# 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

∂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.

## Use @not_implemented appropriately

One can use @not_implemented to mark missing differentials. This is helpful if the function has multiple inputs or outputs, and one has worked out analytically and implemented some but not all differentials.

It is recommended to include a link to a GitHub issue about the missing differential in the debugging information:

@not_implemented(
"""
derivatives of Bessel functions with respect to the order are not implemented:
https://github.com/JuliaMath/SpecialFunctions.jl/issues/160
"""
)

Do not use @not_implemented if the differential does not exist mathematically (use DoesNotExist() instead).

## Code Style

Use named local functions for the pullback in an rrule.

# good:
function rrule(::typeof(foo), x)
Y = foo(x)
function foo_pullback(Ȳ)
return NO_FIELDS, bar(Ȳ)
end
return Y, foo_pullback
end
#== output
julia> rrule(foo, 2)
(4, var"#foo_pullback#11"())
==#

function rrule(::typeof(foo), x)
return foo(x), x̄ -> (NO_FIELDS, bar(x̄))
end
#== output:
julia> rrule(foo, 2)
(4, var"##9#10"())
==#

While this is more verbose, it ensures that if an error is thrown during the pullback 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.

## Use rule definition tools

Rule definition tools can help you write more frules and the rrules with less lines of code.

### @non_differentiable

For non-differentiable functions the @non_differentiable macro can be used. For example, instead of manually defining the frule and the rrule for string concatenation *(String..), the macro call

@non_differentiable *(String...)

defines the following frule and rrule automatically

function ChainRulesCore.frule(var"##_#1600", ::Core.Typeof(*), String::Any...; kwargs...)
return (*(String...; kwargs...), DoesNotExist())
end
function ChainRulesCore.rrule(::Core.Typeof(*), String::Any...; kwargs...)
return (*(String...; kwargs...), function var"*_pullback"(_)
(Zero(), ntuple((_->DoesNotExist()), 0 + length(String))...)
end)
end

Note that the types of arguments are propagated to the frule and rrule definitions. This is needed in case the function differentiable for some but not for other types of arguments. For example *(1, 2, 3) is differentiable, and is not defined with the macro call above.

### @scalar_rule

For functions involving only scalars, i.e. subtypes of Number (no structs, Strings...), both the frule and the rrule can be defined using a single @scalar_rule macro call.

Note that the function does not have to be $\mathbb{R} \rightarrow \mathbb{R}$. In fact, any number of scalar arguments is supported, as is returning a tuple of scalars.

See docstrings for the comprehensive usage instructions.

## Write tests

ChainRulesTestUtils.jl provides tools for writing tests based on FiniteDifferences.jl. Take a look at the documentation or the existing ChainRules.jl tests to see how to write the tests.

Warning

Don't use analytical derivations for derivatives in the tests. Those are what you use to define the rules, and so can not be confidently used in the test. If you misread/misunderstood them, then your tests/implementation will have the same mistake. Use finite differencing methods instead, as they are based on the primal computation.

## CAS systems are your friends.

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

## Which functions need rules?

In principle, a perfect AD system only needs rules for basic operations and can infer the rules for more complicated functions automatically. In practice, performance needs to be considered as well.

Some functions use ccall internally, for example ^. These functions can not be differentiated through by AD systems, and need custom rules.

Other functions can in principle be differentiated through by an AD system, but there exists a mathematical insight that can dramatically improve the computation of the derivative. An example is numerical integration, where writing a rule removes the need to perform AD through numerical integration.

Furthermore, AD systems make different trade-offs in performance due to their design. This means that a certain rule will help one AD system, but not improve (and also not harm) another. Below, we list some patterns relevant for the Zygote.jl AD system.

Rules for functions which mutate its arguments, e.g. sort!, should not be written at the moment. While technically they are supported, they would break Zygote.jl such that it would sometimes quietly return the wrong answer. This may be resolved in the future by allowing AD systems to opt-in or opt-out of certain types of rules.

### Patterns that need rules in Zygote.jl

There are a few classes of functions that Zygote can not differentiate through. Custom rules will need to be written for these to make AD work.

Other patterns can be AD'ed through, but the backward pass performance can be greatly improved by writing a rule.

#### Functions which mutate arrays

For example,

function addone!(array)
array .+= 1
return sum(array)
end

complains that

julia> using Zygote
ERROR: Mutating arrays is not supported

However, upon adding the rrule (restart the REPL after calling gradient)

function ChainRules.rrule(::typeof(addone!), a)
return NO_FIELDS, ones(length(a))
end
end

julia> gradient(addone!, a)
([1.0, 1.0, 1.0],)
Why restarting REPL after calling gradient?

When gradient is called in Zygote for a function with no rrule defined, a backward pass for the function call is generated and cached. When gradient is called for the second time on the same function signature, the backward pass is reused without checking whether an an rrule has been defined between the two calls to gradient.

If an rrule is defined before the first call to gradient it should register the rule and use it, but that prevents comparing what happens before and after the rrule is defined. To compare both versions with and without an rrule in the REPL simultaneously, define a function f(x) = <body> (no rrule), another function f_cr(x) = f(x), and an rrule for f_cr.

#### Exception handling

Zygote does not support differentiating through try/catch statements. For example, differentiating through

function exception(x)
try
return x^2
catch e
println("could not square input")
throw(e)
end
end

does not work

julia> gradient(exception, 3.0)
ERROR: Compiling Tuple{typeof(exception),Int64}: try/catch is not supported.

without an rrule defined (restart the REPL after calling gradient)

function ChainRulesCore.rrule(::typeof(exception), x)
y = exception(x)
function exception_pullback(ȳ)
return NO_FIELDS, 2*x
end
return y, exception_pullback
end
julia> gradient(exception, 3.0)
(6.0,)

#### Loops

Julia runs loops fast. Unfortunately Zygote differentiates through loops slowly. So, for example, computing the mean squared error by using a loop

function mse(y, ŷ)
N = length(y)
s = 0.0
for i in 1:N
s +=  (y[i] - ŷ[i])^2.0
end
return s/N
end

takes a lot longer to AD through

julia> y = rand(30)
julia> ŷ = rand(30)
julia> @btime gradient(mse, $y,$ŷ)
38.180 μs (993 allocations: 65.00 KiB)

than if we supply an rrule, (restart the REPL after calling gradient)

function ChainRules.rrule(::typeof(mse), x, x̂)
output = mse(x, x̂)
function mse_pullback(ȳ)
N = length(x)
g = (2 ./ N) .* (x .- x̂) .* ȳ
return NO_FIELDS, g, -g
end
return output, mse_pullback
end

which is much faster

julia> @btime gradient(mse, $y,$ŷ)
143.697 ns (2 allocations: 672 bytes)

#### Inplace accumulation

Inplace accumulation of gradients is slow in Zygote. The issue, demonstrated in the folowing example, is that the gradient of getindex allocates an array of zeros with a single non-zero element.

function sum3(array)
x = array[1]
y = array[2]
z = array[3]
return x+y+z
end
julia> @btime gradient(sum3, rand(30))
424.510 ns (9 allocations: 2.06 KiB)

Computing the gradient with only a single array allocation using an rrule (restart the REPL after calling gradient)

function ChainRulesCore.rrule(::typeof(sum3), a)
y = sum3(a)
function sum3_pullback(ȳ)
end
julia> @btime gradient(sum3, rand(30))
192.818 ns (3 allocations: 784 bytes)