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Directly call ForwardDiff to compute the hessian #1270

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Directly call ForwardDiff to compute the hessian #1270

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Update grad.jl
Jul 23, 2022
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add testing
Jul 26, 2022
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Jul 26, 2022
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7 changes: 3 additions & 4 deletions src/lib/grad.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -73,8 +73,7 @@ julia> hessian(sin, pi/2)
"""
hessian(f, x) = hessian_dual(f, x)

hessian_dual(f, x::AbstractArray) = forward_jacobian(x -> gradient(f, x)[1], x)[2]

hessian_dual(f, x::AbstractArray) = ForwardDiff.jacobian(x -> gradient(f, x)[1], x)
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The example you test relies on the gradient rule for ForwardDiff.jacobian, which assumes its f is pure. This is true here.

It would be worth trying to look at some examples where this isn't true, i.e. f closes over some parameters (either arrays or numbers). In this case their gradients are not traced. I believe you should get a warning from ForwardDiff.jacobian's rule.

Are there any further surprises? Anything which could be tested?

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Did you mean nn layers from Flux? I will dig in a bit tomorrow.

The scalar indexing in the above comment can be added to testing as well.

hessian_dual(f, x::Number) = ForwardDiff.derivative(x -> gradient(f, x)[1], x)

"""
Expand Down Expand Up @@ -234,11 +233,11 @@ end
diaghessian(f, args...) -> Tuple

Diagonal part of the Hessian. Returns a tuple containing, for each argument `x`,
`h` of the same shape with `h[i] = Hᵢᵢ = ∂²y/∂x[i]∂x[i]`.
`h` of the same shape with `h[i] = Hᵢᵢ = ∂²y/∂x[i]∂x[i]`.
The original evaluation `y = f(args...)` must give a real number `y`.

For one vector argument `x`, this is equivalent to `(diag(hessian(f,x)),)`.
Like [`hessian`](@ref) it uses ForwardDiff over Zygote.
Like [`hessian`](@ref) it uses ForwardDiff over Zygote.

!!! warning
For arguments of any type except `Number` & `AbstractArray`, the result is `nothing`.
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8 changes: 7 additions & 1 deletion test/utils.jl
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Expand Up @@ -3,13 +3,19 @@ using ForwardDiff
using Zygote: hessian_dual, hessian_reverse

@testset "hessian: $hess" for hess in [hessian_dual, hessian_reverse]
function f(x, bias)
hessian = hess(x->sum(x.^3), x)
return hessian * x .+ bias
end

if hess == hessian_dual
@test hess(x -> x[1]*x[2], randn(2)) ≈ [0 1; 1 0]
@test hess(((x,y),) -> x*y, randn(2)) ≈ [0 1; 1 0] # original docstring version
@test gradient(b->sum(f(rand(3),b)),rand(3))[1] ≈ [1, 1, 1]
else
@test_broken hess(x -> x[1]*x[2], randn(2)) ≈ [0 1; 1 0] # can't differentiate ∇getindex
@test_broken hess(((x,y),) -> x*y, randn(2)) ≈ [0 1; 1 0]
@test_broken gradient(b->sum(f(rand(3),b)),rand(3))[1] ≈ [1, 1, 1] # jacobian is not differentiable
end
@test hess(x -> sum(x.^3), [1 2; 3 4]) ≈ Diagonal([6, 18, 12, 24])
@test hess(sin, pi/2) ≈ -1
Expand Down Expand Up @@ -133,7 +139,7 @@ using ForwardDiff
g3(x) = sum(abs2,ForwardDiff.jacobian(f,x))
out,back = Zygote.pullback(g3,[2.0,3.2])
@test back(1.0)[1] == ForwardDiff.gradient(g3,[2.0,3.2])

# From https://github.com/FluxML/Zygote.jl/issues/1218
f1218(x::AbstractVector,y::AbstractVector) = sum(x)*sum(y)
gradf1218(x,y) = ForwardDiff.gradient(x->f1218(x,y), x)[1]
Expand Down