$$ \hat{y}^{(i)} = \beta_0 + \sum_{j=1}^{p} \beta_j x_j^{(i)} $$
$$ J(\beta) = \frac{1}{2n} \sum_{i=1}^{n} \left( y^{(i)} - \hat{y}^{(i)} \right)^2 \;+\; \lambda \sum_{j=1}^{p} |\beta_j| $$
$$ \nabla_{\beta_j} \left[ \frac{1}{2n} \sum_{i=1}^{n} ( y^{(i)} - \hat{y}^{(i)} )^2 \right] = -\frac{1}{n} \sum_{i=1}^{n} ( y^{(i)} - \hat{y}^{(i)} ) x_j^{(i)} $$
$$ \frac{\partial}{\partial \beta_j} |\beta_j| = \begin{cases} +1 & \beta_j > 0 \\ -1 & \beta_j < 0 \\ [-1, +1] & \beta_j = 0 \end{cases} $$
$$ \beta_j := \beta_j - \alpha \left( -\frac{1}{n} \sum_{i=1}^{n} ( y^{(i)} - \hat{y}^{(i)} ) x_j^{(i)} \;+\; \lambda \cdot \text{sign}(\beta_j) \right) $$
$$ \beta_0 := \beta_0 - \alpha \left( -\frac{1}{n} \sum_{i=1}^{n} ( y^{(i)} - \hat{y}^{(i)} ) \right) $$
$$ \text{sign}(\beta_j) = \begin{cases} +1 & \beta_j > 0 \\ 0 & \beta_j = 0 \\ -1 & \beta_j < 0 \end{cases} $$