# What prior would lead to $\ell_\infty$ regularization of model weights?

• Gaussian prior on weights of a GLM lead to Ridge / $$\ell_2$$ squared regularization.
• Laplace prior leads to $$\ell_1$$ regularization

# Question

What prior would lead to $$\ell_\infty$$ regularization ?

Thinking of similar examples, the reason behind the correspondence between these priors and norms is the exponent part. $$f(\beta)\propto \exp(-\lambda||\beta||_1)\rightarrow \ell_1$$ $$f(\beta)\propto \exp(-\lambda||\beta||^2_2)\rightarrow \ell_2 \text{ squared}$$ So, if we have a prior in the form below, it will correspond to $$\ell_\infty$$ because the negative log likelihood will directly include the term $$\lambda||\beta||_{\infty}$$ as a summand. $$f(\beta)\propto \exp(-\lambda||\beta||_\infty)\rightarrow \ell_\infty$$
I don't know if this PDF has a name, but it is a proper one ($$n$$ is the dimension): $$f(\beta)=\frac{\lambda^n}{2^nn!}\exp(-\lambda||\beta||_\infty)$$