Backpropagation

# Flux.Tracker

Backpropagation, or reverse-mode automatic differentiation, is handled by the `Flux.Tracker` module.

``julia> using Flux.Tracker``

Here we discuss some more advanced uses of this module, as well as covering its internals.

In the basics section we covered basic usage of the `gradient` function.

``````using Flux.Tracker

Tracker.gradient((a, b) -> a*b, 2, 3) # (3.0 (tracked), 2.0 (tracked))``````

`gradient` is actually just a thin wrapper around the backpropagator-based interface, `forward`.

``````using Flux.Tracker: forward

y, back = forward((a, b) -> a*b, 2, 3) # (6.0 (tracked), Flux.Tracker.#9)

back(1) # (3.0 (tracked), 2.0 (tracked))``````

The `forward` function returns two results. The first, `y`, is the original value of the function (perhaps with tracking applied). The second, `back`, is a new function which, given a sensitivity, returns the sensitivity of the inputs to `forward` (we call this a "backpropagator"). One use of this interface is to provide custom sensitivities when outputs are not scalar.

``````julia> y, back = forward((a, b) -> a.*b, [1,2,3],[4,5,6])
(param([4.0, 10.0, 18.0]), Flux.Tracker.#9)

julia> back([1,1,1])
(param([4.0, 5.0, 6.0]), param([1.0, 2.0, 3.0]))``````

We can also take gradients in-place. This can be useful if you only care about first-order gradients.

``````a, b = param(2), param(3)

c = a*b # 6.0 (tracked)

Tracker.back!(c)

## Tracked Arrays

The `param` function converts a normal Julia array into a new object that, while behaving like an array, tracks extra information that allows us to calculate derivatives. For example, say we multiply two parameters:

``````julia> W = param([1 2; 3 4])
Tracked 2×2 Array{Float64,2}:
1.0  2.0
3.0  4.0

julia> x = param([5, 6])
Tracked 2-element Array{Float64,1}:
5.0
6.0

julia> y = W*x
Tracked 2-element Array{Float64,1}:
17.0
39.0``````

The output `y` is also a `TrackedArray` object. We can now backpropagate sensitivities to `W` and `x` via the `back!` function, and see the gradients accumulated in the `W` and `x` tracked arrays:

``````julia> Tracker.back!(y, [1, -1])

2×2 Array{Float64,2}:
5.0   6.0
-5.0  -6.0

2-element Array{Float64,1}:
-2.0
-2.0``````

You may sometimes want to drop derivative information and just get the plain value back. You can do this by calling `Tracker.data(W)`.

We can hook in to the processes above to implement custom gradients for a function or kernel. For a toy example, imagine a custom implementation of `minus`:

``minus(a, b) = a - b``

Firstly, we must tell the tracker system to stop when it sees a call to `minus`, and record it. We can do this using dispatch:

``````using Flux.Tracker: TrackedArray, track, @grad

minus(a::TrackedArray, b::TrackedArray) = track(minus, a, b)``````

`track` takes care of building a new `Tracked` object and recording the operation on the tape. We just need to provide a gradient definition.

``````@grad function minus(a, b)
return minus(data(a), data(b)), Δ -> (Δ, -Δ)
end``````

This is essentially just a way of overloading the `forward` function we saw above. We strip tracking from `a` and `b` so that we are calling the original definition of `minus` (otherwise, we'd just try to track the call again and hit an infinite regress).

Note that in the backpropagator we don't call `data(a)`; we do in fact want to track this, since nest AD will take a derivative through the backpropagator itself. For example, the gradient of `*` might look like this.

``@grad a * b = data(a)*data(b), Δ -> (Δ*b, a*Δ)``

We can then calculate the first derivative of `minus` as follows:

``````a = param([1,2,3])
b = param([3,2,1])

c = minus(a, b)  # [-2.0 (tracked), 0.0 (tracked), 2.0 (tracked)]

Tracker.back!(c, 1)
Tracker.grad(a)  # [1.00, 1.00, 1.00]
Tracker.grad(b)  # [-1.00, -1.00, -1.00]``````

For multi-argument functions with custom gradients, you likely want to catch not just `minus(::TrackedArray, ::TrackedArray)` but also `minus(::Array, TrackedArray)` and so on. To do so, just define those extra signatures as needed:

``````minus(a::AbstractArray, b::TrackedArray) = Tracker.track(minus, a, b)
minus(a::TrackedArray, b::AbstractArray) = Tracker.track(minus, a, b)``````

## Tracked Internals

All `Tracked*` objects (`TrackedArray`, `TrackedReal`) are light wrappers around the `Tracked` type, which you can access via the `.tracker` field.

``````julia> x.tracker
Flux.Tracker.Tracked{Array{Float64,1}}(0x00000000, Flux.Tracker.Call{Nothing,Tuple{}}(nothing, ()), true, [5.0, 6.0], [-2.0, -2.0])``````

The `Tracker` stores the gradient of a given object, which we've seen before.

``````julia> x.tracker.grad
2-element Array{Float64,1}:
-2.0
-2.0``````

The tracker also contains a `Call` object, which simply represents a function call that was made at some point during the forward pass. For example, the `+` call would look like this:

``````julia> Tracker.Call(+, 1, 2)
Flux.Tracker.Call{Base.#+,Tuple{Int64,Int64}}(+, (1, 2))``````

In the case of the `y` we produced above, we can see that it stores the call that produced it – that is, `W*x`.

``````julia> y.tracker.f
Flux.Tracker.Call{...}(*, (param([1.0 2.0; 3.0 4.0]), param([5.0, 6.0])))``````

Notice that because the arguments to the call may also be tracked arrays, storing their own calls, this means that `Tracker` ends up forming a data structure that records everything that happened during the forward pass (often known as a tape).

When we call `back!(y, [1, -1])`, the sensitivities `[1, -1]` simply get forwarded to `y`'s call (`*`), effectively calling

``Tracker.back(*, [1, -1], W, x)``

which in turn calculates the sensitivities of the arguments (`W` and `x`) and back-propagates through their calls. This is recursive, so it will walk the entire program graph and propagate gradients to the original model parameters.