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On the hardness of deterministic second-order optimization of functions with Lipschitz gradients

T0 review · 3 major / 3 minor · reviewed 2026-07-31 · deepseek-v4-flash

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Pith's one-line read This paper proves that deterministic second-order optimization of functions with Lipschitz gradients is intractable: zero-respecting algorithms cannot reach Goldstein approximate second-order stationarity in finite queries, and general dete

desk verdict Strong lower-bound paper with a real proof gap in Corollary 3.2: duplicate iterates are not handled, and that needs to be fixed before the general deterministic result stands. read the letter →

arxiv 2607.24120 v1 pith:E25IOBR7 submitted 2026-07-27 math.OC

Jiewen Guan , Anthony Man-Cho So This is my paper · ORCID
classification math.OC MSC 90C2649J5290C60
keywords C^{11}functionsweakconvexitysecond-orderstationarityGoldsteinsubdifferentialoraclecomplexitylowerboundsnonsmoothoptimizationdeterministicalgorithms
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that no deterministic algorithm can reliably find an approximate second-order stationary point of a C^{1,1} function — a differentiable function with Lipschitz-continuous gradients, whose Hessian may be nonsmooth — within a predetermined finite number of second-order oracle queries. For zero-respecting algorithms, which only update coordinates that previous gradients or Hessians have exposed, the obstacle is absolute: no finite number of queries suffices in any dimension n≥2. For general deterministic algorithms, the obstacle is dimension-sensitive: at most n−3 queries are insufficient. Because C^{1,1} functions are automatically weakly convex, this establishes deterministic second-order weakly convex optimization as intractable. The hard example matches a benign-looking oracle at every queried point, yet hides a direction along which every generalized second-order derivative is strongly negative, so no queried point can pass the Goldstein stationarity test.

What carries the argument

The central object is the Goldstein second-order δ-subdifferential, the convex hull of Clarke generalized Hessians sampled within a radius δ of the current point; because it is a set of matrices rather than a single matrix, stationarity is certified through a min-max condition over directions w. The engine of the proof is the scalarization identity ∂²_δ f(x)(w)=∂_δ⟨w,∇f⟩(x), which converts the second-order condition into a first-order Goldstein subdifferential of the directional derivative, together with a chain rule (Corollary 2.9) that lets the two-dimensional counterexample be pulled back along an isometric embedding into any higher dimension without weakening the stationarity refutation.

What would settle it

Run any deterministic zero-respecting algorithm on the paper's explicit two-dimensional hard function f(x,y;ζ) for m iterations and test each iterate against the Goldstein second-order stationarity condition with δ=1/8 and η=0.0092714; if any iterate satisfies it, Theorem 3.1 is false. A simpler check: compute max⟨z,w⟩ over the entire band R×(−1/4,1/4) for w=(10,11)/√221; if this maximum exceeds −0.0092714, the band-refutation bound in Corollary 4.4 fails.

Watch

Extended reading notes

Core claim

For any m≥1 and n≥2, there exists a C^{1,1} function f with gradient Lipschitz constant 4.086 and initial optimality gap 17/256 such that any deterministic zero-respecting second-order algorithm, run for m iterations, produces only iterates x_0,...,x_m that all fail the (ε,η,δ)-Goldstein approximate second-order stationarity test: for each iterate, min_{||w||=1} max_{z∈∂²_{1/8}f(x_k)(w)} ⟨z,w⟩ ≤ −0.0092714. Corollary 3.2 extends the failure to every deterministic algorithm, with dimension n≥m+3. The proof uses a resisting oracle that always returns (0,0,−e₁e₁ᵀ); the hard function matches this oracle at all queried points while elsewhere containing a horizontal band in which every second-orde

Load-bearing premise

The proof that the two-dimensional example can be lifted to any higher dimension uses a chain rule for the Goldstein subdifferential under an isometric embedding, a rule imported from an earlier result and slightly strengthened; if that strengthening fails for matrices with orthonormal columns, the dimension-free lower bound for all deterministic algorithms would collapse.

Editorial extensions

If this is right

  • Any standard deterministic second-order method, including Newton-type, cubic-regularized, and trust-region variants, can be forced to fail the Goldstein second-order stationarity test forever when no additional structure is assumed.
  • The dimension n enters the lower bound for general deterministic algorithms: fewer than n−3 queries are provably insufficient, so dimension-independent deterministic guarantees are impossible.
  • Weak convexity — an automatic property of C^{1,1} functions — is not by itself enough to make second-order deterministic optimization tractable.
  • The negative curvature toward which an algorithm must move cannot be reliably detected at queried points, because the constructed function is C² at those points and matches a Hessian with a zero direction.
  • The Goldstein second-order stationarity concept, which was introduced as a natural target for complexity guarantees, is shown not to admit deterministic tractability in general.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If randomization can bypass these lower bounds, as it does for analogous first-order Goldstein stationarity problems, then stochastic Hessian sampling may be the only route to dimension-independent second-order guarantees.
  • The paper's appendix conjectures that the same impossibility extends to local oracles, which return all derivatives in a neighborhood; verifying that conjecture would rule out any deterministic oracle-based rescue.
  • A natural stress test of the construction: replace the fixed refuting direction w=(10,11)/√221 with adaptive directions; if the negative band persists for all directions, the hard function is a much stronger obstruction than the proof alone requires.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper establishes deterministic oracle complexity lower bounds for computing Goldstein approximate second-order stationary points of functions with Lipschitz-continuous gradients (C^{1,1} functions). Theorem 3.1 shows that no deterministic zero-respecting second-order algorithm can compute such a point in finitely many iterations; Corollary 3.2 shows that no general deterministic second-order algorithm can do so in at most n−3 iterations, where n is the dimension. The proofs construct a resisting oracle with Hessian −e1e1^T, forcing zero-respecting iterates onto the x-axis, then build a hard 2D C^{1,1} function with local patches matching the oracle and an infinite band in which every point refutes Goldstein second-order stationarity. The general case is obtained by an orthogonal embedding using the Goldstein subdifferential chain rule. The appendix provides a detailed, largely self-contained proof with explicit constants: Lipschitz constant 4.086, initial gap 17/256, and stationarity refutation 0.0092714.

Significance. If the results are correct, they are significant: they show a sharp failure of deterministic second-order methods for weakly convex nonsmooth optimization, complementing existing first-order lower bounds and clarifying the limits of oracle-based deterministic algorithms. The paper also introduces the Goldstein second-order δ-subdifferential and proves scalarization and chain-rule tools that are likely to be useful beyond this work. The main theorems are accompanied by explicit constants and a detailed appendix, which makes the central claims verifiable and reproducible in outline. The construction is original and the paper is honest about its reliance on a previously published chain rule.

major comments (3)
  1. [§4.2, Corollary 3.2] The proof as written does not handle algorithms whose projected iterates e1^T x0, ..., e1^T xm are not pairwise distinct. The function f2 constructed in Section 4.1 is defined only when ζ = min{|xi−xj|, 1} > 0, i.e., when the centers are distinct. For an arbitrary deterministic algorithm in A^n_det(m), the transcript under the constant resisting oracle may contain repeated points or equal first coordinates (e.g., an algorithm that always queries the origin). The proof gives no merging/compression argument in Corollary 3.2. As written, the lower bound covers only algorithms with distinct projected query sequences. This is load-bearing for the claimed general deterministic lower bound, and the gap needs to be repaired, for instance by constructing f2 on the set of distinct projected points and observing that oracle compatibility and badness extend to the repeated queries.
  2. [§4.1, WLOG assumption] The same distinctness issue appears in the proof of Theorem 3.1. The text says 'Without loss of generality, we may assume that xi ≠ xj for all i ≠ j' after establishing that zero-respecting iterates lie on the x-axis. This is not justified in the manuscript: a zero-respecting algorithm may query the same point more than once. A repair is possible by taking the set of distinct iterates as the centers of the hard function, but the WLOG must be argued explicitly. Since this is the foundation for Theorem 3.1, the proof should be amended even if the final statement is likely salvageable.
  3. [Lemma 2.8 / Corollary 2.9] The chain rule Lemma 2.8 is central to Corollary 3.2 and is stated for matrices U with orthonormal columns, which the authors note is a strengthening of the result cited from [Tian and So, 2024]. The proof in the manuscript relies on the assertion 'It follows from the proof of [Tian and So, 2024, Theorem 1]' for the orthogonal case. This is an external dependency at a load-bearing step. Please state the precise lemma from [Tian and So, 2024] that is being invoked, or provide a self-contained proof of the orthogonal case. This would also let the reader verify the strengthening to non-square U with orthonormal columns.
minor comments (3)
  1. [Abstract / Corollary 3.2] The abstract phrase 'no more than n−3 calls' should be aligned with the theorem statement: Corollary 3.2 requires n ≥ m+3, so the claimed lower bound for general deterministic algorithms applies up to m ≤ n−3 iterations. This is clear in the body, but a brief qualifier in the abstract would prevent misreading.
  2. [§4.1, relabeling] After sorting the iterates, the paper says 'x0 may no longer be 0 anymore'. This is fine because the hard function construction only needs the set of queried points, but the reader should be reminded that the algorithm's initial point is still the original x0 when verifying membership in F^n_C1(L,G;A). A short clarifying sentence would help.
  3. [Appendix B] Appendix B is explicitly labeled a conjecture with numerical evidence. This is acceptable, but it is somewhat unusual to include a conjecture in the appendix of a paper whose main claims are theorems. Since the conjecture is not used, consider moving this discussion to 'Concluding remarks' or restricting it to one paragraph, to avoid implying that the local-oracle extension is proven.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the lower-bound construction is adversarial and self-contained; the only same-author import is a technical Goldstein chain rule, not the target hardness result.

full rationale

Walking the derivation chain: Theorem 3.1 is established in Sections 4.1 and Appendix A by an explicit adversary construction. The proof fixes the resisting oracle O(x)=(0,0,-e1e1^T), lets A run on it, uses the resulting first coordinates as centers, and defines the hard function f(.,.;zeta) via the C^3 blend g and ambient h. It then verifies C^{1,1}-smoothness, lower boundedness, oracle compatibility, and the band refutation max <z,w> <= -0.0092714. None of these steps invokes the Goldstein second-order stationarity concept as an input; it is only the quantity being refuted. Corollary 3.2 lifts the 2D construction to R^n by f(x)=f2(U^T x) and transfers the refutation with Corollary 2.9. The only import from the authors' prior work is Lemma 2.8, whose proof says "It follows from the proof of [Tian and So, 2024, Theorem 1]" for a first-order Goldstein chain rule. This is a technical calculus lemma used to prove Corollary 2.9; it is not the target intractability theorem and the proof is mostly reproduced. That is a minor self-citation, but it does not make the central claim reduce to its assumptions: the lower bound is not fitted to data, no stationarity predicate is assumed in the construction, and no uniqueness theorem or ansatz is imported from the same authors. The Appendix B local-oracle extension is explicitly labelled a conjecture, so it carries no circular weight. The duplicate-iterates concern raised by the skeptic is a possible correctness/completeness gap in Corollary 3.2, not a circularity, and therefore does not change the circularity score.

Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

The central proof is largely self-contained. The main external dependencies are standard variational analysis facts and one chain-rule lemma from a self-cited earlier work. No free parameters are fitted to data; the constants in the theorems are derived from explicit inequalities in the appendix.

assumptions (3)
  • standard math Clarke generalized Hessian ∂²_C f(x) is nonempty, convex, and compact for C^{1,1} functions, relying on Alexandrov's a.e. twice differentiability theorem.
    Used in Definition 2.2 and in the oracle model; standard result from Clarke 1990 and Rockafellar-Wets 2009.
  • domain assumption Goldstein subdifferential chain rule for orthogonal transformations, imported from the proof of [Tian and So, 2024, Theorem 1].
    Lemma 2.8 relies on this chain rule; Corollary 2.9 and hence Corollary 3.2 depend on it. It is plausible and standard, but not re-derived in full in this paper.
  • standard math Linear maps commute with convex hulls.
    Used in Proposition 2.4 and related scalarization formulas; a basic convex analysis fact.
invented entities (1)
  • Goldstein second-order δ-subdifferential ∂²_δ f(x)(w)
    purpose: Second-order analogue of the Goldstein subdifferential; defines the approximate second-order stationarity used in the lower bounds.
    Introduced in Definition 2.5 as a natural extension of the first-order Goldstein construction. It is a mathematical definition, not an empirically falsifiable entity.

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Cite this review

Pith. "Pith review of On the hardness of deterministic second-order optimization of functions with Lipschitz gradients." pith.science (2026). https://pith.science/paper/E25IOBR7

@misc{pith2026260724120,
  author       = {Pith},
  title        = {Pith review of: On the hardness of deterministic second-order optimization of functions with Lipschitz gradients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E25IOBR7}},
  note         = {Machine review of arXiv:2607.24120}
}
abstract

We show that no deterministic zero-respecting algorithm (resp., (general) deterministic algorithm) can compute Goldstein approximate second-order stationary points of functions with Lipschitz continuous gradients within a finite number of (resp., no more than $n-3$ with $n$ being the input dimension) second-order oracle calls. This, among other consequences, shows that deterministic second-order weakly convex optimization is intractable.

Figures

Figures reproduced from arXiv: 2607.24120 by the authors.

Figure 1
Figure 1. The landscape of p1(•; 1). Recall from Section 4.1 that the local surgery pertains to interpolating between −2 −1∥•∥2 and 3 It is worth noting that the exponent 4 in (4) is necessary and essential, and its purpose is to enforce a rapid decay within a small vicinity near 0, ensuring that certain undesired magnitudes that arise in later constructions remain controllable when gated by p2(•, •; δ); if we replace 4 with,… view at source ↗
Figure 2
Figure 2. The landscape of 1 − p2(•, •; 1). (x, y) 7→ −2 −1x 2 . To this end, consider the smooth blend g(•, •; δ) : R 2 → R defined by g(x, y; δ) := p2(x, y; δ) · Å − 1 2 x 2 ã + [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. The landscape of g(•, •; 1). 17 [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: The landscape of h11(•, •; 1); for visual clarity, we overwrite h11(x, y; 1) = −1 on (B 2 ) ∁ . Proof. To begin, we calculate h11(x, y; 1) = 4y 2 (x 2 + y 2 − 1)2 (7x 2 + y 2 − 1) − 1. In what follows, we shall derive a set of candidate stationary points of h11(•, •; 1…
Figure 5
Figure 5. Figure 5: The landscape of h12(•, •; 1); for visual clarity, we overwrite h12(x, y; 1) = 0 on (B 2 ) ∁ . as h12(x, y; 1) = 0 as soon as x = 0 or y = 0, we further suppose that x ̸= 0 and y = 0 in the sequel. ̸ By setting these partial derivatives to zero and simplifying, we obta…
Figure 6
Figure 6. Figure 6: The landscape of h22(•, •; 1); for visual clarity, we overwrite h22(x, y; 1) = −1 on (B 2 ) ∁ . Proof. In a similar vein, we begin by calculating h22(x, y; 1) = (x 2 + y 2 − 1)4 + 16y 2 (x 2 + y 2 − 1)3 + 4y 2 (x 2 + 7y 2 − 1)(x 2 + y 2 − 1)2 − 1, together with ∂ ∂xh22…
Figure 7
Figure 7. Figure 7: The landscape of q(•; 1). 24 [PITH_FULL_IMAGE:figures/full_fig_p024_7.png]
Figure 8
Figure 8. Figure 8: The landscape of h(•, •; 1). A.2.1 Stationarity refutations By construction, the function h(•, •; γ) enjoys a similar stationarity refutation to Corollary A.6. Proposition A.8 Let γ > 0 be a constant and w := (10, 11)T/ √ 221 ∈ S 2 . We have max ⟨z, w⟩ : z ∈ ∂ 2 Ch(x,…
Figure 9
Figure 9. Figure 9: An illustration of Bk, Pk, and (implicitly) Hk for k = 0, . . . , m when m = 2 and ζ = 1. Proof. We shall first prove Sm k=0 Hk ∪ Sm k=0 Pk = R 2 ; as R 2 = Sm k=0 Bk  ∪ Sm k=0 Bk ∁ , it suffices to cover these two sets by Sm k=0 Pk and Sm k=0 Hk, respectively. By (1…
Figure 10
Figure 10. Figure 10: The landscape of p1 (•; 1, 0.1). Conceivably, as soon as λ is sufficiently small, p1 (•; δ, λ) should almost coincide with p1(•; δ), thereby inheriting all its desired properties; in fact, numerical evidence suggests that setting λ = 0.1 already suffices by far. Howev…

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