ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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A singular Hessian leaves the minimizing branch undetermined

Example

For $θ \in R$ and state $(u, v) \in R^{2}$ , let $f (θ, (u, v)) = \frac{1}{2} (u - θ)^{2}$ and outer objective $g ((u, v), θ) = v$ . The global minimizer set is ${(θ, v) : v \in R}$ and the inner Hessian is $(\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array})$ , which is singular. The two smooth minimizing branches $(θ, 0)$ and $(θ, θ)$ give outer objectives $0$ and $θ$ , with derivatives $0$ and $1$ . Thus without a branch-selection rule the minimizing set determines no unique hypergradient, and the nonsingular-Hessian hypothesis of the local formula fails.

Example references

Stationary branch and outer objective

Nonsingular-Hessian hypothesis of the local formula

Facts & Assumptions

Given: The smooth inner loss and outer objective in the example. Verify the minimizing set, Hessian, and the two branch derivatives.

[F1]

Smooth bilevel stationary problems and their hypergradients: a local stationary branch is a $C^{1}$ selection $w^{*} (θ)$ satisfying the stationarity residual equation.

[F2]

Hypergradient of a smooth local stationary branch: an invertible state Hessian at a stationary base point guarantees a unique local $C^{1}$ stationary branch and its inverse-Hessian hypergradient formula.

Verification

1.1givenalgebra

The inner loss is nonnegative and equals zero exactly when $u = θ$ , independently of $v$ . Therefore its full global minimizer set is ${(θ, v) : v \in R}$ .

1.2givenalgebra

Its state gradient is $(u - θ, 0)^{T}$ and its Hessian is $(\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array})$ , whose kernel contains $(0, 1)^{T}$ and is therefore singular.

2.1F1step 1.1givenalgebra

The maps $w_{0}^{*} (θ) = (θ, 0)$ and $w_{1}^{*} (θ) = (θ, θ)$ are both $C^{1}$ and lie in the minimizing set from step 1.1 for every $θ$ . Their outer objectives are $g (w_{0}^{*} (θ), θ) = 0$ and $g (w_{1}^{*} (θ), θ) = θ$ , so their derivatives are respectively $0$ and $1$ .

3.1F2step 1.2step 2.1algebra∎

Since these two admissible selections yield distinct derivatives and step 1.2 violates the invertibility hypothesis in [F2], the input does not determine one hypergradient through the argmin set, and the local formula does not apply.

Depends on

Used by

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Sources

Stephen Gould et al., On Differentiating Parameterized Argmin and Argmax Problems with Application to Bi-level Optimization, §3.2 (standard reference, not scraped)
J. Lebl, Basic Analysis II, §8.5.1, Theorem 8.5.6 (standard reference, not scraped)