WIP: Issue 408 (#416)

* WIP

* WIP

* WIP; work to close #408

* forward diff extension

* oops

* adjustment for testing

* version bump

.gitignore

@@ -6,3 +6,4 @@ docs/site
 test/benchmarks.json
 Manifest.toml
 TODO.md
+default.profraw

Project.toml

@@ -1,6 +1,6 @@
 name = "Roots"
 uuid = "f2b01f46-fcfa-551c-844a-d8ac1e96c665"
-version = "2.1.1"
+version = "2.1.2"
 
 [deps]
 ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"

ext/RootsForwardDiffExt.jl

@@ -1,32 +1,60 @@
-
 module RootsForwardDiffExt
 
 using Roots
 using ForwardDiff
-import ForwardDiff: Dual, value, partials
+import ForwardDiff: Dual, value, partials, Partials, derivative, gradient!
 
-# For ForwardDiff we add a `solve` method for Dual types
-# TODO (Issue #384) ForwardDiff.hessian fails, but this works:
-#function hess(f, p)
-#    ∇(p) = ForwardDiff.gradient(f, p)
-#    ForwardDiff.jacobian(∇, p)
-#end
+# What works
+# F(p) = find_zero(f, x0, M, p)
+# G(p) = find_zero(𝐺(p), x0, M)
+#                           F                    G
+# ForwardDiff.derivative    ✓                    x (wrong answer, 0.0)
+# ForwardDiff.gradient      ✓                    x (wrong answer, 0.0)
+# ForwardDiff.hessian       ✓                    x (wrong answer, 0.0)
+# Zygote.gradient           ✓                    ✓
+# Zygote.hessian            ✓                    x (wrong answer!)
+# Zygote.hessian_reverse    ✓                    x (MethodError)
 
 function Roots.solve(ZP::ZeroProblem,
                      M::Roots.AbstractUnivariateZeroMethod,
-                     𝐩::Union{Dual{T},
-                        AbstractArray{<:Dual{T,<:Real}}
-                        };
+                     𝐩::Dual{T};
                      kwargs...) where {T}
 
+
+    # p_and_dp = 𝐩
+    p, dp = value.(𝐩), partials.(𝐩)
+
+    xᵅ = solve(ZP, M, p; kwargs...)
+
     f = ZP.F
-    pᵥ = value.(𝐩)
-    xᵅ = solve(ZP, M, pᵥ; kwargs...)
-    𝐱ᵅ = Dual{T}(xᵅ, one(xᵅ))
+    fₓ = derivative(_x -> f(_x, p), xᵅ)
+    fₚ = derivative(_p -> f(xᵅ, _p), p)
 
-    fₓ = partials(f(𝐱ᵅ, pᵥ), 1)
-    fₚ = partials(f(xᵅ, 𝐩))
-    Dual{T}(xᵅ, - fₚ / fₓ)
+    # x and dx
+    dx = - (fₚ * dp) / fₓ
+
+    Dual{T}(xᵅ, dx)
 end
 
+# cf https://discourse.julialang.org/t/custom-rule-for-differentiating-through-newton-solver-using-forwarddiff-works-for-gradient-fails-for-hessian/93002/22
+function Roots.solve(ZP::ZeroProblem,
+                     M::Roots.AbstractUnivariateZeroMethod,
+                     𝐩::AbstractArray{<:Dual{T,R,N}};
+                     kwargs...) where {T,R,N}
+
+
+    # p_and_dp = 𝐩
+    p, dp = value.(𝐩), partials.(𝐩)
+    xᵅ = solve(ZP, M, p; kwargs...)
+
+    f = ZP.F
+    fₓ = derivative(_x -> f(_x, p), xᵅ)
+    fₚ = similar(𝐩)  # <-- need this, not output of gradient(p->f(x,p), p)
+    gradient!(fₚ, _p -> f(xᵅ, _p), p)
+
+    # x_and_dx
+    dx = - (fₚ' * dp) / fₓ
+
+    Dual{T}(xᵅ, Partials(ntuple(k -> dx[k], Val(N))))
+end
 end

src/chain_rules.jl

@@ -3,19 +3,33 @@
 # ∇f = 0 => ∂/∂ₓ f(xᵅ, p) ⋅ ∂xᵅ/∂ₚ + ∂/∂ₚf(x\^α, p) ⋅ I = 0
 # or ∂xᵅ/∂ₚ = - ∂/∂ₚ f(xᵅ, p)  / ∂/∂ₓ f(xᵅ, p)
 
+# There are two cases considered
+# F(p) = find_zero(f(x,p), x₀, M, p) # f a function
+# G(p) = find_zero(𝐺(p), x₀, M)      # 𝐺 a functor
+# For G(p) first order derivatives are working
+# **but** hessian is not with Zygote. *MOREOVER* it fails
+# with the **wrong answer** not an error.
+#
+# (`Zygote.hessian` calls `ForwardDiff` and that isn't working with a functor;
+# `Zygote.hessian_reverse` doesn't seem to work here, though perhaps
+# that is fixable.)
+
+
+# this assumes a function and a parameter `p` passed in
+import ChainRulesCore: Tangent, NoTangent, frule, rrule
 function ChainRulesCore.frule(
     config::ChainRulesCore.RuleConfig{>:ChainRulesCore.HasForwardsMode},
     (_, _, _, Δp),
     ::typeof(solve),
     ZP::ZeroProblem,
-    M::AbstractUnivariateZeroMethod,
+    M::Roots.AbstractUnivariateZeroMethod,
     p;
     kwargs...,
 )
     xᵅ = solve(ZP, M, p; kwargs...)
 
     # Use a single reverse-mode AD call with `rrule_via_ad` if `config` supports it?
-    F = p -> Callable_Function(M, ZP.F, p)
+    F = p -> Roots.Callable_Function(M, ZP.F, p)
     fₓ(x) = first(F(p)(x))
     fₚ(p) = first(F(p)(xᵅ))
     fx = ChainRulesCore.frule_via_ad(config, (ChainRulesCore.NoTangent(), true), fₓ, xᵅ)[2]
@@ -24,23 +38,59 @@ function ChainRulesCore.frule(
     xᵅ, -fp / fx
 end
 
+# Case of Functor carrying parameters
+ChainRulesCore.frule(
+    config::ChainRulesCore.RuleConfig{>:ChainRulesCore.HasForwardsMode},
+    xdots,
+    ::typeof(solve),
+    ZP::Roots.ZeroProblem,
+    M::Roots.AbstractUnivariateZeroMethod,
+    ::Nothing;
+    kwargs...,
+) =
+    frule(config, xdots, solve, ZP, M; kwargs...)
+
+function ChainRulesCore.frule(
+    config::ChainRulesCore.RuleConfig{>:ChainRulesCore.HasForwardsMode},
+    (_, Δq, _),
+    ::typeof(solve),
+    ZP::Roots.ZeroProblem,
+    M::Roots.AbstractUnivariateZeroMethod;
+    kwargs...,
+)
+    # no `p`; make ZP.F the parameter (issue 408)
+    foo = ZP.F
+    zprob2 = ZeroProblem(|>, ZP.x₀)
+    nms = fieldnames(typeof(foo))
+    nt = NamedTuple{nms}(getfield(foo, n) for n ∈ nms)
+    dfoo = Tangent{typeof(foo)}(;nt...)
+
+    return frule(config,
+                 (NoTangent(), NoTangent(), NoTangent(), dfoo),
+                 Roots.solve, zprob2, M, foo)
+end
+
+
+##
+
 ## modified from
 ## https://github.com/gdalle/ImplicitDifferentiation.jl/blob/main/src/implicit_function.jl
+# this is for passing a parameter `p`
 function ChainRulesCore.rrule(
     rc::ChainRulesCore.RuleConfig{>:ChainRulesCore.HasReverseMode},
     ::typeof(solve),
     ZP::ZeroProblem,
-    M::AbstractUnivariateZeroMethod,
+    M::Roots.AbstractUnivariateZeroMethod,
     p;
     kwargs...,
 )
     xᵅ = solve(ZP, M, p; kwargs...)
 
-    f(x, p) = first(Callable_Function(M, ZP.F, p)(x))
+    f(x, p) = first(Roots.Callable_Function(M, ZP.F, p)(x))
     _, pullback_f = ChainRulesCore.rrule_via_ad(rc, f, xᵅ, p)
     _, fx, fp = pullback_f(true)
-    yp = -fp / fx
 
+    yp = -fp / fx
     function pullback_solve_ZeroProblem(dy)
         dp = yp * dy
         return (
@@ -53,3 +103,51 @@ function ChainRulesCore.rrule(
 
     return xᵅ, pullback_solve_ZeroProblem
 end
+
+# this assumes a functor 𝐺(p) for the function *and* no parameter
+ChainRulesCore.rrule(
+    rc::ChainRulesCore.RuleConfig{>:ChainRulesCore.HasReverseMode},
+    ::typeof(solve),
+    ZP::ZeroProblem,
+    M::Roots.AbstractUnivariateZeroMethod,
+    ::Nothing;
+    kwargs...,
+) =
+    ChainRulesCore.rrule(rc, solve, ZP, M; kwargs...)
+
+
+function ChainRulesCore.rrule(
+    rc::ChainRulesCore.RuleConfig{>:ChainRulesCore.HasReverseMode},
+    ::typeof(solve),
+    ZP::ZeroProblem,
+    M::Roots.AbstractUnivariateZeroMethod;
+    kwargs...,
+)
+
+
+    𝑍𝑃 = ZeroProblem(|>, ZP.x₀)
+    xᵅ = solve(ZP, M; kwargs...)
+    f(x, p) = first(Roots.Callable_Function(M, 𝑍𝑃.F, p)(x))
+
+    _, pullback_f = ChainRulesCore.rrule_via_ad(rc, f, xᵅ, ZP.F)
+    _, fx, fp = pullback_f(true)
+
+    yp = NamedTuple{keys(fp)}(-fₚ/fx for fₚ ∈ values(fp))
+
+    function pullback_solve_ZeroProblem(dy)
+        dF = ChainRulesCore.Tangent{typeof(ZP.F)}(; yp...)
+
+        dZP = ChainRulesCore.Tangent{typeof(ZP)}(;
+                                                 F = dF,
+                                                 x₀ = ChainRulesCore.NoTangent()
+                                                 )
+
+        dsolve = ChainRulesCore.NoTangent()
+        dM = ChainRulesCore.NoTangent()
+        dp = ChainRulesCore.NoTangent()
+
+        return dsolve, dZP, dM, dp
+    end
+
+    return xᵅ, pullback_solve_ZeroProblem
+end

test/test_chain_rules.jl

@@ -4,6 +4,13 @@ using Zygote
 using Test
 
 # issue #325 add frule, rrule
+struct 𝐺
+    p
+end
+(g::𝐺)(x) = cos(x) - g.p * x
+G₃(p) = find_zero(𝐺(p), (0, pi/2), Bisection())
+F₃(p) = find_zero((x,p) -> cos(x) - p*x, (0, pi/2), Bisection(), p)
+
 
 @testset "Test frule and rrule" begin
     # Type inference tests of `test_frule` and `test_rrule` with the default
@@ -33,7 +40,7 @@ using Test
     G(p) = find_zero(g, 1, Order1(), p)
     @test first(Zygote.gradient(G, [0, 4])) ≈ [1 / 2, 1 / 4]
 
-    # a tuple of functions 
+    # a tuple of functions
     fx(x, p) = 1 / x
     test_frule(solve, ZeroProblem((f, fx), 1), Roots.Newton(), 1.0; check_inferred=false)
     test_rrule(solve, ZeroProblem((f, fx), 1), Roots.Newton(), 1.0; check_inferred=false)
@@ -67,4 +74,27 @@ using Test
     )
     G2(p) = find_zero((g, gx), 1, Roots.Newton(), p)
     @test first(Zygote.gradient(G2, [0, 4])) ≈ [1 / 2, 1 / 4]
+
+    # test Functor; issue #408
+    x = rand()
+    @test first(Zygote.gradient(F₃, x)) ≈ first(Zygote.gradient(G₃, x))
+    # ForwardDiff extension makes this fail.
+    VERSION >= v"1.9.0" && @test_broken first(Zygote.hessian(F₃, x)) ≈ first(Zygote.hessian(G₃, x))
+    # test_frule, test_rrule aren't successful
+    #=
+    # DimensionMismatch: arrays could not be broadcast to a common size; got a dimension with lengths 3 and 2
+    test_frule(
+        solve,
+        ZeroProblem(𝐺(2), (0.0, pi/2)),
+        Roots.Bisection();
+        check_inferred=false,
+    )
+    # MethodError: no method matching keys(::NoTangent)
+    test_rrule(
+        solve,
+        ZeroProblem(𝐺(2), (0.0, pi/2)),
+        Roots.Bisection();
+        check_inferred=false,
+    )
+    =#
 end

test/test_extensions.jl

@@ -56,19 +56,17 @@ using ForwardDiff
         @test ForwardDiff.derivative(F, p) ≈ 1 / (2sqrt(p))
     end
 
-    # Hessian is *broken*
+    # Hessian is *fixed* for F(p) = find_zero(f, x₀, M, p)
     f = (x, p) -> x^2 - sum(p .^ 2)
     Z = ZeroProblem(f, (0, 1000))
     F = p -> solve(Z, Roots.Bisection(), p)
     Z = ZeroProblem(f, (0, 1000))
     F = p -> solve(Z, Roots.Bisection(), p)
-    hess(f, p) = ForwardDiff.jacobian(p -> ForwardDiff.gradient(F, p), p)
     for p ∈ ([1,2], [1,3], [1,4])
         @test F(p) ≈ sqrt(sum(p .^ 2))
-        @test_throws DimensionMismatch ForwardDiff.hessian(F, p)
         a, b = p
         n = sqrt(a^2 + b^2)^3
-        @test hess(F, p) ≈ [b^2 -a*b; -a*b a^2] / n
+        @test ForwardDiff.hessian(F, p) ≈ [b^2 -a*b; -a*b a^2] / n
     end
 end
 

tmp/Project.toml

@@ -0,0 +1,4 @@
+[deps]
+ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
+Diffractor = "9f5e2b26-1114-432f-b630-d3fe2085c51c"
+Zygote = "e88e6eb3-aa80-5325-afca-941959d7151f"