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Gradient Descent & SGD Solver — Course Implementation

🚀 Professional educational project for understanding and implementing Gradient Descent and Stochastic Gradient Descent (SGD) from scratch in Python.

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🔥 Project Overview

This repository demonstrates the mathematical foundation and practical implementation of:

• Gradient Descent (GD)
• Stochastic Gradient Descent (SGD)
• Mini-Batch Gradient Descent
• Convergence analysis
• Optimization visualization

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Keywords

gradient descent
stochastic gradient descent
sgd solver python
optimization algorithm
machine learning optimization
gradient descent from scratch
mini batch gradient descent
convex optimization
loss function minimization
python optimization implementation

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📚 Mathematical Foundation

Gradient Descent Update Rule

For parameters:

$$ \theta $$

Update rule:

$$ \theta_{t+1} = \theta_t - \eta \nabla J(\theta_t) $$

Where:

• $$\eta$$ — learning rate
• $$J(\theta)$$ — loss function
• $$\nabla J(\theta)$$ — gradient

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Stochastic Gradient Descent

Instead of full dataset gradient:

$$ \theta_{t+1} = \theta_t - \eta \nabla J_i(\theta_t) $$

Where:

• Gradient computed on a single sample
• Faster but noisier updates

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Mini-Batch Version

$$ \theta_{t+1} = \theta_t - \eta \frac{1}{B} \sum_{i \in B} \nabla J_i(\theta_t) $$

Balances:

• Stability
• Speed
• Computational efficiency

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🧠 Why This Project Matters

Gradient-based optimization is used in:

• Neural networks
• Linear regression
• Logistic regression
• Deep learning
• Large-scale optimization

This project explains the algorithm at a mathematical and implementation level.

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🏗 Project Structure

gradient-descent-sgd-solver/
│
├── README.md
├── LICENSE
├── requirements.txt
│
├── src/
│   ├── gradient_descent.py
│   ├── sgd.py
│   ├── loss_functions.py
│   ├── optimizer.py
│
├── examples/
│   └── demo.py
│
├── docs/
│   └── theory.md
│
├── images/
│   └── convergence_plot.png
│
└── index.html

Clean structure improves:

✔ Discoverability
✔ Professional appearance
✔ Portfolio quality

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🐍 Example Implementation — Gradient Descent

import numpy as np

def gradient_descent(X, y, lr=0.01, epochs=1000):
    m, n = X.shape
    theta = np.zeros(n)

    for _ in range(epochs):
        predictions = X @ theta
        error = predictions - y
        gradient = (1/m) * X.T @ error
        theta -= lr * gradient

    return theta

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🚀 Installation

pip install -r requirements.txt

Run example:

python examples/demo.py

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📊 Visualization (Recommended)

Add:

• Loss curve plot
• Parameter convergence graph
• 3D loss surface

Example:

import matplotlib.pyplot as plt

plt.plot(loss_history)
plt.xlabel("Epoch")
plt.ylabel("Loss")
plt.title("Convergence")
plt.show()