On writing good rrule / frule methods

Use Zero() or One() as return value

The Zero() and One() differential objects exist as an alternative to directly returning 0 or zeros(n), and 1 or I. They allow more optimal computation when chaining pullbacks/pushforwards, to avoid work. They should be used where possible.

Use Thunks appropriately

If work is only required for one of the returned differentials, then it should be wrapped in a @thunk (potentially using a begin-end block).

If there are multiple return values, their computation should almost always be wrapped in a @thunk.

Do not wrap variables in a @thunk; wrap the computations that fill those variables in @thunk:

# good:
∂A = @thunk(foo(x))
return ∂A

# bad:
∂A = foo(x)
return @thunk(∂A)

In the bad example foo(x) gets computed eagerly, and all that the thunk is doing is wrapping the already calculated result in a function that returns it.

Do not use @thunk if this would be equal or more work than actually evaluating the expression itself. Examples being:

• The expression being a constant
• The expression is merely wrapping something in a struct, such as Adjoint(x) or Diagonal(x)
• The expression being itself a thunk
• The expression being from another rrule or frule; it would be @thunked if required by the defining rule already.
• There is only one derivative being returned, so from the fact that the user called frule/rrule they clearly will want to use that one.

Be careful with using adjoint when you mean transpose

Remember for complex numbers a' (i.e. adjoint(a)) takes the complex conjugate. Instead you probably want transpose(a), unless you've already restricted a to be a AbstractMatrix{<:Real}.

Code Style

Use named local functions for the pushforward/pullback:

# good:
function frule(::typeof(foo), x)
    Y = foo(x)
    function foo_pushforward(_, ẋ)
        return bar(ẋ)
    end
    return Y, foo_pushforward
end
#== output
julia> frule(foo, 2)
(4, var"#foo_pushforward#11"())
==#

# bad:
function frule(::typeof(foo), x)
    return foo(x), (_, ẋ) -> bar(ẋ)
end
#== output:
julia> frule(foo, 2)
(4, var"##9#10"())
==#

While this is more verbose, it ensures that if an error is thrown during the pullback/pushforward the gensym name of the local function will include the name you gave it. This makes it a lot simpler to debug from the stacktrace.

Write tests

There are fairly decent tools for writing tests based on FiniteDifferences.jl. They are in tests/test_utils.jl. Take a look at existing test and you should see how to do stuff.

CAS systems are your friends.

It is very easy to check gradients or derivatives with a computer algebra system (CAS) like WolframAlpha.