Do you have a simple example of a loop with no sensible derivative? Or rather, of a loop computing something which should have one, but for which AD fails? If

Do you have a simple example of a loop with no sensible derivative? Or rather, of a loop computing something which should have one, but for which AD fails? If your loop is computing something like f(g(...h(x))) i.e. applying functions in sequence, starting with x, then there's no problem -- dual numbers will compute the correct derivative. If the loop is an implementation detail, like I think the poisson distribution p(λ,k) is often computed with a loop up to k... then making λ a dual number still works. Clearly you can't differentiate on k, because it's an integer.