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arxiv: 2606.27082 · v1 · pith:3BUSCQ7Cnew · submitted 2026-06-25 · 💻 cs.LG · cs.DS· math.OC· quant-ph

Finding Stationary Points by Comparisons

Helin Wang , Chenyi Zhang , Xiwen Tao , Yexin Zhang , Tongyang Li This is my paper

Pith reviewed 2026-06-26 05:24 UTC · model grok-4.3

classification 💻 cs.LG cs.DSmath.OCquant-ph
keywords stationary pointscomparison oraclenon-convex optimizationquery complexityHessian estimationquantum algorithmsLipschitz continuity
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The pith

Comparison oracles suffice to find ε-stationary points of non-convex functions in Õ(n²/ε^{1.5}) queries.

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

The paper shows that stationary points of non-convex functions can be located when the only information available is which of two points has the larger function value. It gives a classical algorithm that reaches an ε-stationary point with Õ(n²/ε^{1.5}) comparison queries for twice differentiable functions whose gradient and Hessian are Lipschitz. The work also supplies the first quantum algorithm for the task, which uses Õ(n/ε^{1.5}) queries. A reader would care because many real-world optimization settings supply only comparative feedback rather than exact values or derivatives.

Core claim

For a twice differentiable f from R^n to R with Lipschitz gradient and Hessian, there exists an algorithm that visits an ε-stationary point using Õ(n²/ε^{1.5}) queries to a comparison oracle. The algorithm rests on a subroutine that estimates the normalized Hessian to accuracy δ with Õ(n² log(1/δ)) queries. A quantum algorithm in the superposition query model achieves the same task with Õ(n/ε^{1.5}) queries.

What carries the argument

The normalized Hessian estimation subroutine that approximates the Hessian direction from pairwise comparison outcomes.

If this is right

  • Stationary points are reachable without direct access to function values or gradients.
  • The classical query bound scales quadratically in dimension and as the -3/2 power of the accuracy parameter.
  • The quantum version reduces the dimension factor from quadratic to linear.
  • The same subroutine that estimates the normalized Hessian can be reused for other tasks requiring curvature information.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may extend to other oracles that return only ordering information.
  • Matching lower bounds could be derived to show the stated upper bounds are tight.
  • The quantum speedup hints at advantages when the dimension is large and comparisons can be superposed.

Load-bearing premise

The function is twice differentiable with Lipschitz continuous gradient and Hessian.

What would settle it

A twice differentiable function with Lipschitz gradient and Hessian for which every comparison-oracle algorithm needs asymptotically more than Õ(n²/ε^{1.5}) queries to reach an ε-stationary point.

Figures

Figures reproduced from arXiv: 2606.27082 by Chenyi Zhang, Helin Wang, Tongyang Li, Xiwen Tao, Yexin Zhang.

Figure 1
Figure 1. Figure 1: The structure of our stationary point finding algorithm. In this figure the arrow represents the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The intuition of ComparisonHessVec (Algorithm 1) for computing Hessian-vector products using gradient directions. x ∇f(x+ry) ∥∇f(x+ry)∥ ∇f(x−ry) ∥∇f(x−ry)∥ ∇f(x) ∥∇f(x)∥ direction of ∇2 f(x) · y Then, we use ComparisonHessVec to implement ComparisonHE (Algorithm 2), which estimates the normalized Hessian ∇2f(x)/∥∇2f(x)∥. Because a column of ∇2f(x) is given by multiplying a unit vector by ∇2f(x), we can use… view at source ↗
Figure 3
Figure 3. Figure 3: The intuition for computing the ratio. By Lemma 25, we have the second-order expansion for some small µ, ∇f(x + µv) = g + µHv + rµ, ∥rµ∥ ≤ 1 2 L2µ 2 . (24) Here we choose v as an eigenvector of Hb corresponding to the eigenvalue λv, which is the largest magnitude eigenvalue satisfying arccos D g1 ∥g1∥ , v E ≥ β and ∥Hbv∥ = |λv| ≥ λ. Otherwise, when finding the stationary point, we apply line search on dire… view at source ↗
read the original abstract

We study the problem of finding stationary points of non-convex functions when access to the objective is provided only through a comparison oracle that, given two points, outputs which has the larger function value. For a twice differentiable $f\colon\mathbb R^n\to\mathbb R$ with Lipschitz gradient and Hessian, we develop an algorithm that visits an $\epsilon$-stationary point using $\widetilde O(n^2/\epsilon^{1.5})$ queries. Our approach uses a subroutine that estimates the normalized Hessian to accuracy $\delta$ using $\widetilde O(n^2\log(1/\delta))$ queries. We further study this problem with a quantum comparison oracle model where queries can be made in superpositions, and develop the first quantum algorithm that finds an $\epsilon$-stationary point, which takes $\widetilde O(n/\epsilon^{1.5})$ queries.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper studies finding ε-stationary points of non-convex functions using only a comparison oracle. For twice differentiable f: R^n → R with Lipschitz gradient and Hessian, it claims a classical algorithm achieving Õ(n²/ε^{1.5}) queries via a subroutine that estimates the normalized Hessian to accuracy δ using Õ(n² log(1/δ)) queries. It also presents the first quantum algorithm in the superposition query model achieving Õ(n/ε^{1.5}) queries.

Significance. If correct, the results establish the first explicit query bounds for stationary-point finding under a comparison oracle, extending non-convex optimization theory to settings where only relative ordering of function values is available. The quantum improvement and the explicit dependence on dimension and smoothness parameters would be notable contributions to oracle complexity.

major comments (1)
  1. [Abstract] Abstract: the claimed Õ(n² log(1/δ)) query bound for estimating the normalized Hessian from comparisons is load-bearing for the overall Õ(n²/ε^{1.5}) classical complexity. Twice differentiability together with Lipschitz gradient and Hessian controls local variation but does not by itself supply a procedure that recovers the quadratic form (or its normalized version) from sign(f(x)−f(y)) queries; additional assumptions on the scale of ||H|| or ||∇f|| appear necessary and are not stated.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and for raising this point about the Hessian estimation subroutine. We address it directly below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claimed Õ(n² log(1/δ)) query bound for estimating the normalized Hessian from comparisons is load-bearing for the overall Õ(n²/ε^{1.5}) classical complexity. Twice differentiability together with Lipschitz gradient and Hessian controls local variation but does not by itself supply a procedure that recovers the quadratic form (or its normalized version) from sign(f(x)−f(y)) queries; additional assumptions on the scale of ||H|| or ||∇f|| appear necessary and are not stated.

    Authors: Section 3.2 of the manuscript presents the explicit subroutine. It selects comparison points at distances scaled by the given Lipschitz constants of the gradient and Hessian; these constants directly determine admissible step sizes so that sign(f(x)−f(y)) queries recover directional second-order information. The normalized Hessian is obtained by a separate magnitude estimation step that likewise uses only comparisons and the same Lipschitz bounds. No explicit bounds on ||H|| or ||∇f|| are required because the normalization and scaling are performed internally via the oracle responses. The abstract claim therefore follows from the stated assumptions alone. We will add one clarifying sentence to the abstract describing the role of the Lipschitz constants. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from oracle model

full rationale

The paper presents an algorithmic construction for finding ε-stationary points under a comparison oracle, with query bounds derived directly from twice-differentiability plus Lipschitz gradient/Hessian assumptions. The Hessian estimation subroutine is described as achieving Õ(n² log(1/δ)) queries via the stated oracle model; no equations reduce a claimed prediction to a fitted parameter by construction, no load-bearing self-citation chains appear, and no ansatz or uniqueness result is imported from prior author work. The overall Õ(n²/ε^{1.5}) bound follows from composing the subroutine with standard non-convex optimization steps under the given smoothness, remaining independent of the target result itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the twice-differentiability and Lipschitz conditions stated in the abstract; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption f is twice differentiable with Lipschitz gradient and Hessian
    Explicitly stated as the setting in which the algorithms achieve the claimed query bounds.

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discussion (0)

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