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REVIEW 4 major objections 4 minor 45 references

High-Dimensional Bayesian Optimization Using Both Random and Supervised Embeddings

T0 review · 4 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read EGORSE claims that high-dimensional Bayesian optimization becomes tractable when the optimizer adaptively learns a low-dimensional linear subspace and replaces a costly quadratic feasibility test with a constraint function.

desk verdict A useful incremental HDBO method with one good trick, but the central 'outperforms by far' claim is not supported by the paper's own data. read the letter →

arxiv 2502.00854 v1 pith:R6NDN3NK submitted 2025-02-02 math.OC cs.LGstat.ML

Rémy Priem , Youssef Diouane , Nathalie Bartoli , Sylvain Dubreuil , Paul Saves This is my paper · ORCID
classification math.OCcs.LGstat.ML MSC 90C2690C56
keywords Bayesianoptimizationhigh-dimensionallinearembeddingrandomsupervisedpartialleastsquaresGaussianprocessexpensiveblack-box
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

EGORSE is a Bayesian optimization method aimed at expensive black-box functions defined on high-dimensional spaces, here tested with 10 to 600 design variables. The paper's central claim is that such problems become tractable if the objective is assumed to depend on a low-dimensional linear subspace, with the optimization carried out inside that subspace while the subspace itself is updated as function evaluations accumulate. The method combines random Gaussian and hash embeddings with supervised embeddings built from partial least squares, so the search directions reflect observed variation of the function rather than being purely random. The authors argue that this yields faster convergence both in CPU time and in number of expensive evaluations than existing high-dimensional Bayesian optimization solvers, and they report that the comparison methods become time-intractable beyond roughly one hundred variables. A reader should care because simulation-based design, such as aircraft multidisciplinary optimization, is exactly the regime where expensive black-box evaluations in high dimension are the bottleneck.

What carries the argument

The load-bearing object is the linear embedding subspace $\mathcal{A}^{(t)} = \{ \mathbf{u} = \mathbf{A}^{(t)} \mathbf{x} : \mathbf{x} \in \Omega \}$, where the transfer matrix $\mathbf{A}^{(t)} \in \mathbb{R}^{d_e \times d}$ is produced by one of several dimension-reduction methods. The reduced optimization runs over the smallest hypercube $\mathcal{B}^{(t)}$ containing $\mathcal{A}^{(t)}$. Two backward maps carry the argument: $\gamma_B$ maps $\mathcal{A}^{(t)}$ bijectively onto $\Omega$, and $\gamma_W$ projects every point of $\mathcal{B}^{(t)}$ onto $\Omega$, which lets the objective be defined on all of $\mathcal{B}^{(t)}$. The mechanism that saves computation is the constraint function $g^{(t)}(\mathbf{u}) \ge 0$, which takes the value $1 - \|\gamma_B(\mathbf{u})\|_2^2 / d$ inside the feasible embedded set and $-\|\mathbf{u}^{\mathbf{A}^{(t)}}\|_2^2$ outside it, turning the membership test into a cheap function evaluation instead of a quadratic program solved during acquisition.

What would settle it

Run EGORSE PLS+Gaussian with $d_e=2$ on a 600-variable problem with a known 60-dimensional active subspace and a known global minimum, as in the Rover_600 benchmark, and check whether the best found value approaches the known optimum; in the paper's own run the best value is about 2.4 against an optimum of $-5$, which would falsify any claim that EGORSE finds global optima in such cases.

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Extended reading notes

Core claim

The paper's central claim is that a Bayesian optimizer can work reliably in a reduced linear subspace of dimension $d_e$ much smaller than $d$, provided the subspace is chosen by combining supervised and random dimension reduction and the reduced problem is formulated with a constraint rather than a feasibility oracle. The previous random-embedding approach required solving a quadratic program at every acquisition-function evaluation to decide whether a candidate point has a valid image in the original space; EGORSE removes that bottleneck by defining the objective and a constraint on the whole bounding hypercube, so a standard constrained Bayesian optimization subroutine can be used. At each outer iteration, transfer matrices are rebuilt from all previously evaluated points using partial least squares and a random Gaussian matrix, which is the adaptive part of the method. On the 100-variable benchmark the method is reported to converge fastest in CPU time, and on the 600-variable path-planning problem it reaches a lower objective value than the hash-embedding baseline in comparable time, although the best value it finds is far from the known global optimum.

Load-bearing premise

The objective function truly depends on a low-dimensional linear subspace of dimension $d_e$, and the user-chosen $d_e$ (two in all main experiments) captures enough of that variation for the optimum to lie in the searched subspace.

Editorial extensions

If this is right

  • High-dimensional expensive black-box problems with hundreds of design variables can be optimized within a few thousand evaluations, with the expensive part of each iteration shifted from the acquisition sub-problem to a single quadratic solve per objective evaluation.
  • The standard comparison methods TuRBO, EGO-KPLS, and RREMBO would be impractical in CPU time for problems beyond about one hundred variables, leaving EGORSE as the plausible choice in that regime.
  • The recommended configuration is the PLS+Gaussian variant with an initial design of $d$ points, because the parametric study reports the best convergence speed and robustness for that setting.
  • When the true number of effective directions is much larger than the user-specified $d_e$, EGORSE will produce a valid but not globally optimal answer; the Rover_600 experiment is the paper's own demonstration of this limitation.
  • The same machinery is intended to extend to constrained black-box problems, which the authors identify as the necessary next step for realistic aerospace design.

Reading between the lines

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

  • The constraint-function replacement is a general trick: any embedding method that provides a transfer matrix and a projection can be plugged into the same reduced formulation, so the CPU-time argument is not limited to PLS or Gaussian embeddings.
  • The Rover_600 shortfall suggests a practical diagnostic: estimate the number of effective directions before choosing $d_e$ (for example by looking at the decay of the PLS variance), because a too-small $d_e$ can silently discard the subspace containing the optimum.
  • If the speed gain really comes from removing quadratic solves from the acquisition loop, then the gap between EGORSE and HESBO should widen as the dimension $d$ and the acquisition budget grow; a controlled scaling study would isolate that mechanism.
  • The claim of superiority rests on two benchmark families, both of which are linear embeddings of low-dimensional functions; testing on functions with genuinely high effective dimension or with nonlinear low-dimensional structure would show whether the method's advantage generalizes.
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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

4 major / 4 minor

Summary. The paper proposes EGORSE, a high-dimensional Bayesian optimization method that adaptively combines several low-dimensional linear embeddings, including supervised dimension-reduction techniques (PLS, MGP) and random embeddings, and replaces the quadratic-program feasibility test of RREMBO with a constraint function in the embedded space. The method is evaluated on modified Branin problems with 10 and 100 variables and a 600-variable Rover path-planning problem, against TuRBO, EGO-KPLS, RREMBO, and HESBO. The central claim, stated in Section VIII, is that EGORSE 'outperforms by far all the state-of-the-art HDBO solvers' in both CPU time and number of expensive function evaluations.

Significance. If supported, the main contribution is practically useful: the constraint-based formulation of Eq. (9) avoids solving a quadratic programming problem at every acquisition-function evaluation, and the adaptive combination of supervised and random embeddings is a sensible way to balance exploitation of discovered active directions with exploration. The paper is clearly structured, the algorithms are described in detail, and the experiments include external baseline methods and a problem with 600 variables. However, the evidence supports only a modest version of the claim. The paper's own results in Sections VI and VII undercut the 'outperforms by far' wording, and the fixed effective dimension d_e=2 is a load-bearing assumption that is violated by the non-toy benchmark.

major comments (4)
  1. [Section IV.A and Section VII.B] The method's core assumption is that the objective depends only on d_e effective directions, and d_e is set to 2 in all experiments. On Rover_600 the true effective dimension is 60, so no two-dimensional subspace can contain the active directions; Figure 9b shows the best value near 2.4 versus the known optimum -5, and Section VII.B explicitly concedes that 'EGORSE cannot provide better results.' The conclusion in Section VIII that EGORSE 'outperforms by far all the state-of-the-art HDBO solvers' is therefore unsupported for problems whose effective dimension exceeds d_e. The authors should either restrict the claim to the low-effective-dimension regime or provide experiments in which d_e is comparable to the problem's actual effective dimension.
  2. [Section VI.B, Figures 7a, 7c, and 9a] The reported data do not support the sweeping conclusion. On MB_10, EGO-KPLS and TuRBO converge faster and to lower objective values, both by number of evaluations and by CPU time. On MB_100, only the CPU-time comparison clearly favors EGORSE. On Rover_600, HESBO reaches its final value faster in CPU time. The defensible claims are narrower: EGORSE PLS+Gaussian is CPU-efficient on MB_100 and achieves a lower value than HESBO on Rover_600 in similar wall-clock time. The 'by far all state-of-the-art HDBO solvers' statement should be revised to match these data.
  3. [Section IV.E] The sentence 'The convergence properties, given by Binois et al., are thus preserved' is asserted without proof. The adaptive setting here differs from Binois et al. in two substantive ways: the embedding is re-estimated from all evaluated points at each outer iteration, and the optimization is performed under the constraint function (9) rather than with the modified EI used in RREMBO. No theorem is cited that covers this variant, and the adaptive resampling of the embedding is precisely the part that their analysis does not address. Either provide a proof or a precise reference for this setting, or remove the claim.
  4. [Section VI.A] The comparison protocol is not apples-to-apples for the CPU-time claims. TuRBO uses five trust regions and its own initial design, RREMBO and HESBO are run as 20 concatenated independent optimizations, and EGORSE and EGO-KPLS use a different initial design. The total budget per method, including initialization and hyperparameter fitting, should be reported on a common basis, and the CPU-time comparisons should state all wall-clock components. Without this, the claim that EGORSE is the fastest in CPU time is difficult to interpret.
minor comments (4)
  1. [Abstract and figures] There are several typos: 'high-dimensionnal' in the abstract, 'Hach' instead of 'Hash' in Figures 5 and 6, and 'EGO-KPSL' instead of 'EGO-KPLS' in the caption of Figure 9.
  2. [Eq. (9)] The text says the constraint function is 'normalized to provide values in [-1,1]', but the infeasible branch -||u_A||_2^2 is not normalized as written; please state the normalization step explicitly.
  3. [Section V.C.2] The phrase 'using 10 initial DoE' should be clarified as 'using 10 independent initial designs', since each optimization run uses one initial DoE.
  4. [Section VI.B] Calling TuRBO, EGO-KPLS, and RREMBO 'intractable in time' is too strong given that Figures 7c and 7d show them completing in several hours; 'substantially more expensive in wall-clock time' would be accurate.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: EGORSE's central claims are empirical benchmark comparisons against external baselines, not consequences of fitted parameters or self-citations.

full rationale

The derivation chain in this paper is an algorithmic construction, not a self-referential reduction. The reduced problem (12), constraint (9), backward maps (7)/(10), and adaptive embeddings are all defined directly from the objective function and the chosen subspace; the algorithm never encodes the target optimum or the benchmark answer as an input. The central claim of outperforming state-of-the-art HDBO solvers is supported by experiments against external implementations (TuRBO, RREMBO, HESBO, EGO-KPLS), so the comparison is not a fitted parameter renamed as a prediction. The hyperparameters of EGORSE are selected in Section V on the same MB_10 and MB_100 functions used later for comparison, which is a benchmark-selection weakness rather than a circular derivation; moreover, the paper's own figures show TuRBO and EGO-KPLS converging faster on MB_10, so the conclusion is not forced by the tuning. The statement in Section IV.E that 'The convergence properties, given by Binois et al.[17], are thus preserved' cites external work and is asserted without proof for the modified algorithm; this is an omitted-proof or generality risk, not circularity, and Binois et al. are not the present authors. The explicit limitation in Section VII.B that d_e=2 cannot cover the 60 effective directions of Rover_600, and the concession that 'EGORSE cannot provide better results,' further show that the paper's claims are empirical and conditional rather than tautological. No load-bearing self-citation chain appears: self-citations to SEGOMOE, SMT, and related prior work are tooling and method references, not the justification for the claimed superiority. Overall, the derivation is self-contained and the empirical claims stand or fall on benchmark evidence, not on circular construction.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The method rests on the low-effective-dimension assumption of Section IV.A, the validity of the extended backward mapping and constraint in Section IV.D, and an unproved claim that combining supervised and random embeddings preserves prior convergence guarantees. The free parameters are mostly user-set budgets and a configuration tuned on the same benchmarks used for comparison.

free parameters (5)
  • effective dimension d_e = 2 (user-selected; actual effective dimension is 60 in Rover_600)
    The algorithm requires the user to specify the reduced-space dimension; all main experiments use d_e=2, and the Rover_600 failure is attributed to d_e being far below the true 60.
  • initial DoE size = d points (chosen in sensitivity study)
    Section V.C.3 concludes d points gives the best speed/robustness trade-off compared with 5 and 2d; this choice biases the later comparison because the same benchmark family is used.
  • subspace evaluation budget max_nb_it_sub = 20*d_e
    Set in Section V.C.2 and used throughout; the effectiveness of each subspace optimization depends on this hand-set budget.
  • total iteration budget max_nb_it = 10 or 20 iterations (800 evaluations per run)
    Iterations are doubled for single-embedding variants to equalize evaluation counts; this is a hand-set budget that shapes all convergence plots.
  • dimension reduction method combination = PLS + Gaussian
    Selected by inspecting convergence plots on MB_10 and MB_100; the comparison only reports this tuned configuration, so part of the observed gap versus baselines may come from tuning.
assumptions (4)
  • domain assumption The objective function f is assumed to depend only on a d_e-dimensional linear subspace: f(x)=f_A(Ax) with d_e << d.
    Stated in Section IV.A; all experiments construct test functions with this property, but real problems may not satisfy it, and Rover_600 shows the consequence when d_e is too small.
  • standard math The backward application gamma_B of Eq. (7) and the extended gamma_W of Eq. (10) provide valid maps between A(t) and Omega, and every minimum of f(t) over B(t) corresponds to a minimum of f over Omega; in particular 'one of the objective function minima is necessarily in A(t)'.
    This is the mathematical argument in Sections IV.C-D, relying on A having full row rank and f depending only on Ax; it is plausible but not formally proved in the paper.
  • domain assumption Including a random Gaussian embedding inside EGORSE preserves the convergence properties of RREMBO/Binois et al.
    Section IV.E states 'The convergence properties, given by Binois et al. [17], are thus preserved' without a proof that supervised embeddings and the constraint formulation do not break the guarantee.
  • domain assumption The CBO solver SEGO/SEGOMOE reliably solves the constrained acquisition sub-problem (12) at each iteration.
    Used as a black-box subroutine throughout; no convergence or accuracy guarantees are given for this inner solver.

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Cite this review

Pith. "Pith review of High-Dimensional Bayesian Optimization Using Both Random and Supervised Embeddings." pith.science (2026). https://pith.science/paper/R6NDN3NK

@misc{pith2026250200854,
  author       = {Pith},
  title        = {Pith review of: High-Dimensional Bayesian Optimization Using Both Random and Supervised Embeddings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R6NDN3NK}},
  note         = {Machine review of arXiv:2502.00854}
}
read the original abstract

Bayesian optimization (BO) is one of the most powerful strategies to solve computationally expensive-to-evaluate blackbox optimization problems. However, BO methods are conventionally used for optimization problems of small dimension because of the curse of dimensionality. In this paper, a high-dimensionnal optimization method incorporating linear embedding subspaces of small dimension is proposed to efficiently perform the optimization. An adaptive learning strategy for these linear embeddings is carried out in conjunction with the optimization. The resulting BO method, named efficient global optimization coupled with random and supervised embedding (EGORSE), combines in an adaptive way both random and supervised linear embeddings. EGORSE has been compared to state-of-the-art algorithms and tested on academic examples with a number of design variables ranging from 10 to 600. The obtained results show the high potential of EGORSE to solve high-dimensional blackbox optimization problems, in terms of both CPU time and the limited number of calls to the expensive blackbox simulation.

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Pith tools

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