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REVIEW 2 major objections 6 minor 129 references

System Architecture Optimization Strategies: Dealing with Expensive Hierarchical Problems

T0 review · 2 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read This paper claims that hierarchy-aware Bayesian optimization finds the same jet engine architecture optimum as NSGA-II using 92% fewer function evaluations.

desk verdict Solid toolkit for hierarchical architecture optimization with a plausible but uncontrolled 10x speedup claim that needs a matched NSGA-II rerun or explicit comparability argument. read the letter →

arxiv 2502.00838 v1 pith:EAA3NCF7 submitted 2025-02-02 math.OC cs.DMstat.AP

Jasper H. Bussemaker , Paul Saves , Nathalie Bartoli , Thierry Lefebvre , Rémi Lafage This is my paper · ORCID
classification math.OCcs.DMstat.AP MSC 90C2690C29
keywords systemarchitectureoptimizationBayesianhierarchicaldesignspacesGaussianprocesskernelssamplingNSGA-IIjetenginehiddenconstraints
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

This paper claims that Bayesian Optimization, when given information about the hierarchical structure of a design space, can solve realistic system architecture optimization problems with one order of magnitude fewer function evaluations than evolutionary algorithms. The demonstration case is a jet engine architecting problem where the BO algorithm reaches essentially the same optimal thrust-specific fuel consumption in 300 evaluations that NSGA-II reaches in 3250, a 92% reduction. For this to work, the paper develops three things: metrics that quantify how much of a design space is missing or redundant because of hierarchy, a Gaussian process kernel that handles hierarchical categorical variables, and a sampling scheme that corrects for uneven representation of architecture groups. It also shows experimentally that for BO, the more hierarchy information integrated, the better the optimization results.

What carries the argument

The argument rests on four devices. First, four new metrics: imputation ratio (declared vs valid design space size), correction ratio (declared vs correct size), correction fraction (share of hierarchy due to value constraints), and max rate diversity (imbalance in how often discrete values occur). Second, a hierarchical Gaussian process kernel for categorical conditionally-active variables, built by one-hot encoding with a modified Gower distance that treats inactive levels as distance zero between inactive pairs and as a penalized distance between active and inactive pairs. Third, a hierarchical sampling algorithm that groups all valid discrete design vectors by their active variable sets and weights groups to mitigate rate diversity, then fills continuous variables with Sobol' sampling. Fourth, a set of eager and lazy correction algorithms that repair invalid design vectors.

What would settle it

Re-run the simple jet engine problem in the same SBArchOpt framework: run NSGA-II for 3250 evaluations and the proposed BO algorithm for 300 evaluations under identical initial design-of-experiments and termination settings. If NSGA-II's final TSFC is more than 0.2% below BO's, or if BO cannot reach the reference TSFC value within 300 evaluations, the 92% claim is refuted.

Watch

Extended reading notes

Core claim

The central claim is that hierarchy-aware Bayesian Optimization is a practical solver for expensive, black-box, mixed-discrete, constrained, multi-objective architecture optimization problems. On the jet engine benchmark, the BO algorithm with activeness information reaches a median optimal TSFC of 6.633 g/kNs within 300 evaluations, compared with 6.640 g/kNs for NSGA-II at 3250 evaluations, which the authors describe as 92% fewer function evaluations for the same result. The paper further establishes that integrating hierarchy information through hierarchical sampling and, to a lesser extent, hierarchical Gaussian process kernels improves BO performance, while the level of hierarchy integration has little influence on NSGA-II.

Load-bearing premise

The comparison assumes the NSGA-II result at 3250 evaluations from the authors' earlier work is directly comparable to the BO runs here, with the same problem formulation, evaluation function, termination logic, and cost model.

Editorial extensions

If this is right

  • Bayesian Optimization can replace evolutionary algorithms on expensive architecture problems, cutting evaluation budgets by roughly an order of magnitude.
  • Problem-agnostic eager correction performs close to problem-specific correction, so hierarchy-aware optimizers need not hand-code repair logic.
  • For BO, hierarchical sampling consistently beats non-hierarchical sampling, while hierarchical GP kernels did not help on the tested smooth problems; the paper keeps both for harder cases.
  • The metrics IR, CR, CRF, and MRD give a standard way to characterize how much hierarchy matters before choosing an optimization strategy.
  • All algorithms and test problems are released open-source in SBArchOpt, enabling direct replication.

Reading between the lines

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

  • If the 92% savings transfer to other expensive MDAO problems, the practical bottleneck shifts from evaluation count to GP training and infill search time, which become the dominant costs.
  • The hierarchical kernel's active-to-inactive distance rule suggests a natural way to extend BO to graph-based architectures: define similarity between architecture instances directly instead of through one-hot relaxed variables.
  • The paper's own comparison suggests a sharper test: run NSGA-II and BO on the same problem with identical initial populations and termination criteria; the 10x claim would then be directly quantified rather than inherited from a prior run.
  • On problems with rougher, more discontinuous objectives than the rocket and GNC tests, hierarchical GP kernels should show larger gains; that is a testable prediction of the paper's reasoning.
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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

2 major / 6 minor

Summary. The paper proposes methods for system architecture optimization (SAO) with expensive, hierarchical, mixed-discrete, constrained, multi-objective problems. It introduces metrics (imputation/correction ratios, correction fraction, max rate diversity), a new Gaussian process kernel for hierarchical categorical variables, hierarchical sampling and correction algorithms, and a comparison of hierarchy integration strategies in NSGA-II and Bayesian optimization (BO). The central empirical claim is that the proposed BO algorithm reaches a TSFC of about 6.63-6.65 g/kNs in 300 function evaluations, which is 92% fewer than the 3250 evaluations reportedly needed by NSGA-II for a comparable or slightly worse result (Section 7, Table 19).

Significance. If substantiated, the paper provides a compelling demonstration that BO can reduce evaluation counts by an order of magnitude on realistic SAO problems, and it contributes reusable open-source components (SBArchOpt) and a dataset with experiment code. The new hierarchy metrics and the kernel for categorical hierarchical variables address a real gap in surrogate-based architecture optimization. The strengths include the reproducible experimental pipeline, the explicit statistical ranking procedure, and the careful discussion of when hierarchical GP models help or hurt.

major comments (2)
  1. [Section 7, Table 19] The central 92% claim rests on an uncontrolled cross-paper comparison. The NSGA-II result at 3250 evaluations is taken from prior work [46] and is not re-run under the same conditions as the BO runs in this paper (problem formulation, correction mechanism, hierarchical sampling, constraint handling, and termination logic). Table 19 reports no variance for this baseline and explicitly states that '∆HV regret was not available for NSGA-II with 3250 evaluations'. To support the headline claim, the authors should re-run NSGA-II at both 300 and 3250 evaluations under the present settings, reporting distributions, or else clearly qualify the claim as a comparison to a previously published result rather than a controlled experimental finding.
  2. [Section 4, Eqs. (12)-(14)] The proposed hierarchical categorical kernel is not proven to be positive semi-definite. The text states that '[Φ(Θi)] is a chosen symmetric positive definite parameterization', but this does not by itself guarantee that the correlation matrices defined by Eqs. (12)-(14) are PSD for arbitrary collections of points. Since a valid GP kernel requires PSD, the paper should provide a proof or a precise citation for the PSD property of this hierarchical kernel, or present it as an empirical/heuristic construction with the associated caveat. This is load-bearing for the paper's kernel contribution.
minor comments (6)
  1. [Section 8] The sentence 'The developed BO algorithm is implemented as as ArchSBO in SBArchOpt' contains a duplicated 'as' and should be corrected.
  2. [Table 13 caption] The caption reads 'running NGSA-II', which should be 'NSGA-II'.
  3. [Section 5.2, Table 10] The label 'Hier.' for the non-grouping hierarchical sampler is ambiguous: it could be confused with 'Hierarchical sampling' generally. The table should clarify that this row is the hierarchical sampler without grouping or weighting.
  4. [Section 4, Eq. (12)] Equation (12) writes the exponent as `exp(-√2[Φ(Θi)]_ℓ_i^r,ℓ_i^r - √2[Φ(Θi)]_ℓ_i^s,ℓ_i^s)`, which is not a standard distance-based correlation form; please verify the intended expression and correct any typographical error.
  5. [Appendix A] The ranking procedure uses a 10% significance level for the two-sample t-test without justification; please state why this threshold was chosen or cite a reference for this practice.
  6. [Abstract] The abstract's statement that 'integrating more hierarchy information in the optimization algorithms yields better optimization results for BO algorithms' is too strong in view of Table 18, where the Activeness configuration (hierarchical GP) has a 20% penalty relative to Hierarchical sampling, and Table 19, where Activeness has higher ∆HV regret than Hierarchical sampling. The abstract should include the exception noted in Section 6.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction: the BO-vs-NSGA-II 92% claim is an empirical comparison, though its baseline is a self-cited prior result that is not re-verified here.

full rationale

The paper's central derivation chain is not circular. The new hierarchical categorical GP kernel is explicitly built on earlier kernels and is then tested on externally defined problems; the number of function evaluations, TSFC values, and Delta-HV regrets are measured quantities, not parameters fitted to the claims they support. The selection of hierarchical sampling and correction strategies in Sections 5 and 6 is a ranking exercise on benchmark problems, and the jet engine application in Section 7 is evaluated against a physical pyCycle-based model with a known optimum, giving the central BO result independent content. The headline claim that BO finds the same result in 92% fewer evaluations does rely on the NSGA-II result at 3250 evaluations reported in the authors' own prior work [46]: the paper states 'The previously-found optimum was found with NSGA-II and an evaluation budget of 3250; BO therefore can be considered to be able to find the same result in 92% less function evaluations.' This is a load-bearing self-citation, and the comparison is not fully controlled because the paper does not re-run NSGA-II at that budget, reports no variance for the 3250 baseline, and does not demonstrate identical problem settings, correction logic, or termination criteria. However, this is a verification and comparability weakness rather than a definitional circularity: the NSGA-II value is not an input to the BO algorithm, and the BO result is independently measured against the physical evaluation function. No equation in the paper reduces to its own inputs, and no fitted parameter is renamed as a prediction. Accordingly, no specific circular step can be exhibited, and the appropriate finding is no significant circularity, with a small score reflecting the self-referential benchmark ecosystem and the unverified baseline.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The method depends on enumerable design spaces, discrete-only activeness, deterministic evaluations, and a surrogate-based test suite whose representativeness is assumed. Hand-chosen algorithm settings (RDmin, weighting factors, thresholds) are listed as free parameters; no new physical entities are introduced, so the invented-entity ledger is empty. The most consequential assumption is that the prior NSGA-II baseline is a valid comparison point.

free parameters (5)
  • RDmin = 80%
    Threshold for grouping by high rate-diversity variables in hierarchical sampling (Section 5.2); hand-chosen as a compromise.
  • Correction distance weighting factors = 1.1 to 1.0
    Linear weighting favoring changes on the right side of the design vector in eager and lazy similar correction (Section 5.3); hand-chosen.
  • PoV threshold = 25%
    Minimum probability of viability enforced during infill search for hidden constraints (Section 3.2); default setting.
  • KPLS components nkpls = 10
    Number of PLS components used by KPLS when the design space has more than 10 variables (Section 3.2); default setting.
  • Hierarchical sampling group weights = w=1, nact, sqrt(nx,grp)
    Options for weighting groups in hierarchical sampling; no single best, selected based on experiments (Section 5.2).
assumptions (5)
  • domain assumption Activeness of continuous variables is determined only by discrete variables (Eq. 2).
    Used in IR_c and CR_c definitions and sampling; excludes cases where continuous variables control activeness, as noted via Zaefferer and Horn.
  • domain assumption xvalid,discr can be enumerated in memory and time for the problems considered.
    Hierarchical sampling and eager correction require full enumeration; the paper notes this fails for design spaces above hundreds of millions of vectors (Section 5.2).
  • domain assumption Objective and constraint evaluations are deterministic, and hidden-constraint failures are deterministic.
    Stated in Section 2.2; BO and GP models and the ranking procedure assume no stochastic noise.
  • domain assumption The random-forest emulator Jet SM behaves like the real pyCycle evaluation for the jet engine problem.
    Test problems use Jet SM to compare strategies (Section 5.1); transfer of conclusions to the real evaluation is assumed.
  • ad hoc to paper The NSGA-II result at 3250 evaluations from prior work [46] is a valid and comparable baseline.
    Section 7 and Table 19 use this single prior data point to support the 92% reduction claim without re-running NSGA-II at that budget.

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Cite this review

Pith. "Pith review of System Architecture Optimization Strategies: Dealing with Expensive Hierarchical Problems." pith.science (2026). https://pith.science/paper/EAA3NCF7

@misc{pith2026250200838,
  author       = {Pith},
  title        = {Pith review of: System Architecture Optimization Strategies: Dealing with Expensive Hierarchical Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EAA3NCF7}},
  note         = {Machine review of arXiv:2502.00838}
}
read the original abstract

Choosing the right system architecture for the problem at hand is challenging due to the large design space and high uncertainty in the early stage of the design process. Formulating the architecting process as an optimization problem may mitigate some of these challenges. This work investigates strategies for solving System Architecture Optimization (SAO) problems: expensive, black-box, hierarchical, mixed-discrete, constrained, multi-objective problems that may be subject to hidden constraints. Imputation ratio, correction ratio, correction fraction, and max rate diversity metrics are defined for characterizing hierar chical design spaces. This work considers two classes of optimization algorithms for SAO: Multi-Objective Evolutionary Algorithms (MOEA) such as NSGA-II, and Bayesian Optimization (BO) algorithms. A new Gaussian process kernel is presented that enables modeling hierarchical categorical variables, extending previous work on modeling continuous and integer hierarchical variables. Next, a hierarchical sampling algorithm that uses design space hierarchy to group design vectors by active design variables is developed. Then, it is demonstrated that integrating more hierarchy information in the optimization algorithms yields better optimization results for BO algorithms. Several realistic single-objective and multi-objective test problems are used for investigations. Finally, the BO algorithm is applied to a jet engine architecture optimization problem. This work shows that the developed BO algorithm can effectively solve the problem with one order of magnitude less function evaluations than NSGA-II. The algorithms and problems used in this work are implemented in the open-source Python library SBArchOpt.

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.