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REVIEW 3 major objections 6 minor 31 references

Dimensionality Reduction Techniques for Global Bayesian Optimisation

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Structured latent manifolds from variational autoencoders improve Bayesian optimisation in 100 dimensions, with sequential domain reduction in the latent space accelerating convergence.

desk verdict A useful workshop-level extension—SDR in VAE latent space—but the headline claim is confounded by a missing control cell, and the empirical base is thinner than the prose. read the letter →

arxiv 2412.09183 v1 pith:TTN2V5F5 submitted 2024-12-12 math.OC cs.LGcs.NAmath.NA

Luo Long , Coralia Cartis , Paz Fink Shustin This is my paper · ORCID
classification math.OCcs.LGcs.NAmath.NA MSC 90C2690C56
keywords Bayesianoptimisationlatentspacevariationalautoencoderdimensionalityreductiondeepmetriclearningsequentialdomainhigh-dimensionalrandomembeddings
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 argues that using variational autoencoders to build a low-dimensional latent space makes Bayesian optimisation work in 100-dimensional black-box problems where standard BO and random-embedding methods struggle. The authors build on an earlier VAE-based latent-space BO algorithm, correct parts of its implementation, and add periodic retraining and a deep metric (soft triplet) loss so that the latent space is structured by objective value. They also integrate sequential domain reduction into the latent search, the first such combination, and test with the Matérn-5/2 kernel rather than the Gaussian kernel used in the earlier regret analysis. On full-rank and low-rank $D=100$ benchmarks, the retrained VAE algorithms solve more problems than BO in the original space and than REMBO, and the version with deep metric loss performs best. If this holds, expensive high-dimensional optimisation no longer needs to assume the objective is low-rank to benefit from dimensionality reduction.

What carries the argument

The load-bearing object is the VAE encoder–decoder pair $q_\phi(z|x): \mathcal{X} \to \mathcal{Z}$ and $p_\theta(x|z): \mathcal{Z} \to \mathcal{X}$, which replaces the original problem with $f^* = \min_{z \in \mathcal{Z}} \mathbb{E}_{p_{\theta^*}(x|z)}[f(x)]$. The paper's mechanism for making that replacement effective has three parts: periodic retraining of the VAE on the growing labelled dataset, a soft triplet loss added to the ELBO that pulls latent points with similar objective values together and pushes dissimilar ones apart, and sequential domain reduction that contracts the latent search box as the optimisation proceeds. The soft triplet loss is what creates the structured latent manifolds the paper credits for the performance gain; SDR is what accelerates early convergence; retraining is what keeps the decoder adapted to the region BO actually explores.

What would settle it

Take a benchmark with a known global minimiser $x^*$, train the VAE exactly as in the paper, and compute $\max_z P[x^* \sim p_{\theta^*}(\cdot|z)]$ over the latent space; if this maximum is negligible for the regions the acquisition function explores, the latent-space search cannot locate $x^*$. A cheaper experiment is to place the minimiser far outside the support of the multivariate-normal pretraining distribution and check whether the reported success rate of S-BOVAE collapses.

Watch

Extended reading notes

Core claim

The paper's central claim is that VAE-based latent-space Bayesian optimisation, reformulated for general optimisation rather than molecule generation, outperforms both ambient-space BO and random-embedding BO on 100-dimensional benchmarks. Concretely, the paper reports that applying sequential domain reduction inside the VAE latent space accelerates convergence (Figure 1), and that the algorithm combining VAE retraining with soft triplet deep metric loss, called S-BOVAE, consistently finds better global optima than the vanilla and retraining-only variants (Figure 2). On two test sets, the VAE-based algorithms reach the $\tau = 10^{-1}$ accuracy level on 90–100% of problems, whereas BO-SDR solves 10–20% and REMBO solves 50%, and the advantage persists at $\tau = 10^{-3}$. The paper also finds that small latent dimensions ($d=2,5$) work better than $d=50$, attributing this to VAE generalisation capacity, and it shows the Matérn-5/2 kernel works in this setting despite lacking the Gaussian-kernel regret bound of the earlier framework.

Load-bearing premise

The argument depends on the true best point $x^*$ being reachable through the autoencoder's decoder from some latent point; if the pretraining samples or the decoder's smoothing miss that region, the method is solving a different problem, and the paper gives no guarantee or test for this.

Editorial extensions

If this is right

  • If the central claim is correct, VAE-based LSBO with retraining and deep metric loss solves more $D=100$ benchmark problems than BO-SDR and REMBO at both $\tau = 10^{-1}$ and $\tau = 10^{-3}$ accuracy levels.
  • The structured latent spaces produced by the soft triplet loss are the main reported source of improvement, so the algorithm's success depends on the metric loss being included, not just on dimensionality reduction.
  • Smaller latent dimensions ($d=2,5$) outperform larger ones, which makes the choice of latent dimension a primary tuning lever for VAE-based LSBO at fixed ambient dimension.
  • SDR can be moved into the VAE latent space and accelerates convergence for the vanilla VAE-BO algorithm, although the paper reports that SDR and deep metric loss conflict in the current implementation.
  • The empirical success with the Matérn-5/2 kernel suggests the earlier Gaussian-kernel regret bound is not the practical bottleneck, but a Matérn-specific regret analysis is needed to match the theory to the experiments.

Reading between the lines

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

  • The paper leaves a coverage condition untested: if the unlabelled pretraining distribution or the decoder's smoothing assigns negligible probability to the region containing $x^*$, latent-space BO is optimising a different function; one concrete extension is to compute $\max_z P[x^* \sim p_{\theta^*}(\cdot|z)]$ for known minimisers and reject latent spaces with zero coverage.
  • Because SDR and deep metric loss conflict, a natural scheduling extension is to retrain with soft triplet loss during the exploration phase and switch on SDR only after the latent region has stabilised; the paper identifies the conflict but does not resolve it.
  • The full-rank benchmark success suggests VAE-based LSBO may be a general high-dimensional strategy rather than a low-effective-dimensionality one; a testable consequence is that its advantage over BO without dimensionality reduction should grow as the number of active variables increases.
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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

3 major / 6 minor

Summary. The paper studies latent-space Bayesian optimisation (LSBO) built on variational autoencoders. It proposes a reformulation of the VAE-based BO algorithm of Grosnit et al. for general black-box optimisation, introducing variants that use periodic VAE retraining and deep metric loss (soft triplet loss). It also implements Sequential Domain Reduction (SDR) inside BoTorch, both in the original ambient space and in VAE-generated latent spaces, and compares the resulting BO-VAE algorithms against BO-SDR and REMBO on 100-dimensional test problems. The central claim is that structured latent manifolds improve BO performance and that Algorithm 4 (S-BOVAE) performs best because of its structured latent space.

Significance. If substantiated, the paper would be a useful empirical contribution: it brings VAE-based LSBO beyond its original molecule-generation setting, shows the first SDR-in-latent-space implementation, and provides a direct comparison with random-embedding methods. The authors are transparent that the only theoretical regret bound they borrow does not apply to the Matérn kernel they use. However, the central empirical claim is not currently supported by the experimental design, because the algorithm comparison changes two variables at once and the statistical evidence is thin. The significance of the work therefore depends on whether the missing control experiments and uncertainty quantification can be supplied.

major comments (3)
  1. [Section 4 and Appendix B.3, Algorithm 4 vs Algorithm 1] The comparison between Algorithm 4 (S-BOVAE) and Algorithm 1 (R-BOVAE) changes two variables simultaneously: the training objective (DML objective instead of ELBO) and the presence of SDR (Algorithm 1 includes SDR; Algorithm 4 does not). Appendix B.3 explicitly states that Algorithm 4 is not implemented with SDR because 'SDR and DML methods conflict with each other in excluding the global optimum.' Therefore, the reported superiority of Algorithm 4, attributed in Section 4 to 'its structured latent spaces', could be entirely due to the removal of SDR from the latent-space loop. The experimental design lacks the control cell with DML off and SDR off; Figure 1 compares the vanilla VAE with and without SDR but does not isolate the DML component. Please add this control, or implement Algorithm 4 with SDR, or rephrase the causal claim.
  2. [Section 3, Eq. (1) and Appendix C] The reduction of problem (P) to min_{z in Z} E_{p_theta*(x|z)}[f(x)] is valid only if the decoder's latent representation covers the ambient global minimizer x*. The paper assumes there exists z with P[x* ~ p_theta*(·|z)] > 0 but provides no theoretical or numerical verification. Since the pretraining set D_U is sampled from a multivariate normal with a large covariance matrix (Appendix C) and the benchmark domains are bounded (e.g., [-30,30]^D for Ackley), it is not automatic that the decoder can generate points near x*. The authors note that the regret bound from Grosnit et al. relies on a Gaussian kernel and does not apply to the Matérn kernel used here, so the coverage assumption is the main remaining safety net and it is untested. Please report, for example, the reconstruction error of the known global minimizers for the benchmark problems, or otherwise justify that the latent-space optimum maps back to the ambient optimum.
  3. [Section 4, Tables 1-2 and Figures 1-2] The headline numerical claim rests on very small samples: each algorithm was run twice on each problem in Test Set 1 and twice on each randomized problem in Test Set 2 (10 problems), while Figures 1 and 2 average only 5 runs. No confidence intervals or standard errors are reported for the success rates in Tables 1 and 2; at tau = 10^-3, Table 1 shows 50% for all three VAE variants and Table 2 shows 40%, 30%, and 20% for the three variants. With 10 problems, these differences are not shown to be statistically significant, and the statement that Algorithm 4 'consistently performed best' is not supported. The paper also does not provide code, seeds, or per-problem traces. Please include per-problem results, confidence intervals for success proportions, and make the implementation or at least the per-run data available.
minor comments (6)
  1. [Throughout] The notation 'V AE', 'BO-V AE', and 'V AE-generated' is inconsistently spaced; please use a single consistent form throughout.
  2. [Appendix B.1] There is a typo in the sentence introducing gamma_p: 'Meanwhiel' should be 'Meanwhile'.
  3. [Appendix B.3] The phrase 'background knowlegde' should be 'background knowledge'.
  4. [Section 1] The text 'BoT orchframework' contains a spacing typo and should read 'BoTorch framework'.
  5. [Table 5 and Appendix C] The caption of Table 5 introduces (beta_i, beta_f, beta_s, beta_a) but the notation is not defined before the table; please define each symbol in the caption.
  6. [Appendix A.2] The text says the low-rank functions are scaled to [-1,1]^{d_e}, but the domain column of Table 4 still lists the original domains; this should be clarified to avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the VAE-BO claims rest on external numerical evaluations; the main attribution is confounded by a missing control cell, which is an experimental-design weakness, not circularity.

full rationale

The paper's central claims are empirical and its derivation chain is not self-referential. The VAE is pre-trained on the unlabelled dataset DU and the GP is fitted on latent observations paired with true function values; no benchmark optimum or target function value is used to set algorithm free parameters, and no fitted parameter is renamed as a prediction. The only borrowed theoretical support, Theorem 1 of Grosnit et al., is explicitly qualified: 'the proof relies on the assumption of a Gaussian kernel, limiting its direct applicability when using the Matérn kernel, as we do here,' so it is not used to force the empirical outcome. The self-citations to Cartis et al. supply test problems, the REMBO embedding scaling delta = 2.2 sqrt(d_e), and performance-profile methodology; these are external numerical constructs, not the paper's own fitted predictions. The skeptic's concern is real but is not circularity: Algorithm 4 (S-BOVAE) differs from Algorithm 1 (R-BOVAE) in two respects at once (DML added, SDR removed), and Appendix B.3 states 'Algorithm 4 is not implemented with SDR in the latent space, as experiments have shown that SDR and DML methods conflict with each other in excluding the global optimum.' This missing control cell means the headline attribution 'consistently performed best due to its structured latent spaces' is under-supported experimentally, but the absence of a control is a confound, not a reduction of the prediction to its inputs. The latent-space formulation (1) rests on an unverified coverage assumption, P[x* ~ p_theta*(.|z)] > 0, but an unverified assumption is not a circular reduction. No equation in the paper defines a claimed output in terms of the target, and no uniqueness theorem is imported from same-author work. Accordingly, there is no significant circularity.

Assumptions & free parameters 7 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the decoder-coverage assumption, the safety of SDR contraction, and the quality of the VAE latent space. These are domain assumptions, not derived results. The free parameters are hyperparameters chosen by the authors; none are fitted to the benchmark optima, so they do not make the empirical claim circular, but they do limit generality.

free parameters (7)
  • Latent dimension d = 2 (primary), also 5, 10, 50
    Chosen per experiment; results show larger d degrades performance (Section 4).
  • SDR parameters (gamma_o, gamma_p, eta, t, xi) = 0.7, 1.0, 0.9, 0.5, 1
    Appendix C; standard values selected without sensitivity study.
  • VAE pretraining hyperparameters (epochs, lr, batch, beta schedule, M) = 300, 1e-3, 1024, (0, 1, 10, 0.1), 50000
    Table 5; chosen for training stability; no ablation.
  • VAE retraining setup (epochs, lr, batch) = 2, 1e-3, 256
    Table 6; chosen to adapt VAE during BO, no sensitivity analysis.
  • Soft triplet loss parameters (eta, nu) = eta = 0.01, nu = 0.2
    Appendix C; fixed for Algorithm 4; no study of their effect.
  • Unlabelled data subsampling ratio N = 1% of DU per retraining stage
    Appendix C; selected to reduce computational cost; changes latent training set size.
  • Fixed VAE input space scale = [-3, 3]^D
    Section 4; chosen to normalise problem domains; requires scaling assumption.
assumptions (5)
  • domain assumption Decoder coverage: there exists z in Z such that P[x* ~ p_theta*(.|z)] > 0
    Section 3, after Eq (1). Without it, the latent-space objective (1) does not contain the true global minimiser.
  • domain assumption Sequential Domain Reduction does not prematurely exclude the global optimum
    Section 4 and Appendix B.1; the authors update the region every q iterations to avoid premature exclusion, but no guarantee is given for general f.
  • domain assumption A regret bound for VAE-based LSBO exists in prior work (Grosnit et al.) and justifies the approach
    Section 3; the authors state the proof assumes a Gaussian kernel and is not directly applicable to their Matérn-5/2 setting.
  • standard math GP posterior and EI acquisition in latent space are valid statistical models of the objective
    Section 2; standard Bayesian optimisation machinery with Matérn-5/2 kernel.
  • domain assumption VAE trained on unlabelled dataset DU approximates a useful latent representation of the feasible region
    Appendix C; DU is sampled from a multivariate normal with large covariance, and only 1% is used per retraining stage.

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Cite this review

Pith. "Pith review of Dimensionality Reduction Techniques for Global Bayesian Optimisation." pith.science (2026). https://pith.science/paper/TTN2V5F5

@misc{pith2026241209183,
  author       = {Pith},
  title        = {Pith review of: Dimensionality Reduction Techniques for Global Bayesian Optimisation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TTN2V5F5}},
  note         = {Machine review of arXiv:2412.09183}
}
abstract

Bayesian Optimisation (BO) is a state-of-the-art global optimisation technique for black-box problems where derivative information is unavailable, and sample efficiency is crucial. However, improving the general scalability of BO has proved challenging. Here, we explore Latent Space Bayesian Optimisation (LSBO), that applies dimensionality reduction to perform BO in a reduced-dimensional subspace. While early LSBO methods used (linear) random projections (Wang et al., 2013), we employ Variational Autoencoders (VAEs) to manage more complex data structures and general DR tasks. Building on Grosnit et. al. (2021), we analyse the VAE-based LSBO framework, focusing on VAE retraining and deep metric loss. We suggest a few key corrections in their implementation, originally designed for tasks such as molecule generation, and reformulate the algorithm for broader optimisation purposes. Our numerical results show that structured latent manifolds improve BO performance. Additionally, we examine the use of the Mat\'{e}rn-$\frac{5}{2}$ kernel for Gaussian Processes in this LSBO context. We also integrate Sequential Domain Reduction (SDR), a standard global optimization efficiency strategy, into BO. SDR is included in a GPU-based environment using \textit{BoTorch}, both in the original and VAE-generated latent spaces, marking the first application of SDR within LSBO.

Figures

Figures reproduced from arXiv: 2412.09183 by the authors.

Figure 1
Figure 1. Comparisons of Vanilla BO-VAE algorithm (Algorithm 3) with and without SDR on 100D Ackley and Rosenbrock problems. The means and the standard deviations (shaded areas) of the minimum function values found are plotted across 5 repeated runs [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Comparisons of Algorithm 3 (Vanilla BO-VAE), Algorithm 1 (Retrain BO-VAE), and Algorithm 4 (Retrain DML BO-VAE) in solving 100D Ackley and Rosenbrock problems. The means and the standard deviations (shaded areas) of the minimum function values found are plotted across 5 repeated runs. miny∈Rd g(y), subject to y ∈ Y = [−δ, δ] d . Here A is a D × d Gaussian matrix for random embedding, with d ≪ D. Solving g(y) in the … view at source ↗
Figure 3
Figure 3. Performance Profiles on Test Sets 1 & 2 when τ = 10−1 and 10−3 . 5. Conclusion and Future Work In this work, we have explored dimensionality reduction techniques to enhance the scalability of BO. The use of VAEs offers an alternative and more general approach for GP fitting in low￾dimensional latent subspaces, alleviating the curse of dimensionality. Unlike REMBO, which pri￾marily targets low-rank functions, VAE-bas… view at source ↗

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