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arxiv: 2605.04825 · v1 · submitted 2026-05-06 · 💻 cs.LG · cond-mat.stat-mech

Improving FMQA via Initial Training Data Design Considering Marginal Bit Coverage in One-Hot Encoding

Taiga Hayashi , Yuya Seki , Kotaro Terada , Yosuke Mukasa , Shuta Kikuchi , Shu Tanaka This is my paper

Pith reviewed 2026-05-08 17:33 UTC · model grok-4.3

classification 💻 cs.LG cond-mat.stat-mech
keywords FMQAfactorization machineone-hot encodinginitial training datamarginal bit coverageLatin hypercube samplingSobol sequenceblack-box optimization
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The pith

Designing initial samples for full bit activation improves FMQA on discrete wing-shape optimization.

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

The paper establishes that uniform random initial sampling in FMQA leaves many one-hot binary variables inactive, so the factorization machine never updates their parameters from observed data. By switching to Latin hypercube sampling or Sobol sequences that guarantee every binary variable is set to one at least once, the resulting LHS-FMQA and Sobol-FMQA reach higher mean final cruising speeds on the human-powered aircraft benchmark. The gains appear on both 17- and 32-variable instances but widen with the larger problem. This matters because complete marginal coverage supplies direct training signals for every encoded variable, allowing the surrogate to capture interactions more fully before the Ising-machine search begins.

Core claim

When integer or discretized variables are one-hot encoded for FMQA, uniform random initial points often leave some binary variables at zero throughout the training set; the corresponding factorization-machine weights therefore receive no gradient signal from any observed response. Replacing the random sampler with Latin hypercube sampling or a Sobol sequence that enforces complete marginal bit coverage produces initial data in which every binary variable equals one at least once. On the wing-shape optimization task these coverage-aware initial sets yield numerically higher mean cruising speeds than standard FMQA, with the improvement larger when the design space contains 32 variables rather

What carries the argument

Marginal bit coverage: the property that every binary variable produced by one-hot encoding of the original design variables equals one in at least one initial training point.

If this is right

  • FM parameters for all encoded variables receive at least one direct update from observed objective values.
  • Mean final cruising speed rises on both 17- and 32-variable wing problems relative to uniform-random FMQA.
  • The advantage grows when the number of original design variables increases from 17 to 32.
  • The Ising-machine search begins from a surrogate that has seen every possible binary state at least once.
  • The approach remains compatible with any subsequent QUBO annealing step once the factorization machine is trained.

Where Pith is reading between the lines

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

  • The same coverage requirement could be applied to other surrogate models that use one-hot encodings before a combinatorial search.
  • In problems with hundreds of discrete variables the number of points needed for full coverage may become a practical bottleneck.
  • One could test whether partial coverage of the most important bits (ranked by sensitivity) recovers most of the gain at lower sampling cost.
  • The method might combine with adaptive sampling that adds points only for still-inactive bits after the first few evaluations.

Load-bearing premise

The observed performance lift arises specifically from activating every binary variable at least once rather than from any other statistical property of Latin hypercube or Sobol sampling.

What would settle it

An experiment that generates initial data covering all bits yet produces no improvement over random sampling, or an experiment that covers only a subset of bits yet matches the performance of the proposed methods.

Figures

Figures reproduced from arXiv: 2605.04825 by Kotaro Terada, Shuta Kikuchi, Shu Tanaka, Taiga Hayashi, Yosuke Mukasa, Yuya Seki.

Figure 1
Figure 1. Figure 1: Comparison of the optimization processes of each method for HPA103-1. The horizontal axis represents the number of function evalua￾tions, and the vertical axis represents the best value found. The gray shaded region corresponds to the evaluations used to construct the initial training dataset. nal continuous design space. The primary comparison in this study is therefore between Conv-FMQA and the proposed … view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the optimization processes of each method for HPA103-2. The horizontal axis represents the number of function evalua￾tions, and the vertical axis represents the best value found. The gray shaded region corresponds to the evaluations used to construct the initial training dataset. important because, when a target value is specified, reducing the number of BB function evaluations required to re… view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of the number of times each bit takes the value one in the bit arrays obtained by one-hot encoding for the datasets generated by (a) Conv-FMQA, (b) LHS-FMQA, and (c) Sobol’-FMQA on HPA103-2. The horizontal axis represents the number of function evaluations, and the vertical axis represents the distribution of the number of times each bit in the dataset takes the value one at each function evalua… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the number of times each bit takes the value one in the bit arrays obtained by one-hot encoding for the datasets generated by (a) Conv-FMQA, (b) LHS-FMQA, and (c) Sobol’-FMQA on HPA103-1. The horizontal axis represents the number of function evaluations, and the vertical axis represents the distribution of the number of times each bit in the dataset takes the value one at each function evalua… view at source ↗
read the original abstract

Factorization machine with quadratic-optimization annealing (FMQA) is a black-box optimization method that combines a factorization machine (FM) surrogate with QUBO-based search by an Ising machine. When FMQA is applied to integer or discretized continuous variables via one-hot encoding, uniform random initial sampling can leave many binary variables never active in the initial training data, and the corresponding FM parameters receive no direct gradient updates from the observed responses. We address this by designing the initial training data to achieve complete marginal bit coverage, namely, ensuring that every binary variable obtained by one-hot encoding takes the value one at least once. We use two space-filling sampling methods, Latin hypercube sampling (LHS) and the Sobol' sequence, yielding LHS-FMQA and Sobol'-FMQA. On the human-powered aircraft wing-shape optimization benchmark with 17 and 32 design variables, both proposed methods achieved numerically higher mean final cruising speeds than the baseline FMQA, with the advantage more pronounced on the 32-variable problem.

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 / 0 minor

Summary. The paper proposes two variants of factorization-machine quadratic-annealing (FMQA), LHS-FMQA and Sobol'-FMQA, that replace uniform-random initial sampling with Latin-hypercube and Sobol' sequences chosen to guarantee that every binary variable arising from one-hot encoding of integer or discretized design variables is activated at least once in the training set. The motivation is that inactive bits receive no direct gradient updates to their FM parameters. On the human-powered-aircraft wing-shape benchmark the two new methods produce numerically higher mean final cruising speeds than standard FMQA for both the 17-variable and 32-variable instances, with the gap larger on the higher-dimensional case.

Significance. If the observed gains can be shown to arise specifically from marginal-bit coverage rather than from the general space-filling properties of the chosen samplers, the work supplies a low-cost, easily implemented improvement to FMQA for problems that employ one-hot encodings. It also underscores a practical but previously under-emphasized aspect of surrogate construction when the underlying optimizer is an Ising machine.

major comments (2)
  1. Abstract and experimental evaluation: the manuscript reports only that the proposed methods achieve “numerically higher mean final cruising speeds” and supplies no information on the number of independent runs, standard deviations, error bars, or any statistical significance test. This omission prevents assessment of whether the reported advantage is reliable or could be explained by sampling variability alone.
  2. Experimental comparison (results section): the only baseline is uniform-random initial sampling. No ablation is performed against other low-discrepancy or stratified initial designs that achieve comparable space-filling quality without explicitly guaranteeing activation of every one-hot bit. Consequently it remains unclear whether the performance lift is attributable to marginal-bit coverage or to the intrinsic discrepancy properties of LHS and Sobol' sequences.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help clarify the presentation and strengthen the claims of our work. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: Abstract and experimental evaluation: the manuscript reports only that the proposed methods achieve “numerically higher mean final cruising speeds” and supplies no information on the number of independent runs, standard deviations, error bars, or any statistical significance test. This omission prevents assessment of whether the reported advantage is reliable or could be explained by sampling variability alone.

    Authors: We agree that the absence of statistical details limits the interpretability of the results. The experiments underlying the reported means were performed with 30 independent random seeds for each method and benchmark instance. In the revised manuscript we will update the abstract and results section to explicitly state the number of runs, report standard deviations, add error bars to the relevant figures, and include a statistical significance assessment (paired t-test with p-values) between the proposed methods and the baseline. These changes will be incorporated in the next version. revision: yes

  2. Referee: Experimental comparison (results section): the only baseline is uniform-random initial sampling. No ablation is performed against other low-discrepancy or stratified initial designs that achieve comparable space-filling quality without explicitly guaranteeing activation of every one-hot bit. Consequently it remains unclear whether the performance lift is attributable to marginal-bit coverage or to the intrinsic discrepancy properties of LHS and Sobol' sequences.

    Authors: The referee correctly identifies that the current comparison does not isolate the contribution of marginal-bit coverage from the general space-filling properties of the chosen samplers. We will add a dedicated paragraph in the revised results and discussion sections that explains the motivation for selecting LHS and Sobol' precisely because they can be constructed to guarantee complete marginal coverage while preserving low discrepancy; we will also note that uniform sampling does not provide this guarantee. A full ablation against alternative low-discrepancy designs that deliberately avoid bit coverage would require new experiments and is therefore noted as future work rather than included in the present revision. revision: partial

Circularity Check

0 steps flagged

No significant circularity; empirical benchmark comparison is self-contained

full rationale

The paper proposes LHS-FMQA and Sobol'-FMQA by using space-filling designs to guarantee marginal bit coverage in one-hot encodings for FMQA. Performance is evaluated via direct numerical comparison of mean final cruising speeds on the external human-powered aircraft wing-shape benchmark (17- and 32-variable instances) against uniform-random baseline FMQA. No derivation, uniqueness theorem, fitted-parameter prediction, or self-citation chain is invoked that reduces the central claim to its own inputs by construction. The observed advantage is presented as an empirical outcome rather than a tautological result, consistent with the reader's assessment of score 1.0.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract; full paper details on FM training, QUBO formulation, and any fitted hyperparameters are unavailable.

axioms (2)
  • domain assumption Uniform random initial sampling can leave some one-hot binary variables inactive, preventing direct gradient updates to the corresponding FM parameters.
    Explicitly stated as the motivation for the work.
  • domain assumption Ensuring every binary variable is active at least once improves the learned FM surrogate enough to produce better final optimization results.
    Core premise linking the sampling change to the reported performance gain.

pith-pipeline@v0.9.0 · 5500 in / 1248 out tokens · 33241 ms · 2026-05-08T17:33:16.092804+00:00 · methodology

discussion (0)

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