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Gradient Descent

Introduction to Gradient Descent:

Gradient Descent is an iterative optimization algorithm used to minimize a function by iteratively moving towards the minimum value of the function.

Objective Function:

The function that needs to be minimized, often known as the cost or loss function in machine learning.

Learning Rate:

A hyperparameter that determines the step size at each iteration while moving toward the minimum.

Types of Gradient Descent:

Batch Gradient Descent
Stochastic Gradient Descent (SGD)
Mini-batch Gradient Descent

Convergence:

The process of approaching the minimum value of the function. Proper tuning of the learning rate is crucial for convergence.

Batch Gradient Descent

Overview:

Batch Gradient Descent computes the gradient of the cost function with respect to the parameters for the entire training dataset.

Advantages:

Stable convergence
Deterministic updates

Disadvantages:

Slow for large datasets
High memory requirement


        // Pseudocode for Batch Gradient Descent
        Initialize parameters
        Repeat until convergence {
            Calculate gradient using entire dataset
            Update parameters
        }

Stochastic Gradient Descent (SGD)

Overview:

SGD updates the parameters for each training example, which makes it faster and suitable for large datasets.

Advantages:

Fast convergence
Less memory usage

Disadvantages:

High variance in updates
May overshoot the minimum


        // Pseudocode for Stochastic Gradient Descent
        Initialize parameters
        Repeat until convergence {
            for each training example {
                Calculate gradient
                Update parameters
            }
        }

Mini-batch Gradient Descent

Overview:

Mini-batch Gradient Descent combines the advantages of both batch and stochastic gradient descent by updating parameters using a small batch of training examples.

Advantages:

Efficient computation
Stable convergence

Disadvantages:

Complexity in tuning batch size


        // Pseudocode for Mini-batch Gradient Descent
        Initialize parameters
        Repeat until convergence {
            for each mini-batch {
                Calculate gradient
                Update parameters
            }
        }

Learning Rate and Convergence

Learning Rate:

The learning rate is a crucial hyperparameter that determines the size of the steps taken towards the minimum.

Impact on Convergence:

Too high: May overshoot the minimum
Too low: Slow convergence

Adaptive Learning Rates:

Techniques like AdaGrad, RMSProp, and Adam adjust the learning rate during training for better convergence.


        // Example of adjusting learning rate
        double learningRate = 0.01;
        if (convergenceSlow) {
            learningRate *= 0.5;
        }

Regularization in Gradient Descent

Purpose of Regularization:

Regularization techniques are used to prevent overfitting by adding a penalty to the loss function.

Common Techniques:

L1 Regularization (Lasso)
L2 Regularization (Ridge)


        // Example of L2 Regularization
        double regularizationTerm = lambda * sum(parameters^2);
        loss += regularizationTerm;

Momentum in Gradient Descent

Concept of Momentum:

Momentum helps accelerate gradient descent by considering past gradients to smooth out the update path.

Benefits:

Faster convergence
Reduced oscillation


        // Example of Momentum
        double velocity = 0;
        double momentum = 0.9;
        velocity = momentum * velocity + learningRate * gradient;
        parameter -= velocity;

Adaptive Gradient Descent Methods

Adaptive Methods Overview:

Adaptive gradient descent methods adjust the learning rate based on past gradients for each parameter.

Popular Methods:

AdaGrad
RMSProp
Adam


        // Example of Adam Optimizer
        double beta1 = 0.9, beta2 = 0.999;
        double m = 0, v = 0;
        m = beta1 * m + (1 - beta1) * gradient;
        v = beta2 * v + (1 - beta2) * gradient^2;
        parameter -= learningRate * m / (sqrt(v) + epsilon);