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}}(6.9213122991648e-310,6.9213636645131e-310,6.9213122991656e-310,6.9213636645131e-310), Dual{ForwardDiff.Tag{typeof(Main.f), Float64}}(6.9213122991664e-310,6.9213636645131e-310,6.9213122991672e-310,6.9213636645131e-310), Dual{ForwardDiff.Tag{typeof(Main.f), Float64}}(6.9213518015701e-310,6.9213636645131e-310,6.9213518013045e-310,6.9213636645131e-310)]))

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}}(6.9213281431323e-310,6.92132814214495e-310,6.9213281412082e-310,6.92132814096947e-310), Dual{ForwardDiff.Tag{typeof(Main.f), Float64}}(6.9213281407987e-310,6.92132814038212e-310,6.92132813995446e-310,6.92132813963273e-310), Dual{ForwardDiff.Tag{typeof(Main.f), Float64}}(6.9213281394288e-310,6.9213281392027e-310,6.9213281371616e-310,6.9213281370051e-310)]))

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