Understanding Gradient Descent: The Math Behind Neural Networks

Gradient descent is the optimization algorithm that powers neural network training. Let's break down how it works and why it's so effective.

The Fundamentals

At its core, gradient descent is about finding the minimum of a function. In machine learning, that function is our loss function, and minimizing it means improving our model's predictions.

The Algorithm

1. Initialize parameters randomly
2. Calculate the loss using current parameters
3. Compute gradients (partial derivatives)
4. Update parameters in the direction that reduces loss
5. Repeat until convergence

Mathematical Foundation

The update rule for gradient descent is:

θ = θ - α ∇J(θ)

Where:

• θ represents our model parameters
• α is the learning rate
• ∇J(θ) is the gradient of the loss function

Implementation Example

import numpy as np

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

    for i in range(iterations):
        # Forward pass
        predictions = X.dot(theta)

        # Calculate loss
        loss = (1/(2*m)) * np.sum((predictions - y)**2)

        # Calculate gradients
        gradients = (1/m) * X.T.dot(predictions - y)

        # Update parameters
        theta -= learning_rate * gradients

    return theta

Variants of Gradient Descent

Batch Gradient Descent

Uses the entire dataset to compute gradients. Slow but stable.

Stochastic Gradient Descent (SGD)

Uses one sample at a time. Fast but noisy.

Mini-Batch Gradient Descent

Best of both worlds—uses small batches of data.

Advanced Optimizers

Modern deep learning uses sophisticated variants:

• Adam: Adaptive learning rates for each parameter
• RMSprop: Addresses vanishing gradients
• AdaGrad: Adapts learning rate based on parameter updates

Conclusion

Understanding gradient descent is crucial for debugging training issues and choosing the right optimizer for your problem. Mastering this concept will make you a better ML practitioner.