What are loss functions, and gradient descend?

 

Loss Functions

A loss function is a mathematical function that measures how well a machine learning model's predictions match the actual target values. It quantifies the "error" between the predicted values ($\hat{y}$) and the true values ($y$). The goal of training a machine learning model is to minimize this loss function, thereby improving the model's performance.

Common Loss Functions:

1. Mean Squared Error (MSE): Used for regression problems.

   $$\text{MSE} = \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2$$

2. Mean Absolute Error (MAE): Another regression loss function, less sensitive to outliers than MSE.

   $$\text{MAE} = \frac{1}{n} \sum_{i=1}^{n} |y_i - \hat{y}_i|$$

3. Cross-Entropy Loss: Used for classification tasks.

   $$\text{CrossEntropy} = -\frac{1}{n} \sum_{i=1}^{n} \left[ y_i \log(\hat{y}_i) + (1 - y_i) \log(1 - \hat{y}_i) \right]$$

4. Hinge Loss: Often used for support vector machines.

   $$\text{Hinge Loss} = \frac{1}{n} \sum_{i=1}^{n} \max(0, 1 - y_i \cdot \hat{y}_i)$$

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Gradient Descent

Gradient Descent is an optimization algorithm used to minimize a loss function. It works by iteratively adjusting the model's parameters (weights and biases) in the direction that reduces the loss.

How It Works:

1. Compute Gradients: Calculate the derivative (gradient) of the loss function with respect to each parameter.

   $$\nabla_\theta L = \frac{\partial L}{\partial \theta}$$

2. Update Parameters: Adjust the parameters using the gradient and a learning rate ($\eta$):

   $$\theta = \theta - \eta \nabla_\theta L$$

3. Repeat: Iterate until convergence (loss stops decreasing or reaches a minimum).

Types of Gradient Descent:

1. Batch Gradient Descent: Uses the entire dataset to compute the gradient at each step.

   • Pros: Accurate gradient computation.
   • Cons: Slow for large datasets.

2. Stochastic Gradient Descent (SGD): Uses a single data point to compute the gradient.

   • Pros: Faster updates.
   • Cons: Noisy convergence.

3. Mini-Batch Gradient Descent: Uses a subset (mini-batch) of data to compute the gradient.

   • Pros: Balances speed and accuracy.

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Relationship Between Loss Functions and Gradient Descent:

The loss function provides a metric for model performance, and gradient descent is the method to minimize this loss function by updating model parameters systematically. Together, they form the foundation of training machine learning models.