I have a small notational question:
Ridge regression has a (regularizer) term $\lambda||\beta||^2$
In our scripts this is written as $\lambda\beta^T\beta$. If $\beta$ is (d,1)-dimensional, then $\beta^T\beta$ would be a matrix multiplication - and even though the result is a scalar, is there a particular reason not to use the dot-product and to hence write the regularization as $\lambda\beta\beta$?
When $\beta$ is a column vector $\left\|\beta\right\|^2 = \beta^T\beta = \beta \cdot \beta = \sum_{i=1}^d\beta_i^2$ is already the dot product.
Writing $\beta\beta$ does not make much sense as it would imply multiplying a $d\times 1$ matrix by another $d \times 1$ matrix which is not possible.