[chillee]

If statements aren't really meaningfully differentiable, regardless of how you do it.

Take

    if x == 59:
        return 1000
    else if x > 59:
        return -x
    else:
        return x

How do you optimize this to maximize x, regardless of what language you're in?

It's true that you can get a derivative, but the derivative is essentially meaningless.

[gradientfree]

  if x <= 0:
      return 0
  else:
      return x

Is in the core of most neural networks today (relu activation), so it is definitely useful.

[mlevental]

I don't understand? It's a piecewise differentiable function and you maximize how you maximize any such function: do gradient ascent where it's differentiable and compare against values at the boundary points (ie start, end of interval and points at which there's a removable discontinuity).

[chillee]

I don't disagree that a gradient exists. As the other commenter commented, the gradient/subgradient will usually exist.

What I'm arguing is that this gradient will not allow you to optimize anything of interest for the vast majority of programs.

[alexmlamb]

Yeah sometimes this is called a "subgradient".

[seanmcdirmid]

There is a derivative, but since the function is non continuous, the derivative will likewise be messed up (but just around x=59). You really don’t want to be climbing a gradient around non continuous functions!

[joe_the_user]

The function has a derivative by some notions of derivative.

But the function's derivative can't be derived by an application of the chain rule and the know derivatives of primitive functions, which is what Algorithmic/automatic differentiation ultimately does (though it does this at run time, not compile time, since ordinary, symbolic differentiation explodes in memory for a complicated functions).

Also:

The continuous function

    int f(int x) {
        if(x > 2) 
            return x + x;
        else 
            return x*2;
    }

Is not automatically or symbolically differentiable when represented that way.

[Certhas]

I believe a source to source differentiator could deal with all these (where well defined of course), e.g.:

https://github.com/FluxML/Zygote.jl

    julia> fs = Dict("sin" => sin, "cos" => cos, "tan" => tan);
    
    julia> gradient(x -> fs[readline()](x), 1)
    sin
    0.5403023058681398

[joe_the_user]

Yeah, Automatic differentiation is essentially only usable on functions specified the way mathematicians specify functions; the compositions of a series of primitives (plus operators like the integral and the differential itself, as well inverse relations).

AD is not usable on loops, conditionals or recursive calls.

So basically, whatever way you specify your functions, you are effectively going to have DSL (within a general purpose language or otherwise) since not all the functions you form are going to be differentiable by AD (and there's some confusion between differentiable in the abstract and differentiable by the methods of AD).

Edit: actually, it's pretty easy to extend AD to functions defined piecewise on intervals to be in the class of function amenable of AD. What's hard/impossible is extending functions defined by loops or recursion.

[neilparikh]

AD on loops/recursion works fine when you implement it using dual numbers (see for https://news.ycombinator.com/item?id=20893414 an example). If you used this to implement x^n using a for loop, you would get the correct derivative (n * x ^ (n - 1)).