$L^2$ regularization:

$J_R(w)=J(w$;$D)+||w|{|}_2^2$

$L^1$ regularization:

$J_R(w)=J(w$;$D)+$

where $J(w$;$D)$ is the original cost function $($cost function without regularization$)$ for training

of a general parametric ML model.

Justify the following facts.

$($a$)L^2$ regularization pushes all parameters towards small values $($but not necessarily exactly

zero$).$

$($b$)L^1$ tends to favor so$-$called "sparse" solutions, where only a few of the parameters are

non$-$zero, and the rest are exactly zero.