# Gradient Descent For Linear Regression

2019/04/21

## Linear regression model

Hypothesis: ${h_{\theta}(x)=\theta_{0}+\theta_{1} x}$

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

## Gradient descent for linear regression

$\frac{\partial}{\partial \theta_{j}} J\left(\theta_{0}, \theta_{1}\right)=\frac{\partial}{\partial \theta_{j}} \frac{1}{2 m} \sum_{i=1}^{m}\left(h_{\theta}\left(x^{(i)}\right)-y^{(i)}\right)^{2}$

$j=0$, $\frac{\partial}{\partial \theta_{0}} J\left(\theta_{0}, \theta_{1}\right)=\frac{1}{m} \sum_{i=1}^{m}\left(h_{\theta}\left(x^{(i)}\right)-y^{(i)}\right)$

$j=1$, $\frac{\partial}{\partial \theta_{1}} J\left(\theta_{0}, \theta_{1}\right)=\frac{1}{m} \sum_{i=1}^{m}\left(\left(h_{\theta}\left(x^{(i)}\right)-y^{(i)}\right) \cdot x^{(i)}\right)$