# Internals

## What Zygote Does

These notebooks and the Zygote paper provide useful background on Zygote's transform; this page is particularly focused on implementation details.

Before we think about AD, we'll consider some simple cases. We can start by defining a function that produces pullbacks, J, explicitly for some simple functions.

J(::typeof(sin), x) = sin(x), ȳ -> ȳ*cos(x)
J(::typeof(cos), x) = cos(x), ȳ -> -ȳ*sin(x)
J(::typeof(*), a, b) = a*b, c̄ -> (b*c̄, a*c̄)

Now we can call J to take a gradient.

gradient(f, x...) = J(f, x...)(1)

gradient(*, 2, 3) # (3, 2)

Now consider a composite function that calls two simple ones:

function foo(x)
a = sin(x)
b = cos(a)
return b
end

We can easily differentiate foo if we can differentiate the functions it calls. If we can get pullbacks via J, the pullback for foo looks as follows. Where the forward pass calculates x -> a -> b, the backwards takes b̄ -> ā -> x̄ via the pullbacks.

function J(::typeof(foo), x)
a, da = J(sin, x)
b, db = J(cos, a)
return b, function(b̄)
ā, = db(b̄)
x̄, = da(ā)
return x̄
end
end

gradient(foo, 1) # (-0.403,)

Things get just a little more complex when control flow is involved. You can see that the derived adjoint for pow mirrors the original function, except that the loop runs in reverse. The easiest way to see why it looks like this is to imagine unrolling the loop n times, working out the adjoint, and then turning it back into a loop.

function pow(x, n) # x^n
r = 1
for _ = 1:n
r *= x
end
return r
end

function J(::typeof(pow), x, n)
r = 1
Js = []
for i = 1:n
r, back = J(*, r, x)
push!(Js, back)
end
return r, function(r̄)
x̄ = 0
for i = n:-1:1
r̄, x̄′ = Js[i](r̄)
x̄ += x̄′
end
return (x̄, 0)
end
end

gradient(pow, 2, 3) # (12, 0)

Despite being reasonably fiddly, this is a fully mechanical transformation, so the only remaining thing is to automate it – a small matter of programming.

## Closures

The J function here corresponds to pullback in Zygote. However, pullback is actually a wrapper around the lower level _pullback function.

julia> y, back = Zygote._pullback(sin, 0.5);

julia> back(1)
(nothing, 0.8775825618903728)

Why the extra nothing here? This actually represents the gradient of the function sin. This is often nothing, but when we have closures the function contains data we need gradients for.

julia> f = let a = 3; x -> x*a; end
#19 (generic function with 1 method)

julia> y, back = Zygote._pullback(f, 2);

julia> back(1)
((a = 2,), 3)

This is a minor point for the most part, but _pullback will come up in future examples.

## Entry Points

You might notice that Zygote is, in effect, just a macro. We could happily implement Zygote by writing definitions like

@differentiable foo(x) = sin(cos(x))

which would expand to generate an appropriate overload to J. As long as every function we want to differentiate is annotated, this will work just fine. However, it's obviously not ideal to have to annotate every function inside every Julia package in order to make it differentiable.

This is where generated functions come in. Making J a generated function allows us to apply the Zygote macro on an as-needed basis; calling J(f, x...) looks up the code for f(x...), transforms it, and then behaves as if you had defined J for that specific function ahead of time.

When we look up the code, we actually get lowered (desugared) code rather than an AST.

julia> foo(x) = baz(bar(x))
foo (generic function with 1 method)

julia> @code_lowered foo(1)
CodeInfo(
1 ─ %1 = (Main.bar)(x)
│   %2 = (Main.baz)(%1)
└──      return %2

We convert the code to SSA form using Julia's built-in IR data structure, after which it looks like this.

julia> Zygote.@code_ir foo(1)
1 1 ─ %1 = (Main.bar)(_2)::Any
│   %2 = (Main.baz)(%1)::Any
└──      return %2

(There isn't much difference unless there's some control flow.)

The code is then differentiated by the code in compiler/reverse.jl. You can see the output with @code_adjoint.

julia> Zygote.@code_adjoint foo(1)
1 1 ─ %1  = (Zygote._pullback)(_2, Zygote.unwrap, Main.bar)::Any
│   %2  = (Base.getindex)(%1, 1)::Any
│         (Base.getindex)(%1, 2)::Any
│   %4  = (Zygote._pullback)(_2, %2, _4)::Any
│   %5  = (Base.getindex)(%4, 1)::Any
│         (Base.getindex)(%4, 2)::Any
│   %7  = (Zygote._pullback)(_2, Zygote.unwrap, Main.baz)::Any
│   %8  = (Base.getindex)(%7, 1)::Any
│         (Base.getindex)(%7, 2)::Any
│   %10 = (Zygote._pullback)(_2, %8, %5)::Any
│   %11 = (Base.getindex)(%10, 1)::Any
│         (Base.getindex)(%10, 2)::Any
└──       return %11
1 ─ %1  = Δ()::Any
1 │   %2  = (@12)(%1)::Any
│         (@9)(%3)::Any
│   %6  = (@6)(%4)::Any
│         (@3)(%7)::Any
│   %10 = (Zygote.tuple)(nothing, %8)::Any
└──       return %10
, )

This code is quite verbose, mainly due to all the tuple unpacking (gradindex is just like getindex, but handles nothing gracefully). There are two pieces of IR here, one for the modified pullback pass and one for the pullback closure. The @ nodes allow the closure to refer to values from the pullback pass, and the Δ() represents the incoming gradient ȳ. In essence, this is just what we wrote above by hand for J(::typeof(foo), x).

compiler/emit.jl lowers this code into runnable IR (e.g. by turning @ references into getfields and stacks), and it's then turned back into lowered code for Julia to run.

## Closure Conversion

There are no closures in lowered Julia code, so we can't actually emit one directly in lowered code. To work around this we have a trick: we have a generic struct like

struct Pullback{F}
data
end

We can put whatever we want in data, and the F will be the signature for the original call, like Tuple{typeof(foo),Int}. When the pullback gets called it hits another generated function which emits the pullback code.

In hand written code this would look like:

struct Pullback{F}
data
end

function J(::typeof(foo), x)
a, da = J(sin, x)
b, db = J(cos, a)
return b, Pullback{typeof(foo)}((da, db))
end

function (p::Pullback{typeof(foo)})(b̄)
da, db = p.data, p.data
ā = db(b̄)
x̄ = da(ā)
return x̄
end

## Debugging

Say some of our code is throwing an error.

bad(x) = x

gradient(foo, 1) # error!

Zygote can usually give a stacktrace pointing right to the issue here, but in some cases there are compiler crashes that make this harder. In these cases it's best to (a) use _pullback and (b) take advantage of Zygote's recursion to narrow down the problem function.

julia> y, back = Zygote._pullback(foo, 1);

julia> back(1) # just make up a value here, it just needs to look similar to y

# Ok, so we try functions that foo calls

julia> y, back = Zygote._pullback(sin, 1);

julia> back(1)
(nothing, 0.5403023058681398)

# Looks like that's fine

julia> y, back = Zygote._pullback(bad, 1);

julia> back(1) # ok, here's our issue. Lather, rinse, repeat.
ERROR: bad

Of course, our goal is that you never have to do this, but until Zygote is more mature it can be a useful way to narrow down test cases.