$$ \hat{y}^{(i)} = \beta_0 + \beta_1 x_1^{(i)} + \cdots + \beta_p x_p^{(i)} $$
$$ \hat{y} = X\beta $$
$$ J(\beta) = \frac{1}{2n} \sum_{i=1}^{n} \left( \hat{y}^{(i)} - y^{(i)} \right)^2 $$
$$ \nabla_{\beta} J(\beta) = \frac{1}{n} X^T (X\beta - y) $$
$$ \beta := \beta - \alpha \frac{1}{n} X^T (X\beta - y) $$
$$ \beta_j := \beta_j - \alpha \frac{1}{n} \sum_{i=1}^{n} \left( \hat{y}^{(i)} - y^{(i)} \right) x_j^{(i)} \qquad \text{với } j = 0, 1, \ldots, p $$
$$ \beta_0 := \beta_0 - \alpha \frac{1}{n} \sum_{i=1}^{n} \left( \hat{y}^{(i)} - y^{(i)} \right) $$