Optimization Algorithms

Optimization Algorithms are the backbone of training neural networks, enabling them to learn from data by adjusting weights and biases to minimize the loss function. This process is crucial for improving model accuracy and performance.

Gradient Descent

The most common optimization algorithm, gradient descent, iteratively updates model parameters in the direction of the steepest descent of the loss function. The update rule is given by:

where:

• (\theta) are the model parameters (weights and biases),
• (\eta) is the learning rate,
• (\nabla J(\theta)) is the gradient of the loss function with respect to the parameters.

Understanding the Gradient

Gradient: The rate of change of the function with respect ot its parameters. It points in the direction of the steepest increase of the function. Descent: Moving in the opposite direction of the gradient to minimize the loss function. Iteration: Each update step in the optimization process is called an iteration, where the parameters are adjusted based on the computed gradient.

The Gradient Descent Algorithm

• Initialize parameters (\theta) randomly or with zeros.
• Compute the gradient (\nabla J(\theta)) of the loss function.
• Update the parameters using the update rule.
• Repeat until convergence or a predefined number of iterations.

import numpy as np

def gradient_descent(x, y, learning_rate, iterations):
    m = np.random.randn()
    b = np.random.randn()

    for i in range(iterations):
        # Calc predictions
        predictions = m * x + b
        # Calculate the error
        errors = predictions - y

        # Calculate gradients
        dm = np.mean(errors * x)
        db = np.mean(errors)

        # Update parameters
        m -= learning_rate * dm
        b -= learning_rate * db

    return m, b

Adam

Adam (Adaptive Moment Estimation) is an advanced optimization algorithm that combines the benefits of two other extensions of stochastic gradient descent: AdaGrad and RMSProp. It adapts the learning rate for each parameter based on the first and second moments of the gradients.

• Momentum: Keeps track of the exponentially decaying average of past gradients to smooth out updates.
• Adaptive Learning Rate: Adjusts the learning rate for each parameter based on the magnitude of the gradients, allowing for faster convergence.

import torch
import torch.optim as optim

optimizer = optim.Adam(model.parameters(), lr=0.001)

Others

Other popular optimization algorithms include:

• RMSProp: Adapts the learning rate based on the average of recent gradients.
• Adagrad: Adapts the learning rate for each parameter based on the historical sum of squares of the gradients.
• SGD with Momentum: Adds a fraction of the previous update to the current update, helping to accelerate gradients vectors in the right directions.