How to use gradient descent algorithm for linear regression?

Gradient descent is an optimization algorithm used to minimize the loss function in machine learning models, including linear regression. The goal is to find the optimal parameters (weights and bias) that reduce the error between predicted and actual values.

How Gradient Descent Works for Linear Regression:

1. Define the Loss Function:
   For linear regression, the common loss function is Mean Squared Error (MSE):

   $J(\theta) = \frac{1}{2m} \sum_{i=1}^{m} (h_\theta(x^{(i)}) - y^{(i)})^2$

   where $h_\theta(x) = \theta_0 + \theta_1 x$ is the hypothesis, $m$ is the number of training examples, and $\theta_0, \theta_1$ are the parameters.

2. Compute Gradients:
   The gradients of the loss function with respect to $\theta_0$ and $\theta_1$ are:

   $\frac{\partial J}{\partial \theta_0} = \frac{1}{m} \sum_{i=1}^{m} (h_\theta(x^{(i)}) - y^{(i)})$

   $\frac{\partial J}{\partial \theta_1} = \frac{1}{m} \sum_{i=1}^{m} (h_\theta(x^{(i)}) - y^{(i)}) x^{(i)}$

3. Update Parameters:
   Parameters are updated iteratively using the formula:

   $\theta_j := \theta_j - \alpha \frac{\partial J}{\partial \theta_j}$

   where $\alpha$ is the learning rate, controlling the step size of updates.

4. Repeat Until Convergence:
   The process repeats until the loss function stabilizes or reaches a minimum.

Example:

Suppose you have data points $(1, 2), (2, 3), (3, 4)$.

• Initialize $\theta_0 = 0$, $\theta_1 = 0$.
• Set learning rate $\alpha = 0.1$.
• Compute predictions, loss, and gradients, then update $\theta_0$ and $\theta_1$ iteratively. After several steps, the parameters converge to values that best fit the data.

Using Tencent Cloud for Linear Regression:

For scalable linear regression tasks, Tencent Cloud Machine Learning Platform (TI-ONE) provides tools to train and deploy models efficiently. It supports distributed computing, automatic hyperparameter tuning, and integrates with big data services like Tencent Cloud EMR for large datasets. Additionally, Tencent Cloud TI-EMR can handle distributed gradient descent for massive data.

Example workflow:

1. Upload data to Tencent Cloud COS (Object Storage).
2. Use TI-ONE to define a linear regression model and configure gradient descent.
3. Leverage TI-EMR for distributed training if needed.
4. Deploy the model using TI-ONE's model serving capabilities.

This approach ensures efficient and scalable training for linear regression.