L2 regularization intuition

Posted on Sun 22 January 2023 in Mathematics

A nice intuition for L2 regularization comes from having a prior on the distribution of parameters: the prior assumes that the parameters are close to zero. Let's assume that the prior is $\mathcal{N}(0, \Sigma)$. The MAP estimate of the parameters would then be

$$\begin{align} \theta_{\text{MAP}} &= \text{argmax}_{\theta} \; P(\theta | D) \\ &= \text{argmax}_{\theta} \; P(D | \theta) P(\theta) \\ &= \text{argmax}_{\theta} \; \log P(D | \theta) + \log P(\theta) \end{align}$$

$\log P(D | \theta)$ is simply the log-likelihood of the model, and optimizing that would give you the MLE parameters. However, incorporating $\log P(\theta)$ gives us the L2 regularization parameter

$$\log P(\theta) = -\log((2\pi)^{n/2}|\Sigma|^{1/2}) - \frac{\theta^T \Sigma^{-1} \theta}{2}$$

The constant at the start goes out of the argmax, and we're left with the L2 regularization term. Taking a negative on both sides would give us the negative log-likelihood, which is the unregularized loss function. We'd then minimize $\theta$. The net expression would look something like this:

$$\theta_{\text{MAP}} = \text{argmin}_{\theta} \; - P(\theta | D) + \frac{1}{2} \theta^T \Sigma^{-1} \theta$$

Regularization generally features a strength term $\lambda$: We can think of $\lambda$ as being the inverse of every term in the diagonal of the covariance matrix (if it is a diagonal covariance matrix). We'd then get

$$\theta_{\text{MAP}} = \text{argmin}_{\theta} \; - P(\theta | D) + \frac{1}{2} \lambda\theta^T\theta$$

And this is the familiar L2 reglarized loss function.


I got this from one of the Deep Learning lectures; it's a nice treatment which I couldn't find in other places (maybe because I haven't looked hard enough). Goodfellow certainly doesn't feature it though.