If such a relationship exists in a closed mathematical form, it surely will depend on the order of the closed loop system.
I know that such a relationship exists for a pure second order system.
That means, there is only a limited application for LDOs.
It results in a graph showing the overshoot vs. phase argin and is given by Sergio Fanco (Book: Design with opamps and analog integrated circuits).
(The formulas are rather involved).