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Precision loss in Hessian computation

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#3,104 11 comentários 0 reações 0 responsáveis Ver no GitHub

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@blegat já está trabalhando nisso.

Desde 16/10/2026.

  • #3105 de @blegat — aberto

Avaliação

Dificuldade
5/5
Tempo estimado
Mais de uma semana
Facilidade para iniciantes
15/100
Tipo de issue
Bug
Clareza
Precisa de esclarecimento
Status de atividade
Ativa
Domínio
backend

Direção de pesquisa

Start by running the included MWE, which compares the sparse Hessian from MOI.eval_hessian_lagrangian on MOI.Nonlinear.SparseReverseMode() with ForwardDiff.hessian, and check the entry H[4, 3] (about 1.2e-11 versus 1.2e-11 with a larger absolute error). The cause is the coloring-based Hessian recovery, not the derivative formulas. The maintainers have not chosen a fix among the three options (document, Double64 intermediates, or SparseMatrixColorings star coloring), so decide on the approach with them before writing code.

Escrita pelo modelo de indexação a partir do texto da issue.

Descrição

I'm not sure this is actionable, but it also seems worth doing something about: maybe just documentation, but there are a couple of other options I'll detail below.

This came up while analyzing (with Claude's help) a standard benchmark problem, the CUTEst problem nondquar at n = 5:

(x1 - x2)^2 + (x4 - x5)^2 + Σ_{i=1}^{3} (x_i + x_{i+1} + x_5)^4

Here's the MWE to demonstrate the precision loss:

import MathOptInterface as MOI
import ForwardDiff

# MOI implementation
n = 5
x = [1.0, -1.0, 1.0, -1.0, 1e-6]
v = MOI.VariableIndex.(1:n)
obj = Expr(:call, :+, :(($(v[1]) - $(v[2]))^2), :(($(v[n-1]) - $(v[n]))^2),
           (:(($(v[i]) + $(v[i+1]) + $(v[n]))^4) for i in 1:n-2)...)
model = MOI.Nonlinear.Model()
MOI.Nonlinear.set_objective(model, obj)
ev = MOI.Nonlinear.Evaluator(model, MOI.Nonlinear.SparseReverseMode(), v)
MOI.initialize(ev, [:Hess, :HessVec])

st = MOI.hessian_lagrangian_structure(ev)
vals = zeros(length(st))
MOI.eval_hessian_lagrangian(ev, vals, x, 1.0, Float64[])
H = zeros(n, n)
for ((i, j), w) in zip(st, vals)
    H[i, j] += w
    i == j || (H[j, i] += w)
end

# ForwardDiff implementation
f(z) = (z[1] - z[2])^2 + (z[n-1] - z[n])^2 + sum((z[i] + z[i+1] + z[n])^4 for i in 1:n-2)
Hf = ForwardDiff.hessian(f, x)

Now compare the two Hessians:

julia> H - Hf
5×5 Matrix{Float64}:
 0.0   0.0           0.0         0.0         0.0
 0.0   4.44089e-16  -4.3416e-17  0.0         0.0
 0.0  -4.3416e-17    0.0         4.3416e-17  0.0
 0.0   0.0           4.3416e-17  0.0         0.0
 0.0   0.0           0.0         0.0         4.44089e-16

You might think everything is fine: errors are ~eps. But they are against a baseline of 1e-11:

julia> H[4, 3]
1.2000043416036534e-11

julia> Hf[4, 3]
1.2e-11

And that's a very large error indeed. At x = [1, -1, 1, -1, 1e-6] every inner sum equals 1e-6 exactly in floating point, so the cross partial ∂²f/∂x3∂x4 = 12·(1e-6)² = 1.2e-11; ForwardDiff's answer is the correct one.

It seems pretty clear this comes from the coloring, and because your coloring is both good and important the paths to fix this are narrow. I currently see three possible options:

  • don't fix, just document
  • support Double64, if the user requests it, for intermediate storage
  • support SparseMatrixColorings (xref #3806), which would allow one to use star coloring

Any thoughts on what you prefer? The documentation or Double64 might be something I could try; I'd probably defer to others for SparseMatrixColorings if they'd be willing to pitch in.

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