[throwawayffffas]

Out of my depth really, but I believe the "data driven" approach probably refers to some sort of renormalization.

What renormalization does is your theory produces all sort of infinities but you know there is mechanism that eliminates them but your theory is not able to calculate how that happens and the exact degree so you go out look at real numbers and then tweak your model to predict the correct values .

I may be glossing over misrepresenting stuff but that is the gist of it I think. As far as I understand physists don't love this, Feynman and company had reservations (not objections mid you), but it works and allows you to do additional useful work.

[physicsdude]

The data-driven approach is referring to a more general technique --- it's a less fancy concept than renormalization, which only comes up when doing full-blown relativistic quantum field theory. Specifically, it refers to the use of a couple of well-grounded mathematical tricks to sidestep doing a from-first-principles calculation by, effectively, translating the results of different measurements into a prediction for something new (in this case, a quantity that contributes to the calculation of the magnetic moment).

In non-relativistic quantum mechanics, there is the concept of the "wavefunction," from which you can predict the results of measurements of a particle/system. But in most real-world scenarios, you can't actually compute the wavefunction, and so naively you can't make any predictions. But there is something called the "optical theorem," which relates a single evaluation of the wavefunction to a scattering cross-section, which is something that can be measured. If you're familiar with complex analysis, there is also the "residue theorem" which allows you relate individual function values with integrals of the function in the complex plane. Basically, you can combine those two relations to translate kind one integral (which you need to compute) into a different integral (which can be measured).

This is what was done here --- just instead of a simple QM wavefunction, the relevant concept is called a "vacuum polarization function."