  We perform a numerical computation of the anomalous magnetic moment ($g-2$)
of the electron in QED by using the stochastic perturbation theory. Formulating
QED on the lattice, we develop a method to calculate the coefficients of the
perturbative series of $g-2$ without the use of the Feynman diagrams. We
demonstrate the feasibility of the method by performing a computation up to the
$\alpha^3$ order and compare with the known results. This program provides us
with a totally independent check of the results obtained by the Feynman
diagrams and will be useful for the estimations of not-yet-calculated higher
order values. This work provides an example of the application of the numerical
stochastic perturbation theory to physical quantities, for which the external
states have to be taken on-shell.
