I have never seen a regularization parameter (usually lambda or alpha) be different for each parameter. People consider different regularization parameters, but I believe they penalize all the parameters with equal strength.
Consider a linear regression with intercept and 2 predictors.
A suggestion for the regularization: instead of $\lambda \sum B_i^2$ consider $\sum(\lambda_i * B_i^2)$, from 1 to n, where i is the i-th parameter.
While generally a single $\lambda$ would be applied on all coefficients, we might have a vector of lambda, one for each coefficient (except the intercept). For $B_1$, $\lambda$ might be 5, while $\lambda$ for $B_2$ would be 10.
Have people used different regularization parameters for different fitted parameters, and are there any reasons to do so? When would be such a case?
One could imagine that by theory one would rather shrink one parameter more than another.