diff --git a/docs/make.jl b/docs/make.jl index d7f14d8e..ed6a8c8b 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -18,6 +18,8 @@ makedocs(modules=[Flux, NNlib], "One-Hot Encoding" => "data/onehot.md", "GPU Support" => "gpu.md", "Saving & Loading" => "saving.md", + "Internals" => + ["Backpropagation" => "internals/tracker.md"], "Community" => "community.md"]) deploydocs( diff --git a/docs/src/internals/tracker.md b/docs/src/internals/tracker.md new file mode 100644 index 00000000..b9addc34 --- /dev/null +++ b/docs/src/internals/tracker.md @@ -0,0 +1,156 @@ +# Flux.Tracker + +Backpropagation, or reverse-mode automatic differentiation, is handled by the `Flux.Tracker` module. + +```julia +julia> using Flux.Tracker +``` + +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 +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 +julia> Tracker.back!(y, [1, -1]) + +julia> W.grad +2×2 Array{Float64,2}: + 5.0 6.0 +-5.0 -6.0 + +julia> x.grad +2-element Array{Float64,1}: + -2.0 + -2.0 +``` + +## Internals + +All `Tracked*` objects (`TrackedArray`, `TrackedReal`) are light wrappers around the `Tracked` type, which you can access via the `.tracker` field. + +```julia +julia> x.tracker +Flux.Tracker.Tracked{Array{Float64,1}}(0x00000000, Flux.Tracker.Call{Void,Tuple{}}(nothing, ()), true, [5.0, 6.0], [-2.0, -2.0]) +``` + +The `Tracker` stores the value and gradient of a given object, which we've seen before. + +```julia +julia> x.tracker.data +2-element Array{Float64,1}: + 5.0 + 6.0 + +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 +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 +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 + +```julia +Tracker.back(*, [1, -1], W, x) +``` + +which in turn calculates the sensitivities of the arguments (`W` and `x`) and backpropagates through their calls. This is recursive, so it will walk the entire program graph and propagate gradients to the original model parameters. + +## Custom Gradients + +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`: + +```julia +julia> 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: + +```julia +julia> minus(a::TrackedArray, b::TrackedArray) = Tracker.track(minus, a, b) +minus (generic function with 2 methods) +``` + +`Tracker.track` does two things: (1) it makes sure `minus` is called with *normal* array, not tracked ones (you can use `@show` inside `minus` to verify this), and (2) it uses the result to add a `minus` node to the tape. Look inside the result of calling `minus` to see what happened: + +```julia +julia> a, b = param([6,5,4]), param([1,2,3]) +(param([6.0, 5.0, 4.0]), param([1.0, 2.0, 3.0])) + +julia> c = minus(a, b) +Tracked 3-element Array{Float64,1}: + 5.0 + 3.0 + 1.0 + +julia> c.tracker.f +Flux.Tracker.Call{...}(minus, (param([6.0, 5.0, 4.0]), param([1.0, 2.0, 3.0]))) +``` + +Finally, we have to specify the gradient of `minus`. + +```julia +julia> Tracker.back(::typeof(minus), Δ, a, b) = + (Tracker.@back(a, Δ); Tracker.@back(b, -Δ)) +``` + +`@back(x, Δ)` tells the tracker to continue propagating the sensitivity `Δ` through `x`. Now, AD will work with any program that calls `minus`. + +```julia +julia> Flux.back!(c, 1) + +julia> a.grad +3-element Array{Float64,1}: + 1.0 + 1.0 + 1.0 + +julia> b.grad +3-element Array{Float64,1}: + -1.0 + -1.0 + -1.0 +``` + +## Notes + +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: + +```julia +minus(a::AbstractArray, b::TrackedArray) = Tracker.track(minus, a, b) +minus(a::TrackedArray, b::AbstractArray) = Tracker.track(minus, a, b) +``` + +`@back` *must* be called exactly once on each tracked input argument. You do not need to do any special handling if one of the arguments is not tracked, as `@back` will just become a no-op.