update

doc/src/week39/week39.do.txt

@@ -1148,19 +1148,19 @@ C_{\text{ridge}}(\beta) = \frac{1}{n}||X\beta -\mathbf{y}||^2 + \lambda ||\beta|
 \]
 !et
 
-In order to minimize $C_{\text{ridge}}(\beta)$ using GD we only have adjust the gradient as follows 
+In order to minimize $C_{\text{ridge}}(\beta)$ using GD we adjust the gradient as follows 
 !bt
 \[
 \nabla_\beta C_{\text{ridge}}(\beta)  = \frac{2}{n}\begin{bmatrix} \sum_{i=1}^{100} \left(\beta_0+\beta_1x_i-y_i\right) \\
 \sum_{i=1}^{100}\left( x_i (\beta_0+\beta_1x_i)-y_ix_i\right) \\
-\end{bmatrix} + 2\lambda\begin{bmatrix} \beta_0 \\ \beta_1\end{bmatrix} = 2 (X^T(X\beta - \mathbf{y})+\lambda \beta).
+\end{bmatrix} + 2\lambda\begin{bmatrix} \beta_0 \\ \beta_1\end{bmatrix} = 2 (\frac{1}{n}X^T(X\beta - \mathbf{y})+\lambda \beta).
 \]
 !et
 
 We can easily extend our program to minimize $C_{\text{ridge}}(\beta)$ using gradient descent and compare with the analytical solution given by 
 !bt
 \[
-\beta_{\text{ridge}} = \left(X^T X + \lambda I_{2 \times 2} \right)^{-1} X^T \mathbf{y}.
+\beta_{\text{ridge}} = \left(X^T X + n\lambda I_{2 \times 2} \right)^{-1} X^T \mathbf{y}.
 \]
 !et