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arxiv: 2604.26867 · v2 · pith:EONCAKGCnew · submitted 2026-04-29 · 🧮 math.OC

Function-free Optimization via Comparison Oracles

Katya Scheinberg , Zikai Xiong This is my paper

Pith reviewed 2026-05-21 00:03 UTC · model grok-4.3

classification 🧮 math.OC
keywords comparison oraclepreference optimizationlevel-set optimality gapnormal direction estimationfunction-free optimizationconvex optimizationzeroth-order methods
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The pith

Optimization using only pairwise preference comparisons reaches an ε level-set gap in Õ(d D²/ε²) queries by estimating normal directions to level sets.

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

This paper develops a way to solve optimization when the only available information is answers to which of two points is preferred, without any objective function or derivatives. It defines the level-set optimality gap as how far a current set of equally preferred points lies from the best feasible points and introduces the regularity radius as a certificate of stationarity. Under a regularity assumption on the preference relation in d-dimensional space, an approximate normal direction to the boundary of the preferred set can be recovered to accuracy ε using O(d log(d/ε)) comparisons. Adding convexity and a local growth condition on the regularity radius lets a normal-direction descent procedure reduce the gap to ε using O(D²/ε²) estimation steps and Õ(d D²/ε²) total comparisons, where D is the distance from the starting point to the optimum; this matches known lower bounds for methods that use normal directions.

Core claim

Under regularity of the preference relation in a d-dimensional Euclidean space, normal directions to the boundary of the current preferred set can be estimated to accuracy ε using O(d log(d/ε)) comparisons. When the preference is additionally convex and obeys a local growth condition on the regularity radius, repeated normal-direction descent steps reach an ε level-set optimality gap using at most Õ(d D²/ε²) comparisons over O(D²/ε²) steps, where D is the distance from the initial point to the optimal solutions; this matches the lower bound of Ω(D²/ε²) steps for normal-direction-span methods.

What carries the argument

Normal direction estimation from a comparison oracle, which constructs an approximate normal vector to the boundary of the current preference level set to serve as a descent direction without requiring any function values.

If this is right

  • The method applies even when no underlying objective function exists or when that function is nonsmooth, nonconvex, or discontinuous.
  • The query complexity nearly matches information-theoretic lower bounds for comparison-based methods.
  • Adaptive schemes remove the need to know the distance D or target accuracy ε in advance while preserving the same asymptotic bounds up to logs.
  • The framework directly addresses preference- and ranking-based applications where numeric objective values are unavailable or meaningless.

Where Pith is reading between the lines

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

  • The geometric view of level sets could extend to noisy or stochastic comparisons by repeating queries to control error.
  • Similar normal-direction ideas might apply to other non-numeric oracles such as ranking or tournament feedback in learning settings.
  • The approach suggests a purely comparative route to zeroth-order methods that could be tested on real user-preference data sets.

Load-bearing premise

The preference relation must be regular, convex, and satisfy a local growth condition on the regularity radius near the optimum.

What would settle it

A concrete preference relation in which estimating a normal direction to accuracy ε requires more than O(d log(d/ε)) comparisons, or in which normal-direction descent requires asymptotically more than O(D²/ε²) steps to reach an ε level-set gap.

Figures

Figures reproduced from arXiv: 2604.26867 by Katya Scheinberg, Zikai Xiong.

Figure 1
Figure 1. Figure 1: Graphs and level sets of three functions on [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the outward unit normal direction [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Illustration examples of Lemma 3.1 for nonconvex sublevel sets Sx (left two plots) and Lemma 3.2 for convex sublevel sets Sx (right one plot). outward orientation. In this subsection, we work with the scheme of fixed comparison radius h, while the adaptive comparison radius case will be presented in Section 3.3. Algorithm 1 solves the planar subproblem in span{a, b} and returns an approximately tangential … view at source ↗
read the original abstract

In this work, we study optimization specified only through a comparison oracle: given two points, it reports which one is preferred. We call it function-free optimization because we do not assume access to, nor the existence of, a canonical application-given objective function. The goal is to find the most preferred feasible point, which we call the optimal solution. This model arises in preference- and ranking-based settings where objective values and derivatives are unavailable or meaningless. Even when a representative function exists, it may be nonsmooth, nonconvex, or discontinuous. We develop an analytical and algorithmic framework based on the geometry of preference level sets, which remains well-defined from comparisons alone. We introduce the level-set optimality gap, the distance from a preference level set to the optimal solutions, and the regularity radius, a stationarity certificate. Under regularity of the preference relation in a $d$-dimensional Euclidean space, we estimate normal directions to accuracy $\epsilon$ using $O(d\log(d/\epsilon))$ comparisons, nearly matching a lower bound of $\Omega(d\log(1/\epsilon))$. Under convexity, regularity, and a local growth condition on the regularity radius, the resulting normal direction descent method reaches an $\epsilon$ level-set optimality gap using at most $\widetilde O(dD^2/\epsilon^2)$ comparisons over $O(D^2/\epsilon^2)$ normal direction estimation steps, where $D$ is the distance from the initial point to the optimal solutions. This number of steps matches the lower bound of $\Omega(D^2/\epsilon^2)$ for normal direction span-based methods. Since prior knowledge in practical applications is usually limited, we also develop adaptive schemes for estimating the normal direction and solving the optimization problem. They match the fixed-parameter complexity bounds up to logarithmic factors.

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

2 major / 3 minor

Summary. The manuscript develops a function-free optimization framework that operates exclusively via a comparison oracle reporting which of two points is preferred. It introduces the level-set optimality gap (distance from a preference level set to the optimal solutions) and the regularity radius (a stationarity certificate). Under regularity of the preference relation in d-dimensional Euclidean space, normal directions are estimated to accuracy ε using O(d log(d/ε)) comparisons, nearly matching a lower bound of Ω(d log(1/ε)). Under convexity, regularity, and a local growth condition on the regularity radius, a normal direction descent method reaches an ε level-set optimality gap using at most Õ(d D²/ε²) comparisons over O(D²/ε²) steps, where D is the distance from the initial point to the optima; this iteration count matches the Ω(D²/ε²) lower bound for normal-direction-span methods. Adaptive schemes are also developed that match the fixed-parameter bounds up to logarithmic factors.

Significance. If the stated geometric arguments and complexity derivations hold, the work offers a meaningful contribution to preference-based and ranking-driven optimization by supplying explicit, near-optimal complexity guarantees without requiring an explicit objective function. The matching of both the normal-direction estimation bound and the overall iteration count to corresponding lower bounds is a clear strength, as is the development of adaptive variants. The level-set geometry provides a well-defined alternative to classical stationarity notions when only comparisons are available.

major comments (2)
  1. [§3] §3 (Normal Direction Estimation): The O(d log(d/ε)) comparison bound for estimating the normal direction to accuracy ε is presented as a direct consequence of the regularity assumption on the preference relation, but the manuscript provides only a high-level geometric argument without an explicit derivation or lemma establishing how the sequence of comparisons produces the stated accuracy; this step is load-bearing for both the estimation result and the subsequent optimization complexity.
  2. [§4.2] §4.2 (Normal Direction Descent): The local growth condition on the regularity radius is invoked to obtain the O(D²/ε²) iteration bound and the Õ(d D²/ε²) comparison bound, yet the manuscript does not include a concrete verification or example showing that this condition holds for a non-trivial class of convex preference relations beyond the abstract statement; without this, the claimed iteration complexity remains conditional on an unverified hypothesis.
minor comments (3)
  1. [§2] The definition of the regularity radius is introduced in §2 but is referenced in the complexity statements of §3 and §4 without a brief reminder of its quantitative meaning, which would aid readability.
  2. Notation for the level-set optimality gap is used consistently but could be accompanied by a short table comparing it to classical optimality measures (e.g., function-value gap or gradient norm) to clarify its relation to existing literature.
  3. [§5] The adaptive schemes in §5 are stated to match the fixed-parameter bounds up to log factors, but the precise logarithmic overhead is not quantified in the theorem statements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of the geometric framework and complexity results. We address each major comment below and will incorporate revisions to improve rigor and clarity.

read point-by-point responses

  1. Referee: [§3] §3 (Normal Direction Estimation): The O(d log(d/ε)) comparison bound for estimating the normal direction to accuracy ε is presented as a direct consequence of the regularity assumption on the preference relation, but the manuscript provides only a high-level geometric argument without an explicit derivation or lemma establishing how the sequence of comparisons produces the stated accuracy; this step is load-bearing for both the estimation result and the subsequent optimization complexity.

    Authors: We thank the referee for this observation. The current presentation relies on a geometric argument derived from the regularity assumption to obtain the normal-direction estimation bound. We agree that an explicit lemma would strengthen the exposition and make the load-bearing step fully rigorous. In the revised manuscript we will add a dedicated lemma in Section 3 that specifies the comparison sequence, invokes the regularity condition to control the angular error, and derives the O(d log(d/ε)) guarantee together with its relation to the matching lower bound. This addition will also clarify the link to the optimization results that follow. revision: yes

  2. Referee: [§4.2] §4.2 (Normal Direction Descent): The local growth condition on the regularity radius is invoked to obtain the O(D²/ε²) iteration bound and the Õ(d D²/ε²) comparison bound, yet the manuscript does not include a concrete verification or example showing that this condition holds for a non-trivial class of convex preference relations beyond the abstract statement; without this, the claimed iteration complexity remains conditional on an unverified hypothesis.

    Authors: We appreciate the referee highlighting the need for concrete verification of the local growth condition. The condition is formulated to encompass a natural family of convex preference relations, yet we recognize that an illustrative example would make the applicability of the O(D²/ε²) bound more transparent. In the revision we will include a new example (in Section 4 or an appendix) of a non-trivial convex preference relation—such as one induced by a smooth, strictly convex quadratic-like function—for which the local growth condition on the regularity radius can be verified explicitly. This will demonstrate that the stated iteration and comparison bounds are attained for a concrete, non-degenerate case. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives its complexity bounds directly from external geometric assumptions on the preference relation (regularity, convexity, and local growth condition on the regularity radius). These hypotheses are stated as prerequisites rather than being defined in terms of the target quantities. The normal-direction estimation bound O(d log(d/ε)) and the overall Õ(d D²/ε²) comparison bound are presented as consequences of the geometry, with explicit matching to lower bounds under the same assumptions. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 2 invented entities

The central claims rest on three domain assumptions about the preference relation that are not derived from comparisons alone.

axioms (3)
  • domain assumption Regularity of the preference relation in d-dimensional Euclidean space
    Invoked to guarantee that normal directions can be estimated to accuracy ε with the stated number of comparisons.
  • domain assumption Convexity of the preference relation
    Required for the normal direction descent method to make progress toward the optimal level set.
  • domain assumption Local growth condition on the regularity radius
    Used to obtain the Õ(D²/ε²) iteration bound; without it the complexity guarantee does not hold.
invented entities (2)
  • level-set optimality gap no independent evidence
    purpose: Distance from a preference level set to the optimal solutions
    Newly defined quantity that serves as the convergence measure.
  • regularity radius no independent evidence
    purpose: Stationarity certificate based on the geometry of preference level sets
    Introduced as a new concept to certify progress from comparisons alone.

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