Suppose we have data $D=\{x_i,y_i\}_{i=1}^n$ where $x=(x_{i,1},x_{i,2},1)^T \in \mathbb{R}^3$. An estimator for these data, $$y=f(x;\theta)=(\theta_1 x_1+\theta_2 x_2+\theta_3) ~~~(\theta \in \mathbb{R}^3)$$
To estimate the parameter $\theta$, squared error $$L(\theta,\lambda)=\frac{1}{2}\sum_{i=1}^n (y_i-f(x_i;\theta))^2+\frac{\lambda}{2}\theta^T\theta ~~~(\lambda ≥0)$$ My question is what is the equation that the parameter of estimated value $\theta_{\lambda}$ should satisfy?
Differentiating the above equation wrt $\theta$ and setting it to zero,
For $p = 1,2,3$ we have
$\sum_{i=1}^n (y_i- f(x_i;\theta)) x_{ip} + \lambda \theta_p = 0 $
Using these three equations, get the values of $\theta_{1,2,3}$ in terms of $\lambda$