Field theorist here. You talk about renormalization as a thing which can smooth over unimportant noise, which basically matches my understanding, but you haven’t explicitly named your regulator. A regulator may be a useful concept to have in interpretability, but I have no idea if it is common in the literature.
In QFT, our issue is that we go to calculate things that are measurable and finite, but we calculate horrible infinities. Obviously those horrible infinities don’t match reality, and they often seem to be coming from some particular thing we don’t care about that much in our theory, so we find a way to poke it out of the theory. (To be clear, this means that our theories are wrong, and we’re going to modify them until they work.) The tool by which you remove irrelevant things which cause divergences is called a regulator. A typical regulator is a momentum cutoff. You go to do the integral over all real momenta which your Feynman diagram demands, and you find that it’s infinite, but if you only integrate the momenta up to a certain value, the integral is finite. Of course, now you have a bunch of weird constants sitting around which depend of the value of the cutoff. This is where renormalization comes in. You notice that there are a bunch of parameters, which are generally coupling constants, and these parameters have unknown values which you have to go out into the universe and measure. If you cleverly redefine those constants to be some “bare constant” added to a “correction” which depends on the cutoff, you can do your cutoff integral and set the “correction” to be equal to whatever it needs to be to get rid of all the terms which depend on your cutoff. (edit for clarity: This is the thing that I refer to when I say “renormalization.” Cleverly redefining bare parameters to get rid of unphysical effects of a regulator.) By this two step dance, you have taken your theoretical uncertainty about what happens at high momenta and found a way to wrap it up in the values of your coupling constants, which are the free parameters which you go and measure in the universe anyway. Of course, now your coupling constants are different if you choose a different regulator or a different renormalization scheme to remove it, but physicists have gotten used to that.
So you can’t just renormalize, you need to define a regulator first. You can even justify your regulator. It is a typical justification for a momentum cutoff that you’re using a perturbative theory which is only valid at low energy scales. So what’s the regulator for AI interpretability? Why are you justified in regulating in this way? It seems like you might be pointing at regulators when you talk about 1/w and d/w, but you might also be talking about orders in a perturbation expansion, which is a different thing entirely.
I consider the lattice to be a regulator as well, but, semantics aside, thank you for the example.