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Precision loss in Hessian computation

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@blegat 已经在做这个了。

开始于 2026年10月16日。

  • #3105 来自 @blegat —— 未关闭

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难度
5/5
预计耗时
一周以上
新手友好度
15/100
Issue 类型
缺陷
描述清晰度
需要澄清
活跃度
活跃
领域
backend

调研方向

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.

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描述

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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