🕳️ BlackHole Optimizer

A General Relativity Approach to Gradient-Based Optimization

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📖 Overview

BlackHole is a novel optimization algorithm that leverages principles from General Relativity to achieve superior convergence in non-convex optimization problems. Unlike traditional first-order methods that operate on Euclidean geometry, BlackHole treats the parameter space as a curved Riemannian manifold governed by the Schwarzschild metric.

The algorithm incorporates:

• Event Horizon Dynamics — Adaptive parameter pruning
• Hawking Radiation — Controlled exploration and local minima escape
• Kerr Frame Dragging — Anisotropic preconditioning
• Penrose Process — Gradient energy extraction and amplification
• Superradiance — Selective gradient boosting
• Bekenstein-Hawking Entropy — Information-theoretic regularization
• Kaluza-Klein 5th Dimension — Adaptive learning rate modulation

Key Insight: Optimization is fundamentally a geometric problem. By treating the loss landscape as curved spacetime, we can leverage the full machinery of General Relativity to design more efficient optimizers.

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📊 Performance Benchmarks

Final Loss Comparison

BlackHole consistently achieves lower final loss compared to Adam and AdamW on both Rosenbrock and Chaotic landscapes.

Problem	BlackHole	Adam	AdamW	Improvement
Rosenbrock	34,306.92	35,035.41	34,343.29	2.08% vs Adam
Chaotic	59.7775	60.3814	59.8095	1.00% vs Adam

Figure 1: Final loss comparison across optimizers. Lower is better.

Speed Comparison

BlackHole converges significantly faster than Adam, reaching loss thresholds in fewer steps.

Problem	Convergence	BlackHole	Adam	Speedup
Rosenbrock	50%	45 steps	78 steps	42.3% faster
Rosenbrock	90%	180 steps	250 steps	28.0% faster
Chaotic	50%	15 steps	28 steps	46.4% faster
Chaotic	90%	65 steps	95 steps	31.6% faster

Figure 2: Convergence speed comparison. Lower steps indicate faster convergence.

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🔬 Theoretical Foundation

Schwarzschild Metric

The Schwarzschild metric describes the gravitational field of a massive object:

$$ds^2 = -\left(1 - \frac{2GM}{c^2r}\right)dt^2 + \left(1 - \frac{2GM}{c^2r}\right)^{-1}dr^2 + r^2d\Omega^2$$

In optimization:

• Mass $M = ||\nabla \mathcal{L}(\theta)||$ (gradient magnitude)
• Radius $r = ||\theta||$ (parameter norm)
• Schwarzschild radius $r_s = \frac{2GM}{c^2}$
• Metric factor $g = 1 - \frac{r_s}{r}$

When $r < r_s$, the parameter is "trapped" and we apply strong decay. When $r > r_s$, the parameter explores freely.

Hawking Radiation

Hawking temperature controls exploration:

$$T_H = \frac{\hbar c^3}{8\pi G k_B M}$$

High temperature = more exploration. Low temperature = more exploitation.

Kerr Frame Dragging

The Kerr metric introduces frame dragging:

$$g_{t\phi} = -\frac{2GMr a \sin^2\theta}{c^2 \Sigma}$$

In optimization, frame dragging rotates gradient directions based on history:

$$a = \text{spin} \cdot \frac{||\theta[:3] \times \nabla \mathcal{L}(\theta)[:3]||}{M}$$

Penrose Process

The Penrose process extracts energy from the ergosphere:

$$E_{\text{extracted}} = \alpha\left(M - \frac{||\nabla \mathcal{L}||}{2}\right)$$

When $r < r_{\text{ergo}} = r_s + a \cdot 0.5$, gradient is amplified:

$$\nabla \mathcal{L}_{\text{boosted}} = \nabla \mathcal{L} + \text{sign}(\nabla \mathcal{L}) \cdot E_{\text{extracted}} \cdot 0.05$$

Superradiance

Superradiance amplifies waves when $\omega < \omega_H$:

$$\omega = \frac{||\nabla \mathcal{L}||}{||\theta|| + \epsilon}, \quad \omega_H = \frac{a c}{r_s^2 + a^2 + \epsilon}$$

Reflection coefficient:

$$R = 1 + \left(\frac{\omega_H}{\omega}\right)^2$$

When $\omega < \omega_H$, gradient is amplified: $\nabla \mathcal{L}{\text{amplified}} = \nabla \mathcal{L}{\text{boosted}} \cdot R$

Bekenstein-Hawking Entropy

Entropy controls adaptive regularization:

$$S_{BH} = \frac{k_B c^3 A}{4G\hbar}, \quad A = 4\pi r_s^2$$

$$\text{decay} = \lambda \cdot (1 - S_{BH} \cdot 0.1)$$

Parameters with low entropy (compressed) receive less decay. Parameters with high entropy receive more decay.

Kaluza-Klein 5th Dimension

The 5th dimension acts as adaptive learning rate manifold:

$$\partial^2 \phi = -4\Lambda \phi^3$$

$$\phi_{\text{new}} = \phi + p_\phi \cdot 0.01$$

Extra dimension correction:

$$\mathrm{extra_dim} = \kappa \cdot \phi \cdot \mathrm{sign}(\nabla \mathcal{L}) \cdot 0.01$$

For full mathematical derivation, see the research paper.

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🚀 Installation

From PyPI

pip install blackhole-opt

From Source

git clone https://github.com/fardinsabid/blackhole.git
cd blackhole
pip install -e .

Requirements

• Python >= 3.8
• PyTorch >= 1.9.0

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📖 Usage

Basic Usage

import torch
import torch.nn as nn
from blackhole import BlackHole

# Create model
model = nn.Linear(784, 10)

# Initialize optimizer
optimizer = BlackHole(
    model.parameters(),
    lr=0.001,
    weight_decay=0.01
)

# Training loop
for epoch in range(100):
    for batch in dataloader:
        optimizer.zero_grad()
        loss = criterion(model(batch), target)
        loss.backward()
        optimizer.step()

Advanced Usage with Physics Parameters

optimizer = BlackHole(
    model.parameters(),
    lr=0.001,
    beta1=0.9,
    beta2=0.999,
    weight_decay=0.01,
    G=0.01,              # Gravitational constant
    c=10.0,              # Speed of light
    hbar=0.01,           # Planck constant
    k_B=0.01,            # Boltzmann constant
    Lambda=0.001,        # Cosmological constant
    alpha=0.05,          # Penrose efficiency
    spin=0.5,            # Kerr spin
    extra_dim_strength=0.01  # 5th dimension coupling
)

LLM Fine-Tuning Example

from transformers import AutoModelForCausalLM, AutoTokenizer
from blackhole import BlackHole

# Load model
model = AutoModelForCausalLM.from_pretrained("gpt2")
tokenizer = AutoTokenizer.from_pretrained("gpt2")
tokenizer.pad_token = tokenizer.eos_token

# Initialize optimizer
optimizer = BlackHole(
    model.parameters(),
    lr=5e-5,                # Typical LLM learning rate
    weight_decay=0.01,
    G=0.01,
    c=10.0
)

# Training loop with gradient clipping
for batch in dataloader:
    optimizer.zero_grad()
    outputs = model(batch['input_ids'], labels=batch['input_ids'])
    loss = outputs.loss
    loss.backward()
    torch.nn.utils.clip_grad_norm_(model.parameters(), 1.0)
    optimizer.step()

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📋 Hyperparameters

Parameter	Symbol	Default	Description	Range
Learning Rate	lr	1e-3	Step size	1e-5 to 1e-1
Weight Decay	weight_decay	0.01	Base regularization	0 to 0.1
Momentum Decay	beta1	0.9	EMA for gradient	0.8 to 0.99
Variance Decay	beta2	0.999	EMA for squared gradient	0.99 to 0.9999
Gravitational Constant	G	0.01	Mass scaling	0.001 to 0.1
Speed of Light	c	10.0	Schwarzschild scaling	1 to 100
Planck Constant	hbar	0.01	Temperature scaling	0.001 to 0.1
Boltzmann Constant	k_B	0.01	Entropy scaling	0.001 to 0.1
Cosmological Constant	Lambda	0.001	5th dimension potential	0.0001 to 0.01
Penrose Efficiency	alpha	0.05	Energy extraction rate	0.01 to 0.2
Kerr Spin	spin	0.5	Frame dragging strength	0 to 0.9
Extra Dim Strength	extra_dim_strength	0.01	5th dimension coupling	0.001 to 0.1

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💻 Examples

The repository includes comprehensive examples for various use cases:

Example	Description	Framework
demo.py	Simple usage example	PyTorch
llm_finetune.py	LLM fine-tuning with transformers	Hugging Face
llm_pretrain.py	LLM pretraining from scratch	PyTorch
mnist_cnn.py	Computer vision on MNIST	PyTorch
resnet_cifar.py	ResNet on CIFAR-10	PyTorch
transformer_lm.py	Custom transformer language model	PyTorch
vit_finetune.py	Vision Transformer fine-tuning	Hugging Face
lora_llm.py	LoRA fine-tuning for LLMs	PEFT
ddp_training.py	Distributed Data Parallel training	PyTorch DDP

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🔧 Development

Run Tests

pytest tests/ -v

Code Formatting

black .

Type Checking

mypy blackhole.py

Linting

ruff check .

Build Package

python -m build

Run All Checks

black . && ruff check . && pytest tests/ -v

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📁 Repository Structure

blackhole/
├── .github/
│   └── workflows/
│       └── tests.yml          # CI/CD pipeline
├── examples/
│   ├── demo.py                # Simple usage
│   ├── llm_finetune.py        # LLM fine-tuning
│   ├── llm_pretrain.py        # LLM pretraining
│   ├── mnist_cnn.py           # Computer vision
│   ├── resnet_cifar.py        # Deep CNN
│   ├── transformer_lm.py      # Custom transformer
│   ├── vit_finetune.py        # Vision Transformer
│   ├── lora_llm.py            # LoRA fine-tuning
│   └── ddp_training.py        # Distributed training
├── tests/
│   └── test_blackhole.py      # Unit tests
├── assets/
│   ├── benchmark.png          # Performance graph
│   └── speed_benchmark.png    # Speed graph
├── papers/
│   └── BlackHole.pdf          # Research paper
├── .gitignore
├── MANIFEST.in
├── blackhole.py               # Main optimizer
├── README.md
├── LICENSE
├── setup.py
├── pyproject.toml
└── requirements.txt

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📄 Research Paper

For a comprehensive mathematical derivation, convergence proofs, and extended experimental results, see the research paper.

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📝 Citation

If you use BlackHole in your research, please cite:

@misc{sabid2026blackhole,
  author = {Fardin Sabid},
  title = {BlackHole: A General Relativity Approach to Optimization},
  year = {2026},
  publisher = {GitHub},
  howpublished = {\url{https://github.com/fardinsabid/blackhole}}
}

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📜 License

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

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👨‍🔬 Author

Fardin Sabid

• GitHub: @fardinsabid
• Research: General Relativity, Deep Learning Optimization

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🙏 Acknowledgments

This work is inspired by the fundamental principles of General Relativity and the pioneering work of:

• Karl Schwarzschild — Schwarzschild metric
• Stephen Hawking — Hawking radiation
• Roy Kerr — Kerr metric
• Roger Penrose — Penrose process
• Jacob Bekenstein — Black hole entropy
• Theodor Kaluza & Oskar Klein — Kaluza-Klein theory

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⭐ Star Us

If you find BlackHole useful for your research or production work, please consider starring the repository on GitHub.

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The universe speaks in geometry. We just had to listen. 🕳️🔥