In that last form, we see that the loss function can be interpreted as a negative log-likelihood and the regularization penalty term as a negative log-prior over the hypothesis.
if we assume a multivariate Gaussian prior on the feature weight vector of mean vector 0 and covariance matrix Σ = σ 2 I (i. e. independent features of same prior standard deviation σ), we do obtain the L2 regularization: And similarly, if we assume a multivariate Laplacian prior on the feature weight vector (i. e. θ i ∼ Laplace(0, 1 λ )), we obtain L 1 -regularization.