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REVIEW 2 major objections 5 minor 1 cited by

Alpha Entropy Search for New Information-based Bayesian Optimization

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Alpha Entropy Search builds a Bayesian optimization acquisition function from the alpha-divergence and shows an ensemble over alpha values matches or beats KL-based information methods.

desk verdict AES is a clean, genuinely new generalization of JES to alpha-divergence, with a sound derivation and honest experiments; the main soft spot is that the approximation is validated only in 1D. read the letter →

arxiv 2411.16586 v1 pith:35JZRMDI submitted 2024-11-25 stat.ML cs.LG

Daniel Fernández-Sánchez , Eduardo C. Garrido-Merchán , Daniel Hernández-Lobato This is my paper · ORCID
classification stat.MLcs.LG
keywords AlphaEntropySearchBayesianoptimizationalpha-divergenceacquisitionfunctionJointGaussianprocesstruncatedapproximationensembleoffunctions
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

Alpha Entropy Search (AES) is a new class of acquisition functions for Bayesian optimization built on Amari's $\alpha$-divergence instead of the Kullback-Leibler divergence. The paper argues that the dependency between a candidate observation $y$ and the global optimum $\{x^\star, y^\star\}$ can be measured by the $\alpha$-divergence, and that the resulting acquisition function can be evaluated efficiently with a truncated-Gaussian approximation plus a closed-form integral. Because no single $\alpha$ value wins everywhere, the paper proposes an ensemble of eleven $\alpha$ values that normalizes and sums the individual acquisitions. Across synthetic, benchmark, and real-world hyperparameter-tuning experiments, the ensemble matches or beats Joint Entropy Search, Max-value Entropy Search, and Predictive Entropy Search, with the clearest gains in noiseless settings. The point of the work is that a tunable divergence family can improve how BO spends its expensive evaluations.

What carries the argument

The load-bearing object is Amari's $\alpha$-divergence, $D_\alpha(p\|q)=\frac{1}{(1-\alpha)\alpha}\left(1-\int q^{1-\alpha}p^\alpha\right)$, which generalizes the KL divergence (recovered as $\alpha\to 1$) and the reversed KL divergence (as $\alpha\to 0$). The paper plugs into it the joint distribution of the observation $y$ and the optimizer $\{x^\star, y^\star\}$ versus the product of marginals, defining the acquisition. To make that tractable, the conditional distribution $p(y|D_{t-1},x,\{y^\star,x^\star\})$ is approximated as a Gaussian whose mean and variance come from a truncated Gaussian (Eqs. 11--12), and the integral over $y$ is computed in closed form using the log-normalizer $g(\eta)$ of the Gaussian (Eqs. 13--15). Samples of $\{x^\star,y^\star\}$ are drawn from the GP posterior via random Fourier features, and Monte Carlo averages the closed-form integrand. The ensemble variant sums each $\alpha$'s normalized acquisition over eleven equally spaced $\alpha$ values, reusing one set of samples.

What would settle it

Evaluate Eq. (10) exactly in a low-dimensional problem (e.g., a 2D or 4D sample from the GP prior) using fine Monte Carlo and quadrature, then compare the maximizer of the approximate Eq. (15) with the maximizer of the exact acquisition for alpha near 0.01; if the approximate maximizer is consistently far from the exact one and the gap translates into worse optimization regret, the truncated-Gaussian approximation is the failing link.

Watch

Extended reading notes

Core claim

The central claim is that information-based Bayesian optimization does not have to be tied to the KL divergence: replacing the mutual-information/KL objective of Joint Entropy Search with Amari's $\alpha$-divergence yields a family of acquisition functions, called AES, that retain the same sampling machinery and achieve competitive or better optimization performance. The paper derives $$a_{\mathrm{AES}}(x)=D_\$\alpha$\bigl(p(y,\{y^\star,x^\star\}|D_{t-1},x)\,\|\,p(\{y^\star,x^\star\}|D_{t-1})p(y|D_{t-1},x)\bigr),$$ where $D_\alpha$ is Amari's $\alpha$-divergence, and supplies an approximation: the conditional $p(y|D_{t-1},x,\{y^\star,x^\star\})$ is replaced by a Gaussian with the truncated-Gaussian mean and variance from Eqs. (11)--(12), and the integral over $y$ is evaluated in closed form through the exponential-family form of the Gaussian, giving Eq. (15). With $\alpha$ approaching 1 the acquisition is close to, though not identical to, JES; smaller $\alpha$ values change the trade-off between mode-seeking and global coverage. Since no single $\alpha$ dominated, the paper's recommended procedure averages eleven normalized AES acquisitions over $\alpha\in(0,1)$. The experiments support the claim that this ensemble is especially effective in noiseless evaluations and remains competitive when observations are noisy.

Load-bearing premise

The load-bearing premise is that the conditional distribution of the objective at a candidate point, given the location and value of the optimum, is well approximated by a Gaussian with the mean and variance of a truncated Gaussian; the paper's own 1D comparison shows this approximation underestimates the exact alpha-divergence, especially for alpha near zero, so if the bias grows in higher dimensions the method's claimed competitive performance could degrade.

Editorial extensions

If this is right

  • If the central claim holds, practitioners can replace KL-based acquisitions with an alpha-ensemble at the same sampling cost and get equal or better performance in noiseless low-to-moderate-dimensional BO problems.
  • The ensemble's fewer local maxima (12.53 vs 17.60 on average in the paper's 1D study) make acquisition optimization less likely to get stuck, which explains why it avoids JES's degradation as the number of optimizer samples S grows.
  • AES with alpha near 0.999 gives a new, slightly different approximation to JES, so it can serve as a drop-in alternative in BO libraries that already support JES.
  • In the noisy setting, gains over JES shrink and JES can win in high dimensions (e.g., Cosine-8D), so the ensemble is best viewed as a complement rather than a universal replacement.
  • Because the alpha parameter changes the acquisition's exploration-exploitation profile, the ensemble provides a principled way to avoid choosing alpha by hand.

Reading between the lines

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

  • A natural extension the paper leaves implicit is to use the same alpha-divergence construction with only $x^\star$ or only $y^\star$, which would yield alpha-generalizations of PES and MES rather than JES.
  • The smoothness benefit suggests that alpha-ensembling could help any acquisition function whose ruggedness scales with the number of optimizer samples, such as multi-objective joint entropy search.
  • One testable implication is that exact (or better-approximated) alpha-divergence values for small alpha would further improve performance, since the paper's own Figure 4 shows the truncated-Gaussian approximation underestimates the true acquisition most there.
  • The ensemble idea could be combined with adaptive weighting across BO iterations, giving more weight to alpha values that have recently produced better evaluations, rather than the fixed equal weights used here.
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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 / 5 minor

Summary. This paper introduces Alpha Entropy Search (AES), a Bayesian optimization acquisition function that replaces the KL divergence used in Joint Entropy Search with Amari's alpha-divergence. AES measures the dependence between a candidate observation y at x and the joint optimum {x*,y*}, leading to an intractable acquisition (Eq. 10). The authors approximate the conditional predictive distribution p(y|D,x,{y*,x*}) by a Gaussian whose moments come from a truncated Gaussian (Eqs. 11-12), which yields a closed-form integral (Eq. 13) and a Monte Carlo acquisition estimate (Eq. 15). Because no single alpha is uniformly best, they also propose an ensemble over eleven alpha values (Eq. 16). The paper provides a BOTorch implementation and extensive experiments on synthetic, benchmark, and real-world tasks, reporting that the ensemble is competitive with JES, MES, and PES, particularly in noiseless settings.

Significance. If the approximation is trustworthy, AES is a principled and useful generalization of JES: it introduces a tunable divergence family into information-based BO, and the ensemble provides a robust acquisition that is often better than any single alpha and than JES in noiseless settings. The paper is careful about its limitations: it reports that no single alpha dominates, that the approximation underestimates the exact acquisition, and that the benefits are smaller under noise. The appendices contain detailed derivations, and the authors provide code. The observation that the ensemble's performance is insensitive to the number of samples S, while JES's degrades with S in low-dimensional noiseless problems, is an interesting empirical finding. However, the empirical claims rest on the fidelity of the Gaussian approximation to the true alpha-divergence, which is only demonstrated in one dimension.

major comments (2)
  1. [Section 5.1, Figure 4, Eq. (15)] The approximation that yields Eq. (15) is validated only on a 1D toy problem. Figure 4 shows that the approximate AES systematically underestimates the exact acquisition and that the error is larger for small alpha (alpha=0.001 and 0.1). Because the ensemble in Eq. (16) includes these values, the normalization weights w_alpha = max_x \tilde{a}_AES(x;alpha) are computed from underestimated maxima whose bias depends on alpha. The paper does not test, in dimensions above one, whether the maximizer of \tilde{a}_AES(x;alpha) coincides with the maximizer of the exact acquisition, which is the quantity that actually determines the next query. If the approximation changes the ranking of candidate points, the method optimizes a different functional than the alpha-divergence claimed. Please add a 2D or 3D study comparing the approximate and exact maximizers (using the Section 5.1 exact estimator restricted to a candidate set) and report the alpha-dependent bias in w_alpha.
  2. [Section 3.3, Eq. (16)] The ensemble is described as giving equal weight to each alpha value by normalizing each acquisition by its maximum. However, the normalizing constant w_alpha is the maximum of the approximate acquisition, not of the exact acquisition. Since the approximation error is alpha-dependent (Section 5.1), the normalized approximate acquisitions do not correspond to equal weights on the exact AES objectives. This matters for the paper's central empirical claim that the ensemble is better than any single alpha: part of the benefit may be an artifact of unequal weighting induced by the approximation. The authors should either demonstrate that the relative approximation error at the maximizers is roughly constant across alpha, or use a bias-corrected estimator of the maxima for normalization.
minor comments (5)
  1. [Section 3.1] In the bullet list near Eq. (9), 'DKL(p(x) ∥ x(x))' should read 'DKL(p(x) ∥ q(x))'.
  2. [Section 6] In the conclusion, 'arxvic' appears to be a typo for 'arXiv' or 'also'.
  3. [Figure 8 caption] The caption says 'log hyper-volume relative difference', but the y-axis and the text describe a log relative difference to the maximum objective value; the caption should be consistent with the text.
  4. [Table 1 and Section 3.3] The claim that the difference in the number of local maxima is 'statistically significant' is not supported by a significance test; the table reports only means and standard errors.
  5. [Section 5.2] The statement that 'the ensemble method outperforms AES for all values of alpha' is stronger than the 4D results in Figure 5 show, where the curves largely overlap; please qualify this claim.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: AES's alpha-divergence acquisition is derived in closed form from Eqs. (10)-(15) with no fitted constants; the only self-citation cluster, justifying the alpha grid, is not load-bearing.

full rationale

The derivation chain is self-contained. Eq. (10) defines the AES acquisition as Amari's alpha-divergence between the joint p(y,{y*,x*}|D,x) and the product of marginals; this is a definition, not a result fitted to data. Section 3.2 approximates the intractable conditional with the same truncated-Gaussian moments used by JES (Eqs. 11-12), and Eq. (13) follows by exponential-family algebra; Eq. (15) is the Monte Carlo estimator. No parameter is fitted to the benchmark outcomes and then renamed a prediction: alpha is user-chosen, the ensemble weights w_alpha in Eq. (16) are fixed normalizers equal to each approximate acquisition's maximum, not performance-fitted weights, and S is shared across alpha values. The 1D exact-vs-approximate check in Section 5.1 (Figure 4) is an approximation-quality check, not a circular validation; underestimation for small alpha is a correctness concern, not a reduction of the prediction to its inputs. The only minor self-citation cluster is the justification of the alpha grid in Section 3.3 ('This range of values has been employed before in the literature of approximate inference ... [26,27,23]'), where all three cited papers share the author D. Hernandez-Lobato; this is a design-choice justification and the central derivation and benchmark comparisons do not depend on it. Hence the paper has no significant circularity, only a minor non-load-bearing self-citation cluster.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard GP assumptions, an untested-in-general truncated-Gaussian approximation, and a handpicked alpha ensemble. No new entities are introduced.

free parameters (2)
  • alpha_divergence_parameter_alpha = Not fitted; user-selected or ensemble over {0.001, 0.1, 0.2, ..., 0.9, 0.999}
    The parameter alpha controls the behavior of the alpha-divergence. The paper does not fit alpha to data; it proposes an ensemble over a handpicked grid. This is a free parameter of the method, not fitted to the optimization targets.
  • mc_sample_count_S = 32
    The number of Monte Carlo samples of {x*, y*} used to approximate the expectation in the acquisition function. Chosen by hand; the paper shows results are robust to S, but it is a hand-set value.
assumptions (5)
  • domain assumption The objective function f is a sample from a Gaussian process prior.
    Assumed in Section 2; the GP posterior is the source of all predictive distributions used in the acquisition.
  • domain assumption The conditional predictive distribution p(y|D, x, {y*, x*}) can be approximated by a Gaussian with the same mean and variance as a truncated Gaussian.
    Used in Section 3.2 and Eqs. (11)-(12). This is the key approximation enabling the closed-form integral; it is not proven but empirically tested in 1D.
  • domain assumption Random features can approximate the GP posterior for sampling {x*, y*}.
    Used in Section 3.2, following [25] and prior information-based BO. It assumes the GP kernel is compatible with a random feature expansion.
  • standard math The Gaussian distribution is in the exponential family, so the integral in Eq. (19) has a closed form via log-normalizers.
    Used in Appendix C, Eqs. (24)-(25).
  • standard math The alpha-divergence is non-negative and zero iff distributions coincide, and its limiting cases match the KL divergences.
    Used in Section 3.1, from [17,18]. Standard property.

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Pith. "Pith review of Alpha Entropy Search for New Information-based Bayesian Optimization." pith.science (2026). https://pith.science/paper/35JZRMDI

@misc{pith2026241116586,
  author       = {Pith},
  title        = {Pith review of: Alpha Entropy Search for New Information-based Bayesian Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/35JZRMDI}},
  note         = {Machine review of arXiv:2411.16586}
}
read the original abstract

Bayesian optimization (BO) methods based on information theory have obtained state-of-the-art results in several tasks. These techniques heavily rely on the Kullback-Leibler (KL) divergence to compute the acquisition function. In this work, we introduce a novel information-based class of acquisition functions for BO called Alpha Entropy Search (AES). AES is based on the {\alpha}-divergence, that generalizes the KL divergence. Iteratively, AES selects the next evaluation point as the one whose associated target value has the highest level of the dependency with respect to the location and associated value of the global maximum of the optimization problem. Dependency is measured in terms of the {\alpha}-divergence, as an alternative to the KL divergence. Intuitively, this favors the evaluation of the objective function at the most informative points about the global maximum. The {\alpha}-divergence has a free parameter {\alpha}, which determines the behavior of the divergence, trading-off evaluating differences between distributions at a single mode, and evaluating differences globally. Therefore, different values of {\alpha} result in different acquisition functions. AES acquisition lacks a closed-form expression. However, we propose an efficient and accurate approximation using a truncated Gaussian distribution. In practice, the value of {\alpha} can be chosen by the practitioner, but here we suggest to use a combination of acquisition functions obtained by simultaneously considering a range of values of {\alpha}. We provide an implementation of AES in BOTorch and we evaluate its performance in both synthetic, benchmark and real-world experiments involving the tuning of the hyper-parameters of a deep neural network. These experiments show that the performance of AES is competitive with respect to other information-based acquisition functions such as JES, MES or PES.

Figures

Figures reproduced from arXiv: 2411.16586 by the authors.

Figure 1
Figure 1. GP fit of the objective function (top of images) and the associated acquisition function [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. The Gaussian distribution q(x) is fitted to p(x) by minimizing Amari’s divergence with different values of α. When α → −∞, α tries to match one mode of p(x), and as α increases, q(x) starts covering more of the entire distribution. Finally, when α → ∞, q(x) covers p(x) entirely. Reproduced from [18]. In most practical applications of α-divergences for approximate inference only values of α in the interval (0, 1) are… view at source ↗
Figure 3
Figure 3. (bottom) Comparison of AES for different [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: (top-left) GP predictive distribution for the objective. From (top-right) to (bottom [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: Average logarithm relative difference between the objective at each method’s recom [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: Average logarithm relative difference between the objective at each method’s recom [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: Average logarithm relative difference between the objective at each method’s recom [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]
Figure 8
Figure 8. Figure 8: Average log hyper-volume relative difference between the recommendation of each [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]
Figure 9
Figure 9. Figure 9: Average log relative difference between the objective at each method’s recommendation [PITH_FULL_IMAGE:figures/full_fig_p022_9.png]
Figure 10
Figure 10. Figure 10: (bottom) Comparison of AES for different [PITH_FULL_IMAGE:figures/full_fig_p030_10.png]
Figure 11
Figure 11. Figure 11: Average logarithm relative difference between the objective at each method’s recom [PITH_FULL_IMAGE:figures/full_fig_p031_11.png]
Figure 12
Figure 12. Figure 12: Average logarithm relative difference between the objective at each method’s recom [PITH_FULL_IMAGE:figures/full_fig_p031_12.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Information-theoretic Bayesian Optimization: Survey and Tutorial

    cs.LG 2025-01 conditional

    A survey and tutorial of information-theoretic acquisition functions for Bayesian optimization, covering entropy-based methods and their extensions, with no new algorithms or experiments.

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