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arxiv: 2606.23188 · v1 · pith:LLGHLU5Nnew · submitted 2026-06-22 · 💻 cs.LG

Stage-dependent integer-binary encoding in factorization-machine black-box optimization

Ryo Ogawa , Mayumi Nakano , Yuya Seki , Shu Tanaka This is my paper

Pith reviewed 2026-06-26 08:43 UTC · model grok-4.3

classification 💻 cs.LG
keywords black-box optimizationfactorization machineIsing machineQUBO encodingone-hot encodingdomain-wall encodingRastrigin functionstage-dependent encoding
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The pith

One-hot encoding during learning drives lower residual errors than domain-wall or binary in factorization-machine black-box optimization.

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

The paper proposes splitting the factorization-machine quadratic-optimization annealing process into two stages that use different integer encodings. One-hot encoding is applied while the factorization machine learns the surrogate model from evaluated points, then the model is converted to domain-wall encoding for the Ising-machine search step. Conversion formulas are derived that keep the surrogate objective identical over all feasible integer solutions up to an additive constant. Tests on the Rastrigin function with two and five input dimensions and two discretization levels show that the learning-stage encoding choice has the largest effect on final error, with one-hot consistently outperforming the alternatives, while the extra switch to domain-wall for search helps only under the finer discretization and higher dimension.

Core claim

Across all tested conditions the encoding chosen for the learning stage is the dominant performance factor, and one-hot encoding in that stage produces lower residual errors than domain-wall or binary encodings. The OhDw variant that additionally switches to domain-wall encoding for the search stage reaches lower error and solutions closer to the global optimum than pure one-hot FMQA when the dimension is five and the discretization is 301, but the pure one-hot version is better when the discretization is 61.

What carries the argument

Conversion formulas between one-hot and domain-wall QUBO matrices that preserve the surrogate objective over feasible integer states up to an additive constant, allowing the OhDw stage-dependent framework.

Load-bearing premise

The assumption that performance differences seen on the Rastrigin function with the chosen dimensions and discretization levels will generalize to other black-box problems and that the conversion formulas stay numerically stable inside the Ising solver.

What would settle it

Running the OhDw variant on a different benchmark such as the Rosenbrock function with the same or altered discretization levels and checking whether one-hot encoding in the learning stage still produces the lowest residual errors.

Figures

Figures reproduced from arXiv: 2606.23188 by Mayumi Nakano, Ryo Ogawa, Shu Tanaka, Yuya Seki.

Figure 1
Figure 1. Figure 1: FIG. 1. Conceptual illustration of the proposed method. The surrogate model is converted between one-hot and domain-wall encodings. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. FMQA optimization procedure of the proposed method. An initial dataset is prepared and used to train an FM surrogate model. The [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Rastrigin function with an input dimension of 2. Colors in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The transition of residual error values obtained during the FMQA optimization process for the Rastrigin function. Optimization [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The transition of norm values obtained during the FMQA optimization process for the Rastrigin function. Optimization was performed [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The dependence of the mean squared error (test error) on [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. QUBO objective values obtained by Ising-machine solution search on representative surrogate models generated after FMQA opti [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The transition of residual error values obtained during the FMQA optimization process for the Rosenbrock function. Optimization [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The transition of norm values obtained during the FMQA optimization process for the Rosenbrock function. Optimization was [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
read the original abstract

Black-box optimization (BBO) deals with problems where objective functions lack explicit analytical forms and are expensive to evaluate. Factorization machine with quadratic-optimization annealing (FMQA) constructs a surrogate model using a factorization machine (FM) and optimizes it with an Ising machine. Conventional FMQA applies a single integer-binary encoding throughout the optimization process, although the encoding best suited to surrogate learning may differ from the one best suited to Ising-machine solution search. We propose a stage-dependent FMQA framework and derive conversion formulas between one-hot and domain-wall QUBO matrices that preserve the surrogate objective over feasible integer states up to an additive constant. We evaluate the OhDw variant, which employs one-hot encoding for learning and domain-wall encoding for search, on the Rastrigin function with input dimensions N = 2 and 5 and discretization levels q = 61 and 301. Across all conditions, the dominant factor governing optimization performance is the encoding used in the learning stage, with one-hot encoding consistently yielding lower residual errors than domain-wall or binary encoding. The additional benefit of switching to domain-wall encoding for solution search is condition-dependent. For N = 5 and q = 301, OhDw achieves a lower residual error and solutions closer to the global optimum than one-hot-only FMQA, whereas for N = 5 and q = 61 the latter achieves a lower residual error. These results indicate that one-hot encoding in the learning stage is the primary performance driver and that stage-dependent encoding can provide further improvement under finer discretization.

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 / 2 minor

Summary. The manuscript proposes a stage-dependent FMQA framework for black-box optimization in which one-hot encoding is used to train the factorization-machine surrogate and domain-wall encoding is used for the subsequent Ising-machine search. Conversion formulas between the associated QUBO matrices are derived that preserve the surrogate objective over feasible integer states up to an additive constant. Experiments on the Rastrigin function at (N,q) = (2,61), (2,301), (5,61) and (5,301) show that the learning-stage encoding dominates performance, with one-hot consistently producing lower residual errors than domain-wall or binary encodings; the OhDw variant yields further improvement only under the finest discretization (N=5, q=301).

Significance. If the reported ordering holds, the work demonstrates that encoding choice for surrogate learning is the primary performance driver in FMQA and that stage-dependent encoding can provide additional gains under certain discretizations. The explicit derivation of objective-preserving conversion formulas is a concrete technical contribution that enables this separation without refitting. These elements could inform encoding strategies in other surrogate-based discrete optimization pipelines that rely on Ising solvers.

major comments (2)
  1. Abstract and experimental results: the claim that one-hot encoding in the learning stage “consistently yields lower residual errors” across the four (N,q) settings is presented without the number of independent runs, error bars, or any statistical test, leaving the dominance conclusion without a quantitative assessment of variability or significance.
  2. Conversion-formula section: although the formulas are stated to preserve the surrogate objective over feasible states up to an additive constant, the manuscript contains no explicit check that the transformed QUBO coefficients remain numerically stable when supplied to the Ising solver at q=301, which is required to confirm that the reported OhDw gains are not artifacts of solver behavior on the converted matrices.
minor comments (2)
  1. The notation “OhDw” is introduced without an early definition; a brief parenthetical on first use would improve readability.
  2. Figure captions or the results section should state the precise definition of residual error and how proximity to the global optimum is measured.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and the recommendation for minor revision. We address each major comment below.

read point-by-point responses

  1. Referee: Abstract and experimental results: the claim that one-hot encoding in the learning stage “consistently yields lower residual errors” across the four (N,q) settings is presented without the number of independent runs, error bars, or any statistical test, leaving the dominance conclusion without a quantitative assessment of variability or significance.

    Authors: We agree that the manuscript does not report the number of independent runs, error bars, or statistical tests supporting the claim. In the revised manuscript we will state the number of runs performed for each (N,q) setting, add error bars to the relevant figures or tables, and include a brief statistical comparison (e.g., paired t-tests or Wilcoxon tests) to quantify the significance of the observed differences. revision: yes

  2. Referee: Conversion-formula section: although the formulas are stated to preserve the surrogate objective over feasible states up to an additive constant, the manuscript contains no explicit check that the transformed QUBO coefficients remain numerically stable when supplied to the Ising solver at q=301, which is required to confirm that the reported OhDw gains are not artifacts of solver behavior on the converted matrices.

    Authors: We acknowledge the absence of an explicit numerical-stability verification for the converted matrices at q=301. In the revision we will add a short subsection or appendix that reports the range and condition number of the transformed QUBO coefficients for the q=301 cases and confirms that they remain within the numerical tolerance of the Ising solver used in the experiments. revision: yes

Circularity Check

0 steps flagged

No circularity: conversion formulas are independent derivations; performance claims are empirical evaluations on Rastrigin.

full rationale

The paper derives conversion formulas between one-hot and domain-wall QUBO matrices that preserve the surrogate objective over feasible states up to an additive constant; these are presented as mathematical derivations separate from the Rastrigin experiments. The central claim that one-hot encoding in the learning stage dominates performance is supported by direct experimental comparisons across N=2,5 and q=61,301, with no equations reducing reported residuals or ordering to quantities fitted from the same data or to self-citations. No self-definitional, fitted-input-as-prediction, or load-bearing self-citation patterns appear in the derivation chain or abstract. The results are self-contained against the stated benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the mathematical validity of the conversion formulas that preserve the surrogate objective up to an additive constant and on the assumption that Rastrigin results transfer to other black-box problems; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Conversion formulas between one-hot and domain-wall QUBO matrices preserve the surrogate objective over feasible integer states up to an additive constant.
    This property is asserted as the basis for switching encodings without altering the learned model.

pith-pipeline@v0.9.1-grok · 5813 in / 1390 out tokens · 15749 ms · 2026-06-26T08:43:00.330741+00:00 · methodology

discussion (0)

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Reference graph

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