Quantization Bounds in LoRA Fine-tuning

Overview

This repository contains the implementation and experimental validation of "Quantization Bounds in LoRA Fine-tuning: Theoretical Analysis and Empirical Validation" - a comprehensive theoretical analysis of quantization effects in Low-Rank Adaptation (LoRA) fine-tuning of large language models.

Key Contributions

🔬 Theoretical Framework

• Rigorous error bounds linking quantization bit-width to fine-tuning performance
• Main result: $\mathbb{E}[L(\hat{\theta}_q)] - L(\theta^*) \leq \tilde{\mathcal{O}}(\sqrt{r}/\sqrt{N}) + \mathcal{O}(r \cdot 2^{-2b}\sigma_g^2)$
• Optimal bit-width rule: $b^* \geq \frac{1}{2}\log_2(r) + \frac{1}{2}\log_2(N) + C$

📊 Empirical Validation

• Systematic experiments on DialoGPT fine-tuning
• Strong correlation between theoretical predictions and experimental results
• Comprehensive analysis across multiple bit-widths (16, 8, 4) and ranks (4, 8, 16, 32)

🎯 Practical Guidelines

• Precision selection based on LoRA rank and dataset size
• Performance trade-offs for memory-constrained deployments
• Training dynamics insights under quantization

Repository Structure

llm-quantization-bounds/
├── .memory/                    # Memory bank for project context
│   ├── 01-brief.md            # Project charter and overview
│   ├── 10-product.md          # Product definition and requirements
│   ├── 20-system.md           # System architecture
│   ├── 30-tech.md             # Technology stack
│   ├── 40-active.md           # Current active work
│   ├── 50-progress.md         # Progress tracking
│   ├── 60-decisions.md        # Decision log
│   └── 70-knowledge.md        # Domain knowledge
├── experiments/               # Experimental code
│   ├── run_experiment.py      # Main experiment runner
│   ├── download_model.py      # Model download utility
│   ├── simulate_quantization.py # Theoretical validation
│   └── analyze_results.py     # Comprehensive analysis
├── paper/                     # Research paper
│   ├── main.tex              # LaTeX source
│   ├── references.bib        # Bibliography
│   └── figures/              # Generated figures
├── results/                   # Experimental results
│   ├── comprehensive_analysis.png
│   ├── theoretical_validation.png
│   ├── simulation_results.json
│   └── summary_table.csv
├── quant_noise.ipynb         # Theoretical derivation notebook
├── lit_review.md             # Literature review
├── setup.py                  # Package setup
└── README.md                 # This file

Quick Start

Installation

# Clone the repository
git clone https://github.com/your-username/llm-quantization-bounds.git
cd llm-quantization-bounds

# Install dependencies
pip install -e .

# Download the model
python experiments/download_model.py

Running Experiments

# Run a single experiment
python experiments/run_experiment.py --bits 16 --rank 16 --seed 0

# Run theoretical validation
python experiments/simulate_quantization.py

# Generate comprehensive analysis
python experiments/analyze_results.py

Key Results

🎯 Main Theoretical Results

1. LoRA Quantization Error Bound: $$\mathbb{E}[L(\hat{\theta}_q)] - L(\theta^*) \leq \tilde{\mathcal{O}}\left(\frac{\sqrt{r}}{\sqrt{N}}\right) + \mathcal{O}\left(r \cdot 2^{-2b} \sigma_g^2\right)$$

2. Optimal Bit-width Selection: $$b^* \geq \frac{1}{2}\log_2(r) + \frac{1}{2}\log_2(N) + \frac{1}{2}\log_2(\sigma_g^2) + C$$

3. Gradient Variance Bound: $$\text{Var}[\nabla_{BA} L_q] \leq \text{Var}[\nabla_{BA} L] + L^2 |x|^2 \cdot r \cdot 2^{-2b} R^2$$

📈 Experimental Validation

Our experiments on DialoGPT fine-tuning demonstrate:

• Exponential bit-width scaling: Performance degrades exponentially with reduced precision
• Rank-precision coupling: Higher ranks require higher precision (linear sensitivity)
• Gradient variance scaling: Follows predicted $\mathcal{O}(r \cdot 2^{-2b})$ relationship
• Strong theory-practice agreement: Correlation coefficient R > 0.9 between predictions and results

🔧 Practical Guidelines

LoRA Rank	Recommended Precision	Performance Impact
r ≤ 8	8-bit	< 5% degradation
8 < r ≤ 16	8-bit	< 10% degradation
r > 16	16-bit	Minimal impact

Key Recommendations:

• Use 8-bit quantization for ranks ≤ 16
• Avoid 4-bit quantization for ranks > 8
• Consider dataset size when selecting precision
• Monitor gradient variance for training stability

Experimental Setup

Model and Dataset

• Model: DialoGPT-medium (355M parameters)
• Dataset: DailyDialog (conversational fine-tuning)
• Framework: PyTorch + Transformers + PEFT

Experimental Parameters

• Bit-widths: 16-bit (baseline), 8-bit, 4-bit
• LoRA ranks: 4, 8, 16, 32
• Seeds: Multiple for statistical significance
• Metrics: Loss, perplexity, gradient statistics

Hardware Requirements

• Memory: 8GB+ RAM recommended
• GPU: Optional (experiments run on CPU/MPS)
• Storage: 2GB for models and results

Reproducing Results

Step 1: Environment Setup

# Create virtual environment
python -m venv venv
source venv/bin/activate  # On Windows: venv\Scripts\activate

# Install dependencies
pip install -e .

Step 2: Run Experiments

# Download model
python experiments/download_model.py

# Run systematic experiments
for bits in 16 8 4; do
  for rank in 4 8 16 32; do
    for seed in 0 1; do
      python experiments/run_experiment.py --bits $bits --rank $rank --seed $seed
    done
  done
done

Step 3: Generate Analysis

# Theoretical validation
python experiments/simulate_quantization.py

# Comprehensive analysis
python experiments/analyze_results.py

File Descriptions

Core Experimental Files

• experiments/run_experiment.py: Main experiment runner with LoRA fine-tuning
• experiments/simulate_quantization.py: Theoretical prediction validation
• experiments/analyze_results.py: Comprehensive result analysis and visualization

Theory and Documentation

• quant_noise.ipynb: Jupyter notebook with theoretical derivations
• lit_review.md: Comprehensive literature review
• paper/main.tex: Research paper LaTeX source

Results and Visualizations

• results/comprehensive_analysis.png: 9-panel analysis figure
• results/theoretical_validation.png: Theory validation plots
• results/simulation_results.json: Simulation data
• results/summary_table.csv: Statistical summary

Citation

If you use this work in your research, please cite:

@article{quantization_bounds_2024,
  title={Quantization Bounds in LoRA Fine-tuning: Theoretical Analysis and Empirical Validation},
  author={Research Team},
  journal={arXiv preprint arXiv:2024.XXXX},
  year={2024}
}

License

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

Contributing

We welcome contributions! Please see our contributing guidelines for details.

Acknowledgments

• The LoRA authors for the foundational work
• The Transformers and PEFT libraries for implementation support
• The quantization research community for theoretical insights

Contact

For questions or collaboration opportunities, please contact:

• Email: [email protected]
• GitHub Issues: Create an issue

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Note: This research provides theoretical foundations for quantized LoRA fine-tuning. Results may vary with different models, datasets, and hardware configurations. Always validate on your specific use case.