# Design Notes: The many-to-many relationship between differential types and primal types

ChainRules has a system where one primal type (the type having its derivative taken) can have multiple possible differential types (the type of the derivative); and where one differential type can correspond to multiple primal types. This is in-contrast to the Swift AD efforts, which has one differential type per primal type (Swift uses the term associated tangent type, rather than differential type).

differential and associated tangent type

The use of “associated tangent type” in AD is not technically correct, as differentials naturally live in the cotangent plane instead of the tangent plane. However it is often reasonable for AD to treat the cotangent plane and tangent plane as the same thing, and this was an intentional choice by the Swift team. Here we will just stick to the ChainRules terminology and only say “differential type” instead of “tangent type”.

One thing to understand about differentials is that they have to form a vector space (or something very like them). They need to support addition to each other, they need a zero which doesn't change what it is added to, and they need to support scalar multiplication (this isn't really required, but it is handy for things like gradient descent). Beyond being a vector space, differentials need to be able to be added to a primal value to get back another primal value. Or roughly equivalently a differential is a difference between two primal values.

One thing to note in this example is that the primal does not have to be a vector. As an example, consider DateTime. A DateTime is not a vector space: there is no origin point, and DateTimes cannot be added to each other. The corresponding differential type is any subtype of Period, such as Millisecond, Hour, Day etc.

## Natural differential

For a given primal type, we say a natural differential type is one which people would intuitively think of as representing the difference between two primal values. It tends to already exist outside of the context of AD. So Millisecond, Hour, Day etc. are examples of natural differentials for the DateTime primal.

Note here that we already have a one primal type to many differential types relationship. We have Millisecond and Hour and Day all being valid differential types for DateTime. In this case we could convert them all to a single differential type, such as Nanoseconds, but that is not always a reasonable decision: we may run in to overflow, or lots of allocations if we need to use a BigInt to represent the number of Nanosecond since the start of the universe. For types with more complex semantics, such as array types, these considerations are much more important.

Natural differential types are the types people tend to think in, and thus the type they tend to write custom sensitivity rules in. An important special case of natural differentials is when the primal type is a vector space (e.g. Real,AbstractMatrix) in which case it is common for the natural differential type to be the same as the primal type. One exception to this is getindex. The ideal choice of differential type for getindex on a dense array would be some type of sparse array, due to the fact the derivative will have only one non-zero element. This actually further brings us to a weirdness of differential types not actually being closed under addition, as it would be ideal for the sparse array to become a dense array if summed over all elements.

## Structural differential types

AD cannot automatically determine natural differential types for a primal. For some types we may be able to declare manually their natural differential type. Other types will not have natural differential types at all - e.g. NamedTuple, Tuple, WebServer, Flux.Dense - so we are destined to make some up. So beyond natural differential types, we also have structural differential types. ChainRules uses Tangent{P, <:NamedTuple} to represent a structural differential type corresponding to primal type P. Zygote v0.4 uses NamedTuple.

Structural differentials are derived from the structure of the input. Either automatically, as part of the AD, or manually, as part of a custom rule.

Consider the structure of DateTime:

julia> dump(now())
DateTime
instant: UTInstant{Millisecond}
periods: Millisecond
value: Int64 63719890305605

The corresponding structural differential is:

Tangent{DateTime}(
instant::Tangent{UTInstant{Millisecond}}(
periods::Tangent{Millisecond}(
value::Int64
)
)
)
One must be allowed to take derivatives of integer arguments

This brings up another contrast to Swift. In Swift Int is considered non-differentiable, which is quite reasonable; it doesn’t have a very good definition of the limit of a small step (as that would be some floating/fixed point type). Int is intrinsically discrete. It is commonly used for indexing, and if one takes a gradient step, say turning x[2] into x[2.1] then that is an error. However, disallowing Int to be used as a differential means we cannot handle cases like DateTime having an inner field of milliseconds counted as an integer from the unix epoch or other cases where an integer is used as a convenience for computational efficiency. In the case where a custom sensitivity rule claims that there is a non-zero derivative for an Int argument that is being used for indexing, that code is simply wrong. We can’t handle incorrect code and trying to is a path toward madness. Julia, unlike Swift, is not well suited to handling rules about what you can and can’t do with particular types.

So the structural differential is another type of differential. We must support both natural and structural differentials because AD can only create structural differentials (unless using custom sensitivity rules) and all custom sensitivities are only written in terms of natural differentials, as that is what is used in papers about derivatives.

## Semi-structural differentials

Where there is no natural differential type for the outermost type but there is for some of its fields, we call this a "semi-structural" differential.

Consider if we had a representation of a country's GDP as output by some continuous time model like a Gaussian Process, where that representation is as a sequence of TimeSamples structured as follows:

julia> struct TimeSample
time::DateTime
value::Float64
end

We can look at its structure:

julia> dump(TimeSample(now(), 2.6e9))
TimeSample
time: DateTime
instant: Dates.UTInstant{Millisecond}
periods: Millisecond
value: Int64 63720043490844
value: Float64 2.6e9

Thus we see the that structural differential would be:

Tangent{TimeSample}(
time::Tangent{DateTime}(
instant::Tangent{UTInstant{Millisecond}}(
periods::Tangent{Millisecond}(
value::Int64
)
)
),
value::Float64
)

But instead in the custom sensitivity rule we would write a semi-structured differential type. Since there is not a natural differential type for TimeSample but there is for DateTime.

Tangent{TimeSample}(
time::Day,
value::Float64
)

So the rule author has written a structural differential with some fields that are natural differentials.

Another related case is for types that overload getproperty such as SVD and QR. In this case the structural differential will be based on the fields, but those fields do not always have an easy relation to what is actually used in math. For example, the QR type has fields factors and t, but we would more naturally think in terms of the properties Q and R. So most rule authors would want to write semi-structural differentials based on the properties.

To return to the question of why ChainRules has Tangent{P, <:NamedTuple} whereas Zygote v0.4 just has NamedTuple, it relates to semi-structural derivatives, and being able to overload things more generally. If one knows that one has a semi-structural derivative based on property names, like Tangent{QR}(Q=..., R=...), and one is adding it to the true structural derivative based on field names Tangent{QR}(factors=..., τ=...), then we need to overload the addition operator to perform that correctly. We cannot happily overload similar things for NamedTuple since we don't know the primal type, only the names of the values contained. In fact we can't actually overload addition at all for NamedTuple as that would be type-piracy, so have to use Zygote.accum instead.

Another use of the primal being a type parameter is to catch errors. ChainRules disallows the addition of Tangent{SVD} to Tangent{QR} since in a correctly differentiated program that can never occur.

## Differentials types for computational efficiency

There is another kind of unnatural differential. One that is for computational efficiency. ChainRules has Thunks and InplaceableThunks, which wrap the computation of a derivative and delays that work until it is needed, either via the derivative being added to something or being unthunked manually, thus saving time if it is never used.

Another differential type used for efficiency is ZeroTangent which represents the hard zero (in Zygote v0.4 this is nothing). For example the derivative of f(x, y)=2x with respect to y is ZeroTangent(). Add ZeroTangent() to anything, and one gets back the original thing without change. We noted that all differentials need to be a vector space. ZeroTangent() is the trivial vector space. Further, add ZeroTangent() to any primal value (no matter the type) and you get back another value of the same primal type (the same value in fact). So it meets the requirements of a differential type for all primal types. ZeroTangent can save on memory (since we can avoid allocating anything) and on time (since performing the multiplication ZeroTangent and Thunk are both examples of a differential type that is valid for multiple primal types.

## Conclusion

Now, you have seen examples of both differential types that work for multiple primal types, and primal types that have multiple valid differential types. Semantically we can handle these very easily in julia. Just put in a few more dispatching on +. Multiple-dispatch is great like that. The down-side is our type-inference becomes hard. If you have exactly 1 differential type for each primal type, you can very easily workout what all the types on your reverse pass will be - you don't really need type inference - but you lose so much expressibility.

## Appendix: What Swift does

I don't know how Swift is handling thunks, maybe they are not, maybe they have an optimizing compiler that can just slice out code-paths that don't lead to values that get used; maybe they have a language built in for lazy computation.

They are, as I understand it, handling ZeroTangent by requiring every differential type to define a zero method – which it has since it is a vector space. This costs memory and time, but probably not actually all that much. With regards to handling multiple different differential types for one primal, like natural and structural derivatives, everything needs to be converted to the canonical differential type of that primal.

As I understand it, things can be automatically converted by defining conversion protocols or something like that, so rule authors can return anything that has a conversion protocol to the canonical differential type of the primal.

However, it seems like this will run into problems. Recall that the natural differential in the case of getindex on an AbstractArray was a sparse array. But for say the standard dense Array, the only reasonable canonical differential type is also a dense Array. But if you convert a sparse array into a dense array you do giant allocations to fill in all the other entries with zero.

So this is the story about why we have many-to-many differential types in ChainRules.