I’m not saying gradient descent was empirically discovered, I’m saying that its use in machine learning (or elsewhere, I suppose) is itself a form of empiricism in that what it is is essentially a repeated observe/measure error/adjust cycle.
The data is the input, the output is to generally find the lowest amount of a loss function. It’s a greedy approach because brute forcing is inefficient.
It’s no more empirical than a greedy algorithm for scheduling.
Right, GP is drawing a distinction between search, ie mechanical exploration of a space, with understanding, ie having a map of the territory such that you don’t need trial and error.
"Empiricism" implies that the technique is based on observable, but not mathematically proven foundations. If a problem space is convex, gradient descent is guaranteed to converge to a global optimal solution, regardless of whether you know the exact formulation of the space.
Applying it when you don't understand if a space is convex is another question, but that's not a fault of gradient descent.