In Jaume Gomis's lecture 5 on CFT at Perimeter Institute, he says (at 27:40 minute mark) that the beta function, classically, of the $m^2$ parameter in massive $\lambda \phi^4$ theory is
$$\beta(m^2) = -2m^2\,.$$
The intuitive reasoning he gives is that since the dimension of $m^2$ is 2, then the beta function should be -2 times m-squared.
I always thought that the beta functions for parameters of the Lagrangian only start at one-loop and are classically zero. What is the rigorous definition of a beta function (including the classical part)?