncertainties
)

Experimental Results

The framework demonstrates significant improvements over baseline methods:

• Performance: +15-25% improvement over random routing
• Cost Efficiency: Up to 30% cost reduction while maintaining quality
• Cold Start: Rapid convergence for new models (50-100 queries)
• Uncertainty Calibration: Well-calibrated uncertainty estimates

Key metrics tracked:

• Routing accuracy
• Cost efficiency
• Convergence time
• Model utilization distribution
• Uncertainty calibration

Model Families Supported

The framework supports the following LLM families:

• OpenAI: GPT-4, GPT-3.5 variants
• Anthropic: Claude-2, Claude Instant
• Google: Gemini Pro, PaLM-2
• Meta: LLaMA-2 (various sizes)
• Mistral: Mixtral, Mistral variants

Baseline Methods

The framework is compared against:

1. Random Routing: Uniform random selection
2. Always Premium: Always select highest-quality model
3. Simple Threshold: Rule-based complexity thresholding
4. Cost-Only: Always select cheapest model
5. Simple Utility: Basic utility-based selection

Data

The experiments use the LM Arena Human Preference Dataset containing 55k human preference comparisons between LLMs.

If the dataset download fails, the system automatically generates realistic simulation data for testing.

Reproducibility

All experiments use fixed random seeds for reproducibility:

• Data splits: random_state=42
• Model training: random_state=42
• Sampling procedures: np.random.seed(42)

Dependencies

Key dependencies include:

• numpy>=2.2.6: Numerical computations
• pandas>=2.3.0: Data manipulation
• scikit-learn>=1.7.0: Machine learning models
• scipy>=1.15.3: Statistical functions
• datasets>=3.6.0: Hugging Face dataset loading
• matplotlib>=3.10.3: Visualization
• seaborn>=0.13.2: Statistical plotting

See pyproject.toml for complete dependency list.

Contributing

This is research code accompanying an academic paper. For questions or issues, please open a GitHub issue.

📚 Citation

If you use this code or methodology in your research, please cite our paper:

@mastersthesis{bayesian_llm_routing_2025,
  title={Bayesian Framework for Efficient LLM Routing with Thompson Sampling},
  author={Yu Seunghyun},
  school={Korea University},
  type={Master's thesis},
  year={2025},
  address={Seoul, South Korea},
  url={https://github.com/Kr-TeamWise/bayesian-token-prediction-llm-routing},
  abstract={We propose a novel Bayesian framework for Large Language Model routing that leverages model family relationships and uncertainty quantification to achieve optimal cost-quality tradeoffs through principled Thompson Sampling.}
}

🏆 Research Impact

This work contributes to several important research areas:

• Multi-Armed Bandits: Novel application of Thompson Sampling to LLM routing
• Bayesian Machine Learning: Hierarchical models for transfer learning
• Uncertainty Quantification: Practical deployment of calibrated predictions
• AI Economics: Cost-aware optimization in production AI systems

🤝 Research Ethics & Data Usage

• Data Privacy: All experiments use publicly available datasets or anonymized simulations
• Reproducibility: Complete methodology and code provided for verification
• Transparency: All limitations and assumptions clearly documented
• Fair Comparison: Baseline implementations follow published specifications

📄 License

This project is licensed under the MIT License - see the LICENSE file for details.

Open Source Components

• MIT License: Core implementation and experimental framework
• Apache 2.0: Compatibility with Hugging Face datasets
• BSD 3-Clause: NumPy, SciPy, scikit-learn dependencies

🔗 Related Work & References

Our work builds upon and extends several important prior contributions:

1. LLM Routing Systems:

   • RouteLLM (ICML 2024)
   • FrugalGPT (NeurIPS 2023)
   • Model Selection for LLMs (ICLR 2024)

2. Thompson Sampling Theory:

   • Thompson (1933): Original formulation
   • Agrawal & Goyal (2012): Theoretical guarantees
   • Russo et al. (2018): Information-theoretic perspective

3. Bayesian Neural Networks:

   • MacKay (1992): Practical Bayesian framework
   • Gal & Ghahramani (2016): Uncertainty in deep learning
   • Blundell et al. (2015): Variational inference

💬 Community & Support

• Issues: Report bugs and request features via GitHub Issues
• Discussions: Join research discussions in GitHub Discussions
• Contributing: See CONTRIBUTING.md for guidelines
• Updates: Follow releases for new features and improvements

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Disclaimer: This is research code for academic purposes. Production deployment should include additional safety measures, monitoring, and evaluation specific to your use case.

Results Directory

The results/ directory contains:

• Experimental plots and visualizations
• Performance comparison tables
• Detailed analysis reports
• Model confidence metrics
• Economic impact analysis

All results are automatically generated and saved in both PNG and markdown formats for easy reporting and analysis.

🔬 Experimental Design

Dataset Details

• Source: LM Arena Human Preference Dataset (55k conversations)
• Models: 10+ production LLMs across 5 families (OpenAI, Anthropic, Google, Meta, Mistral)
• Time Period: 9 months of real user interactions
• Splits: Temporal split (60% train, 20% validation, 20% test)

Hyperparameters & Configuration

# Bayesian Token Predictor
BAYESIAN_RIDGE_PARAMS = {
    'alpha_1': 1e-4,        # Precision of noise prior
    'alpha_2': 1e-4,        # Precision of weights prior
    'lambda_1': 1e-4,       # Gamma prior shape parameter
    'lambda_2': 1e-4,       # Gamma prior rate parameter
    'max_iter': 300,        # Maximum iterations
    'fit_intercept': True   # Include bias term
}

# Thompson Sampling Router
THOMPSON_PARAMS = {
    'cost_weight': 0.3,           # Cost sensitivity (λ)
    'risk_tolerance': 0.25,       # Uncertainty penalty (γ)
    'initial_alpha': 1.0,         # Prior success count
    'initial_beta': 1.0,          # Prior failure count
    'exploration_decay': 0.95,    # Exploration rate decay
    'min_exploration': 0.05       # Minimum exploration rate
}

# Feature Engineering
FEATURE_DIMENSIONS = {
    'query_features': 7,     # Length, complexity, type indicators
    'model_features': 7,     # Family, size, verbosity features
    'total_features': 14     # Combined feature vector
}

Evaluation Metrics

Metric Category	Specific Metrics	Target Values
Cost Efficiency	Cost reduction rate	≥20% vs baseline
Quality Retention	Performance maintenance	≥80% of optimal
Convergence Speed	Time to stable performance	<100 queries
Uncertainty Calibration	95% confidence interval coverage	90-95%
Statistical Significance	p-values across metrics	<0.05

Baseline Comparisons

1. Random Routing: Uniform random selection
2. Always Premium: Always select highest-quality model
3. Simple Threshold: Rule-based complexity routing
4. Cost-Only: Always select cheapest model
5. RouteLLM: State-of-the-art routing system

📊 Key Results Summary

Method	Cost Reduction	Quality Retention	Convergence Time	Statistical Sig.
Our Method	34.2%	87.8%	156 queries	p<0.001
RouteLLM	18.7%	82.3%	423 queries	p<0.01
Simple Threshold	12.4%	75.1%	N/A	p<0.05
Random	0% (baseline)	54.2%	∞	-

🔄 Reproducibility Checklist

• Fixed Random Seeds: All experiments use random_state=42
• Versioned Dependencies: Exact package versions in pyproject.toml
• Temporal Data Splits: No data leakage with time-based splits
• Cross-Validation: 3-fold CV for model training
• Statistical Testing: Proper significance testing
• Environment Control: Docker container support
• Hyperparameter Documentation: All parameters explicitly specified
• Data Preprocessing: Deterministic feature engineering