Automatic Differentiation Does Incur Truncation Errors (kinda)

Griewank and Walther’s 0th Rule of algorithmic differentiation (AD) states:

Algorithmic differentiation does not incur truncation error.

(2008, “Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation”, Andreas Griewank and Andrea Walther.)

In this blog post I will show you a case that looks like it does in fact incur truncation error. Though this case will arguably be a misinterpretation of that rule. This blog post will thus highlight why careful interpretation of the rule is necessary. Further it will motivate why we need to often add more custom sensitivity rules (custom primitives) to our AD systems, even though you can AD anything with just a few basic rules.

Credit to Mike Innes who pointed this out to me at JuliaCon 2018.

1. Implement an AD, for demonstration purposes

We will start by implementing a simple forwards mode AD. This implementation is based on ForwardDiffZero from the ChainRules docs., but without ChainRules support. Though it is also the simplest most stock-standard implementation once can conceive of.

struct Dual <: Number

primal(d::Dual) = d.primal
partial(d::Dual) = d.partial

primal(d::Number) = d
partial(d::Number) = 0.0

function Base.:+(a::Union{Dual, T}, b::Union{Dual, T}) where T<:Real
    return Dual(primal(a)+primal(b), partial(a)+partial(b))

function Base.:-(a::Union{Dual, T}, b::Union{Dual, T}) where T<:Real
    return Dual(primal(a)-primal(b), partial(a)-partial(b))

function Base.:*(a::Union{Dual, T}, b::Union{Dual, T}) where T<:Real
    return Dual(
        partial(a)*primal(b) + primal(a)*partial(b)

function Base.:/(a::Union{Dual, T}, b::Union{Dual, T}) where T<:Real
    return Dual(
        (partial(a)*primal(b) - primal(a)*partial(b)) / primal(b)^2

# needed for `^` to work from having `*` defined
Base.to_power_type(x::Dual) = x

"Do a calculus. `f` should have a single input."
function derv(f, arg)
    duals = Dual(arg, 1.0)
    return partial(f(duals...))

We can try out this AD and see that it works.

julia> foo(x) = x^3 + x^2 + 1;

julia> derv(foo, 20.0)

julia> 3*(20.0)^2 + 2*(20.0)

2. Implement Sin and Cos, for demonstration purposes

Now we are going to implement the sin and cos functions for demonstration purposes. JuliaLang has an implementation of sin and cos in Julia, that could be ADed though by a source to source AD (like Zygote.jl). But because it is restricted to Float32 and Float64 an operator overloading AD like ours can’t be used with it. That’s Ok we will just code up a simple one using Taylor polynomials. we know that eventually the code run does have to look something like this, since all operations are implemented in terms of +, * and ^, /, bit-shifts and control-flow. (Technically x86 assembly does have a primitive for sin and cos but as far as I know no LibM actually uses them. There is a discussion of why LLVM doesn’t ever emit them if you go looking) The real code would include control flow to wrap around large values and stay close to zero, but we can skip that and just avoid inputting large values.

So using Taylor polynomials of degree 12 for each of them:

my_sin(x) = x - x^3/factorial(3) + x^5/factorial(5) - x^7/factorial(7) + x^9/factorial(9) - x^11/factorial(11)  # + 0*x^12

my_cos(x) = 1 - x^2/factorial(2) + x^4/factorial(4) - x^6/factorial(6) + x^8/factorial(8) - x^10/factorial(10) + x^12/factorial(12)

Check the accuracy we know that sin(π/3) == √3/2, and that cos(π/3) == 1/2 (note that yes, π and √3/2 are approximating here but they are very accurate one. And this doesn’t change the result that follows.)

julia> my_sin(π/3)

julia> √3/2

julia> abs(√3/2 - my_sin(π/3))

julia> my_cos(π/3)

julia> abs(0.5 - my_cos(π/3))

This is not terrible. cos is slightly more accurate than sin. We have a fairly passable implementation of sin and cos.

3. Now lets do AD.

We know the derivative of sin(x) is cos(x). So if we take the derivative of my_sin(π/3) we should get my_cos(π/3)≈0.5. It should be as accurate as the original implementation, right? because Griewank and Walther’s 0th Rule:

Algorithmic differentiation does not incur truncation error.

julia> derv(my_sin, π/3)

Wait a second. That doesn’t seem accurate, we expected 0.5, or at least something pretty close to that. my_cos was accurate to $2 \times 10^{-11}$. my_sin was accurate to $3 \times 10^{-10}$ How accurate is this:

julia> abs(derv(my_sin, π/3) - 0.5)

What went wrong?

4. Verify

Now, I did implement an AD from scratch there. So maybe you are thinking that I screwed it up. Maybe a reverse mode AD would not suffer from this problem; or maybe one that uses source to source? Lets try some of Julia’s many AD systems then.

julia> import ForwardDiff, ReverseDiff, Nabla, Yota, Zygote, Tracker, Enzyme;

julia> ForwardDiff.derivative(my_sin, π/3)

julia> ReverseDiff.gradient(x->my_sin(x[1]), [π/3,])
1-element Vector{Float64}:

julia> Nabla.(my_sin)(π/3)

julia> Yota.grad(my_sin, π/3)[2][1]

julia> Zygote.gradient(my_sin, π/3)

julia> Tracker.gradient(my_sin, π/3)
(0.4999999963909432 (tracked),)

julia> Enzyme.autodiff(my_sin, Active(π/3))

Ok, I just tried 7 AD systems based on totally different implementations. I mean Enzyme is reverse mode running at the LLVM level. Totally different from ForwardDiff which is the more mature version of the forward mode operator overloading AD I coded above. Every single one agreed with my result, up to 1 ULP. I think that last digit changing is probably to do with order of addition (IEEE floating point math is funky), but that is another blog-post. So I think we can reliably say that this is what an AD system will output when asked for the derivative of my_sin at π/3.

5. Explanation

Why does AD seem to be incurring truncation errors? Why is the derivative of my_sin much less accurate than my_cos?

The AD system is (as you might have surmised) not incurring truncation errors. It is giving us exactly what we asked for, which is the derivative of my_sin. my_sin is a polynomial. The derivative of the polynomial is:

d_my_sin(x) = 1 - 3x^2/factorial(3) + 5x^4/factorial(5) - 7x^6/factorial(7) + 9x^8/factorial(9) - 11x^10/factorial(11)

which indeed does have

julia> d_my_sin(π/3)

d_my_sin is a lower degree polynomial approximation to cos than my_cos was, so it is less accurate. Further, you can see that while n-derivative of sin is always defined as sin(x+n*π/2), as we keep taking derivatives of the polynomial approximations terms keep getting dropped. AD is making it smoother and smoother til it is just a flat 0.

The key take away here is that the map is not the territory. Most nontrivial functions on computers are implemented as some function that that approximates (the map) the mathematical ideal (the territory). Automatic differentiation gives back a completely accurate derivative of the that function (the map) doing the approximation. Furthermore, the accurate derivative of an approximation to the idea (e.g d_my_sin), is less accurate than and approximation to the (ideal) derivative of the ideal (e.g. my_cos). There is no truncation error in the work the AD did; but there is a truncation error in the sense that we are now using a more truncated approximation that we would write ourselves.

So what do? Well-firstly, do you want to do anything? Maybe the derivative of the approximation is more useful. (I have been told that this is the case for some optimal control problems). But if we want to fix it can we? Yes, the answer is to insert domain knowledge, telling the AD system directly what the approximation to the derivative of the ideal is. The AD system doesn’t know its working with an approximation, and even if it did, it doesn’t know what the idea it is approximating is. The way to tell it is with a custom primitive i.e. a custom sensitivity rule. This is what the ChainRules project in Julia is about, being able to add custom primitives for more things.

Every real AD system already has a primitive for sin built in (which is one of the reasons I had to define my own above). but it won’t have one for every novel system you approximate. E.g. for things defined in terms of differential equation solutions or other iterative methods.

We can define in our toy AD at the start this custom primitive via:

function my_sin(x::Dual)
    return Dual(my_sin(primal(x)), partial(x) * my_cos(primal(x)))

and it does indeed fix it.

julia> derv(my_sin, π/3)

julia> abs(derv(my_sin, π/3) - 0.5)

Bonus: will symbolic differentiation save me?

Most symbolic differentiation systems will have a rule just like the custom primitive for sin built in. It basically has to do something like this, where-as a forward/reverse AD system could do as we did and fall back to + and *. But, it certainly would not help you to apply symbolic AD to an approximation, that would give exactly the result we derived for d_sin above.

More interestingly, languages where symbolic differentiation is common tend also th have interesting representations of functions in the first place. This does open up avenues for interesting solutions. A suitably weird language could be using representation of sin that is a lazily evaluated polynomial of infinite degree underneath. And in that case there is a rule for its derivative, expressed in terms of changes to its coefficient generating function; which would also give back a lazily evaluated polynomial. I don’t know if anyone does that though; I suspect it doesn’t generalized well. Further, for a lot of things you want to solving systems via iterative methods, and these work for concrete numbers not lazy terms. Maybe there is a cool solution though, I have no real expertise here. Symbolic differentiation in general has problems scaling to large problems.