Gradients

The supertype Gradient comprises different ways of taking gradients:

• GradientFunction,
• GradientAutodiff,
• GradientFiniteDifferences.

We first start by showing GradientAutodiff:

f(x::AbstractArray) = sum(x .^ 2)
x = rand(3)
grad = GradientAutodiff(f, x)

GradientAutodiff{Float64, typeof(Main.f), ForwardDiff.GradientConfig{ForwardDiff.Tag{typeof(Main.f), Float64}, Float64, 3, Vector{ForwardDiff.Dual{ForwardDiff.Tag{typeof(Main.f), Float64}, Float64, 3}}}}(Main.f, ForwardDiff.GradientConfig{ForwardDiff.Tag{typeof(Main.f), Float64}, Float64, 3, Vector{ForwardDiff.Dual{ForwardDiff.Tag{typeof(Main.f), Float64}, Float64, 3}}}((Partials(1.0, 0.0, 0.0), Partials(0.0, 1.0, 0.0), Partials(0.0, 0.0, 1.0)), ForwardDiff.Dual{ForwardDiff.Tag{typeof(Main.f), Float64}, Float64, 3}[Dual{ForwardDiff.Tag{typeof(Main.f), Float64}}(6.93185735838823e-310,6.93185735837637e-310,6.93188586195233e-310,6.93188586194284e-310), Dual{ForwardDiff.Tag{typeof(Main.f), Float64}}(6.9318573583906e-310,6.93185735837163e-310,6.931857358374e-310,6.93185735837875e-310), Dual{ForwardDiff.Tag{typeof(Main.f), Float64}}(6.93185735839298e-310,6.9318858317562e-310,6.9318858317582e-310,0.0)]))

Instead of calling GradientAutodiff(f, x) we can equivalently do:

grad = Gradient(f, x; mode = :autodiff)

GradientAutodiff{Float64, typeof(Main.f), ForwardDiff.GradientConfig{ForwardDiff.Tag{typeof(Main.f), Float64}, Float64, 3, Vector{ForwardDiff.Dual{ForwardDiff.Tag{typeof(Main.f), Float64}, Float64, 3}}}}(Main.f, ForwardDiff.GradientConfig{ForwardDiff.Tag{typeof(Main.f), Float64}, Float64, 3, Vector{ForwardDiff.Dual{ForwardDiff.Tag{typeof(Main.f), Float64}, Float64, 3}}}((Partials(1.0, 0.0, 0.0), Partials(0.0, 1.0, 0.0), Partials(0.0, 0.0, 1.0)), ForwardDiff.Dual{ForwardDiff.Tag{typeof(Main.f), Float64}, Float64, 3}[Dual{ForwardDiff.Tag{typeof(Main.f), Float64}}(NaN,0.0,6.9318831454952e-310,5.0e-324), Dual{ForwardDiff.Tag{typeof(Main.f), Float64}}(3.12e-321,5.0e-324,6.93188314550154e-310,5.0e-324), Dual{ForwardDiff.Tag{typeof(Main.f), Float64}}(NaN,0.0,6.93188314550786e-310,5.0e-324)]))

When calling an instance of Gradient we can use the functions gradient and gradient!^[1]:

gradient(x, grad)

3-element Vector{Float64}:
 1.042427591070766
 1.1736135149066969
 1.7817573961855622