Interpretation of `nothing` from Zygote backend likely incorrect?

[MilesCranmer]

I think the solution in #667 (fixing #662) likely isn't correct, because now it means functions that do not explicitly depend on the input will hit errors.

While nothing is used by Zygote to indicate that a rule has not been implemented, it might ALSO simply mean that there's no dependence. Thus I think throwing an error on every nothing is wrong. For example, this means we can't differentiate f(x) = 0.0 in Zygote, though other backends work:

julia> import Zygote, ForwardDiff; using DifferentiationInterface

julia> DifferentiationInterface.gradient(x -> 0.0, AutoForwardDiff(), [0.0])
1-element Vector{Float64}:
 0.0

julia> DifferentiationInterface.gradient(x -> 0.0, AutoZygote(), [0.0])
ERROR: Zygote failed to differentiate function `#11` with argument types `(Vector{Float64},)` (the pullback returned `nothing`).
Stacktrace:
 [1] check_nothing(::Nothing, f::Function, x::Vector{Float64}, contexts::Tuple{})
   @ DifferentiationInterfaceZygoteExt ~/.julia/packages/DifferentiationInterface/TFo7D/ext/DifferentiationInterfaceZygoteExt/DifferentiationInterfaceZygoteExt.jl:31

I think the best behavior is to just return nothing - it's not perfect, but it will avoid flagging cases like this. Even differentiating the function Base.zero is now an error, because of the fact that

julia> Zygote.gradient(zero, 1.0)
(nothing,)

Ran into this for MilesCranmer/SymbolicRegression.jl#404

[MilesCranmer]

A slightly less trivial example – failing to differentiate a relu for x <= 0

julia> relu(x) = x > 0.0 ? x : 0.0
relu (generic function with 1 method)

julia> DifferentiationInterface.gradient(x -> sum(relu, x), AutoReverseDiff(), [0.0])
1-element Vector{Float64}:
 0.0

julia> DifferentiationInterface.gradient(x -> sum(relu, x), AutoForwardDiff(), [0.0])
1-element Vector{Float64}:
 0.0

julia> DifferentiationInterface.gradient(x -> sum(relu, x), AutoZygote(), [0.0])
ERROR: Zygote failed to differentiate function `#31` with argument types `(Vector{Float64},)` (the pullback returned `nothing`).
Stacktrace:

making a safely-typed relu doesn't fix this either, though we can differentiate for x > 0:

julia> relu(x) = x > zero(x) ? x : zero(x)
relu (generic function with 1 method)

julia> DifferentiationInterface.gradient(x -> sum(relu, x), AutoZygote(), [1.0])
1-element Vector{Float64}:
 1.0

julia> DifferentiationInterface.gradient(x -> sum(relu, x), AutoZygote(), [-1.0])
ERROR: Zygote failed to differentiate function `#5` with argument types `(Vector{Float64},)` (the pullback returned `nothing`).
Stacktrace:

[MilesCranmer]

Also there's inconsistent behavior with nothing returned for only a single element or field:

julia> relu(x) = x > zero(x) ? x : zero(x)
relu (generic function with 1 method)

julia> DifferentiationInterface.gradient(x -> sum(relu, x), AutoZygote(), [1.0, 0.5, -1.0])
3-element Vector{Union{Nothing, Float64}}:
 1.0
 1.0
  nothing

which is the current return value.

I think it should just leave all the nothings in rather than only flagging top-level ones.

[gdalle]

Those are all very valid points, I'll roll back the changes.

[gdalle]

Although I'll avoid the cryptic error message