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REVIEW 4 major objections 8 minor 39 references

Scalable Batch Bayesian Optimization Via Subspace Acquisition Functions

T0 review · 4 major / 8 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper claims that batch Bayesian optimization scales to large batches by maximizing expected improvement inside random axis-aligned subspaces, and that the resulting points stay competitive with sequential EI and seven earlier batch…

desk verdict A simple subspace-sampling batch BO heuristic that genuinely helps at moderate batch sizes, but the large-q 'speedup' claim leans on an iteration axis that conflates parallelism with efficiency. read the letter →

arxiv 2411.16206 v3 pith:4KECLBQX submitted 2024-11-25 cs.LG cs.AIcs.NE

Dawei Zhan , Zhaoxi Zeng , Shuoxiao Wei , Ping Wu This is my paper · ORCID
classification cs.LGcs.AIcs.NE
keywords BayesianoptimizationbatchevaluationexpectedimprovementsubspaceacquisitionGaussianprocessparallelcomputingexpensiveblack-boxCEC2017
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

Standard Bayesian optimization proposes one query at a time, and most batch extensions lose quality as the batch grows. This paper claims a simple fix: draw a batch of random axis-aligned subspaces of the design space, maximize the expected-improvement acquisition function inside each subspace with all other coordinates pinned to the current best point, and evaluate the resulting points in parallel. The resulting method, called Expected SubSpace Improvement (ESSI), has no extra hyperparameters and scales to batch sizes as large as 128. On the 58 rotated and shifted CEC2017 problems, the authors report that ESSI beats or matches sequential EI on most problems and outperforms seven earlier batch expected-improvement methods on a majority of problems at batch sizes 4, 16, and 128. The paper also notes that extending the idea to expensive constraints is left for future work.

What carries the argument

The load-bearing object is the Expected SubSpace Improvement (ESSI) function, $$\mathrm{ESSI}(y) = (f_{\min} - \mu(z))\Phi\!\left(\frac{f_{\min} - \mu(z)}{\$\sigma$(z)}\right) + \$\sigma$(z)\varphi\!\left(\frac{f_{\min} - \mu(z)}{\$\sigma$(z)}\right),$$ where $z$ is obtained from the current best point $x_{\min}$ by overwriting the coordinates of the chosen axis-aligned subspace with $y$, and $\Phi$ and $\varphi$ are the standard normal cumulative distribution and density functions. This is exactly the expected-improvement formula, but confined to a slice of the design space through the current best point. The machinery works by solving $q$ such slice problems independently, one per randomly selected subspace, so a batch arises from diversity of subspaces rather than from penalizing or hallucinating points; the paper notes that $s=d$ recovers standard EI and $s=1$ recovers the expected coordinate improvement criterion. The random subspace selection, with dimension $s$ drawn uniformly from $1$ to $d$, is what keeps the method parameter-free and gives each acquisition subproblem an average dimension of $(d+1)/2$.

What would settle it

Run ESSI and sequential EI on rotated versions of separable benchmark problems, such as rotated Rosenbrock or rotated Ackley in 10 and 30 dimensions, using the same 512 additional evaluations and 30 repeats as the paper, and compare simple regret; if ESSI does not match or beat sequential EI on the majority of these rotated problems, or if its advantage disappears specifically when the optimum requires simultaneous movement of many coordinates, the central claim fails. A simpler check: on a problem whose minimum lies far from $x_{\min}$ along a direction not aligned with any single coordinate, ESSI's batch points all share $x_{\min}$ in most coordinates, so they should cluster near $x_{\min}$ and stall unless the subspaces eventually cover the needed direction.

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Extended reading notes

Core claim

The central claim is that the bottleneck of batch Bayesian optimization is the acquisition side, not the search geometry: rather than optimizing one joint acquisition function over a $d\times q$-dimensional space, one can give each batch point its own low-dimensional improvement problem. For an axis-aligned subspace $Y$ of dimension $s$, the ESSI function is the usual expected-improvement expression evaluated at the point $z$ obtained by taking the current best solution $x_{\min}$ and replacing its $s$ subspace coordinates by the candidate coordinates $y$. Maximizing ESSI for $q$ randomly chosen subspaces yields $q$ different query points, each differing from $x_{\min}$ in only a few coordinates, and these points can be evaluated in parallel. The authors argue this is why the method scales: the average subspace dimension is $(d+1)/2$, so each inner acquisition problem is cheaper than the original $d$-dimensional EI problem, and the $q$ inner problems are independent and can be solved concurrently.

Load-bearing premise

The load-bearing premise is that improving expected improvement inside a randomly chosen slice of coordinates through the current best point yields points that improve the objective over the whole space, with no theoretical guarantee and the most risk on rotated or tightly coupled functions.

Editorial extensions

If this is right

  • If the claim holds, batch Bayesian optimization no longer degrades as the batch reaches 128: ESSI reports better simple regret than seven batch EI baselines on 46-56 of the 58 CEC2017 problems at $q=128$, while beating sequential EI on 24 and matching it on 18.
  • Wall-clock speedup becomes nearly linear when expensive evaluations dominate, because $q$ evaluations run in parallel and acquisition optimization time itself drops; the paper reports 32-73x speedups in acquisition optimization time on 10-dimensional problems at $q=128$.
  • High-dimensional problems benefit more from the subspace decomposition, since each ESSI acquisition problem has dimension averaging $(d+1)/2$ rather than $d$; the authors observe larger advantages over competitors on 30-dimensional than on 10-dimensional problems.
  • The method does not need artificial penalization, fake objective values, multi-objective solvers, or Monte Carlo batch integration, so it can be dropped into an existing EI-based Bayesian optimization loop with essentially no tuning beyond the batch size itself.

Reading between the lines

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

  • A testable implication the paper does not address: because each batch point differs from $x_{\min}$ only in the coordinates of one random subspace, ESSI behaves like a randomized coordinate-descent explorer, so on landscapes whose optima require simultaneous coordinated changes of many variables, such as strongly rotated valleys, its advantage over full-dimensional EI should shrink or reverse.
  • The subspace mechanism is acquisition-agnostic: the same draw-random-axis-aligned-subspaces-and-pin-the-rest construction could be applied to probability of improvement, lower confidence bound, or knowledge-gradient acquisition, producing batch variants by the same argument.
  • One could isolate the source of the gains by comparing ESSI's random subspaces with fixed, length-scale-informed, or greedily selected subspaces; if random selection is already near-optimal, the decomposition itself is what matters, whereas sensitivity to subspace choice would point to tuning opportunities.
  • Because the paper reports under-linear parallel speedups in acquisition optimization (about 1.6-2.7x on 30-dimensional problems at small to medium batch sizes), implementation overhead can dominate when objective evaluations are cheap; on many-core machines this suggests task scheduling or granularity changes are a natural next step.
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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 / 8 minor

Summary. The paper proposes ESSI (Expected SubSpace Improvement), a batch Bayesian optimization method. At each iteration, q axis-aligned subspaces are selected randomly; in each subspace, a candidate is found by maximizing an EI-like acquisition function in which all coordinates outside the subspace are pinned to the current best solution x_min. The resulting q points are evaluated in parallel. The method is tested on 58 CEC2017 benchmark problems (d=10 and d=30) with 30 runs, compared against sequential EI with q from 2 to 128 and against seven batch EI baselines at q=4, 16, and 128, with Wilcoxon signed-rank tests. Results are reported in Tables 1-4 and convergence curves in Figures 4, 6, 7, and 8.

Significance. If the claims were fully supported, the method would be a valuable contribution to batch BO: it is simple, has essentially no additional hyperparameters beyond the batch size, scales to q=128, and beats the included batch EI baselines on a majority of benchmark problems. The empirical program is extensive (58 problems, 30 runs, three batch sizes, statistical tests), and the paper provides a public Matlab implementation. However, the central claim of 'significant speedup' over sequential EI is overstated: the main evidence for it is plotted on an iteration-count axis, which conflates parallelism with per-evaluation sample efficiency. On the per-evaluation axis (Table 1), ESSI at q=128 is comparable to sequential EI rather than better. The subspace heuristic itself is reasonable but has no theoretical grounding, and the paper does not discuss when it could fail. The method's genuine strength is its competitive and often superior performance against other batch EI methods at the same batch size.

major comments (4)
  1. [Section 4.2, Figure 4, Table 1] The claim of 'significant speedup' over sequential EI is established on an iteration-count axis that conflates parallelism with per-evaluation efficiency. In Figure 4, the horizontal axis is 'number of iterations'; ESSI with q=128 performs 512 evaluations in only 4 iterations, whereas the q=1 EI baseline performs 4 evaluations in 4 iterations. A lower regret at iteration 4 therefore reflects the 128 evaluations already consumed, not improved sample efficiency. Table 1, which fixes the total number of evaluations at 512, is the correct per-evaluation comparison: at q=128, ESSI is significantly better than EI on 24/58 problems, significantly worse on 16/58, and similar on 18/58. The abstract's 'speedups the convergence significantly' and the Section 4.2 statement that 'the convergence speed of the proposed ESSI approach can be improved as we increase the batch size from 2 to 128' are not supported for large q on a per-evaluation basis. Please re-plot the sequential-EI comparison on an evaluation-count axis (or wall-clock time with q parallel workers) and revise the claims to separate parallel wall-clock speedup from sample efficiency.
  2. [Section 3.1, Eq. (9)] The ESSI acquisition function fixes all non-subspace coordinates at the current best x_min. Consequently, maximizing ESSI in a subspace is a heuristic that is guaranteed to improve only the lower-dimensional subspace, not necessarily the full-dimensional objective. The paper does not discuss conditions under which this heuristic could fail, such as strongly non-separable or rotated objectives where no axis-aligned subspace contains an improving direction from x_min. Since the CEC2017 test suite is composed of rotated and composition functions, the empirical success is nontrivial, but the absence of any analysis of this assumption is a gap. Please add a paragraph explaining the heuristic, its limitations, and the intuition (e.g., coordinate-descent-like behavior, random subspace coverage over iterations) for why it works in practice.
  3. [Section 4.3, q-EI implementation] The q-EI baseline is computed with only 1000 Monte Carlo samples for batch sizes up to q=128. In dimension d=30 with q=128, a 128-dimensional integral is approximated by 1000 samples; this estimator has high variance and is known to be biased for large q, which can disadvantage q-EI in the comparison. Please justify the MC sample count by reporting estimator variance (e.g., across different MC seeds) or by using a more accurate q-EI approximation (e.g., the analytical formula for small q and the Fq-EI decomposition for large q), and show that the ranking of methods is not sensitive to this choice.
  4. [Section 4.2, Figure 5] The acquisition optimization time comparison in Figure 5 measures the wall-clock time for ESSI with parallel parfor across q acquisition tasks versus sequential EI with one task per iteration. The text phrase 'the speedup of our proposed ESSI' attributes the reduction to the method itself, but a large part of the reduction comes from parallelizing q independent acquisition optimizations; any batch method with independent acquisition subproblems could obtain a similar speedup on the same hardware. The claim 'our proposed ESSI is able to reduce the acquisition optimization time significantly' should be rephrased to describe the parallel implementation, and the text should clarify that the linear speedup for expensive objective evaluation is a property of batch evaluation in general rather than specific to ESSI.
minor comments (8)
  1. [Section 4.3] The text says 'We also use Wilson signed rank test' where it should be 'Wilcoxon signed rank test'; the same typo appears in the captions of Tables 2-4. Section 4.2 uses the correct spelling.
  2. [Section 3.3 / Algorithm 2] Section 3.3 suggests eliminating duplicate subspaces by checking after each selection, but Algorithm 2 (Step 4-8) does not include such a check. Please either implement the rejection step in the algorithm or explicitly state that duplicate subspaces are allowed and quantify how often they occur.
  3. [Section 4.1] Minor wording: 'the number of acquisition samples is set 512' should be 'set to 512', and 'the maximal number of generation' should be 'generations'.
  4. [Abstract and Section 5] The abstract uses 'speedups the convergence'; this should be 'speeds up the convergence'. Also, Section 5's claim of 'no additional parameter' should be qualified, since the batch size q is itself a parameter, as are the GA population size and generation limit used for acquisition optimization.
  5. [Section 4.2, Figure 5] In Figure 5, the numeric values on the bars are not labeled with units in the bar chart itself; the y-axis is already labeled 'time in seconds', but the reader must infer that the numeric labels are seconds. Please make the unit explicit in the figure or caption.
  6. [Section 3.1, Eq. (9)] The definition of z in Eq. (9) is informal; writing it as a vector with the subspace coordinates inserted at their original positions would remove ambiguity about the ordering of coordinates in the Gaussian process input.
  7. [Section 4, general] The convergence curves in Figures 4, 6, 7, and 8 show medians and quartiles, but the corresponding simple regret tables report only the average of 30 runs. Reporting the median (or both) in the tables would make the tables and figures consistent.
  8. [References] Reference [33] is to the authors' own ECI work, which is disclosed in the text as the s=1 special case of ESSI; this is appropriate, but it should be mentioned explicitly that ECI is used as a self-comparison baseline in Section 3.1.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: ESSI is an independently defined acquisition rule validated on external benchmarks.

full rationale

The derivation chain is self-contained. ESSI in Eq. (9) is defined as the standard closed-form EI of Eq. (8) evaluated at the point z obtained by replacing the coordinates of the current best solution xmin with the subspace coordinates y. This is a definition, not a fitted or derived prediction: the batch points are then produced by maximizing these q independent ESSI functions (Algorithm 2), and the central claim is tested against the external CEC 2017 benchmark suite with fixed evaluation budgets and statistical tests. There is no parameter fitted to the benchmark results and no quantity is renamed as a prediction. The only self-citation, ECI [33], is explicitly disclosed as the s=1 special case of ESSI and is not used to justify the method; the other self-citations ([15], [32]) name the authors' earlier methods only as comparators or reviews. The heuristic nature of the subspace-pinning assumption in Eq. (9) is a correctness risk, not a circularity, and the iteration-count versus evaluation-count framing in Figure 4 is a presentation issue, not a circular reduction.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The paper introduces a new acquisition function, not a new physical entity; no new particles, forces, dimensions, or conserved quantities are postulated. The free parameters listed are experimental settings and baseline implementation details, not parameters fitted to the target result.

free parameters (6)
  • batch size q = 2, 4, 8, 16, 32, 64, 128
    The user-chosen number of parallel evaluations; the paper's scalability claim depends on it.
  • GA population size = 10d
    Experimental setting for optimizing acquisition functions in Section 4.1; a larger population would improve acquisition optimization but cost more.
  • GA generation limit = 100
    Experimental setting in Section 4.1; fixes the quality of acquisition function maximization.
  • initial sample count = 10d
    Latin hypercube design size in Section 4.1; determines initial GP quality.
  • q-EI Monte Carlo samples = 1000
    Baseline implementation detail in Section 4.3; approximates the q-EI acquisition function and may weaken that baseline.
  • MSMR length-scale count = 2q
    Baseline implementation detail for MSMR in Section 4.3.
assumptions (6)
  • domain assumption The objective function is a realization of a Gaussian process with constant mean and squared exponential kernel; hyperparameters are estimated by maximum likelihood (Section 2.1).
    This is the standard BO modeling assumption; violations could bias the reported comparisons.
  • domain assumption The objective is single-objective, bound-constrained, deterministic, and expensive to evaluate (Section 2, Eq. 1).
    Defines the problem class; the method is not tested on noisy, constrained, or multi-objective problems.
  • domain assumption CEC2017 benchmark functions are representative of expensive black-box optimization problems (Section 4.1).
    All conclusions are empirical and transfer to real problems only if these benchmarks are representative.
  • ad hoc to paper Random axis-aligned subspace selection with uniform s in {1,...,d} is a reasonable exploration strategy (Section 3.3).
    This is the paper's own heuristic choice; its effectiveness is the core empirical claim, and there is no theoretical justification.
  • domain assumption The EI acquisition function is the right building block for the proposed ESSI (Section 2.2).
    The method inherits EI's trade-off properties; other acquisition functions are not tested.
  • standard math Wilcoxon signed-rank test assumptions hold for the 30 independent runs (Section 4.1).
    Used to declare significant differences; no effect sizes or p-values are reported.

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Cite this review

Pith. "Pith review of Scalable Batch Bayesian Optimization Via Subspace Acquisition Functions." pith.science (2026). https://pith.science/paper/4KECLBQX

@misc{pith2026241116206,
  author       = {Pith},
  title        = {Pith review of: Scalable Batch Bayesian Optimization Via Subspace Acquisition Functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4KECLBQX}},
  note         = {Machine review of arXiv:2411.16206}
}
read the original abstract

Extending Bayesian optimization to batch evaluation can enable the designer to make the most use of parallel computing technology. However, most of current batch approaches do not scale well with the batch size. That is, their optimization efficiencies often deteriorate as the batch size increases. To address this issue, we propose a simple and efficient approach to extend Bayesian optimization to large-scale batch evaluation in this work. Different from existing batch approaches, the idea of the new approach is to draw a batch of axis-aligned subspaces of the original problem and select one point from each subspace using existing acquisition functions. Numerical experiments show that our proposed approach speedups the convergence significantly when compared with the sequential Bayesian optimization algorithm, and performs very competitively when compared with ten batch Bayesian optimization algorithms. The implementation of our proposed approach is available at https://github.com/zhandawei/SubSpace_Acquisition_Functions.

Figures

Figures reproduced from arXiv: 2411.16206 by the authors.

Figure 1
Figure 1. Gaussian process approximation and the corresponding expected improvement of [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Six ESSI functions of a 3-dimensional problem. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Acquisition optimization results on a 3-dimensional problem. [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Convergence curves of the sequential EI and the proposed ESSI on 10-D and 30-D problems. [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: Acquisition optimization time of the proposed ESSI and the EI functions. [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: Convergence curves of the compared batch EI approaches when [PITH_FULL_IMAGE:figures/full_fig_p024_6.png]
Figure 7
Figure 7. Figure 7: Convergence curves of the compared batch EI approaches when [PITH_FULL_IMAGE:figures/full_fig_p026_7.png]
Figure 8
Figure 8. Figure 8: Convergence curves of the compared batch EI approaches when [PITH_FULL_IMAGE:figures/full_fig_p028_8.png]

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