{"id":"9d99854a-de4f-4e6a-8e6c-e91caa52c83b","arxiv_id":"2411.16586","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A new family of information-based acquisition functions for Bayesian optimization is derived from the alpha-divergence, with an efficient Gaussian approximation and an ensemble variant that matches or beats existing methods in most tested settings.","lead":"This paper introduces Alpha Entropy Search (AES), a new Bayesian optimization acquisition function that measures how informative an evaluation would be using the alpha-divergence instead of the standard Kullback-Leibler divergence. The method generalizes existing information-based acquisition functions and is shown to be competitive with them, especially when an ensemble over alpha values is used.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The truncated-Gaussian approximation of p(y|D,x,{x*,y*}) is validated only in 1D, and the paper does not check whether the approximate AES acquisition mis-ranks candidate points relative to the exact alpha-divergence in higher dimensions; the ensemble's performance could ride on an unverified proxy.","rationale":"The paper's central claim is that AES, especially as an ensemble over α values, is competitive with JES, MES, and PES in noiseless settings. For that claim to hold, the approximate acquisition in Eq. (15) must faithfully reflect the true alpha-divergence objective, at least in terms of the ordering of candidate points. The reader identified the truncated-Gaussian approximation of the conditional predictive distribution as the weakest assumption; this stress-test agrees and sharpens the concern. The paper provides direct evidence in Figure 4 that the approximation underestimates the exact acquisition, with larger discrepancies for α near 0, but only in a single 1D problem and without error bars. No test addresses whether the argmax of the approximate acquisition, which drives BO's decisions, matches the argmax of the exact acquisition. In higher dimensions, the global-optimizer conditioning implicit in {x*,y*} is even cruder, since the event 'x* is the global maximizer' imposes a joint constraint across the whole input space that a per-point truncated Gaussian cannot capture. The alpha-divergence for small α approaches the reverse KL divergence, which is sensitive to tail behavior and to regions where the conditional density is small, making the approximation error more consequential. At the same time, the paper has independent support: a public BOTorch implementation, a 1D validation showing near-identical ensemble curves, and high-dimensional synthetic and benchmark experiments in which the ensemble performs well. These experiments indirectly suggest the approximation does not catastrophically mislead the search, but they do not isolate approximation fidelity from the heuristic value of the acquisition. The concern is therefore a conditional risk rather than a demonstrated failure: it warrants requiring a rank- or maximizer-focused validation of the approximation in a setting where exact evaluation is still tractable, but it does not justify rejecting the paper. The reader's CONDITIONAL verdict is well calibrated, and this stress-test does not change it.","tokens_in":30109,"tokens_out":16067,"duration_ms":143158,"concrete_test":"In a 2D or 3D GP-sample problem where the exact acquisition can be estimated by high-accuracy Monte Carlo (e.g., 50k samples and quadrature in y), compute the approximate Eq. (15) and the exact counterpart for α ∈ {0.001, 0.1, 0.5, 0.999} at a set of candidate points (e.g., a fine grid and the argmax of the approximate acquisition) over several GP posteriors. Report the fraction of times the argmax of the approximate acquisition is within the top 5% of exact-acquisition values, and the Kendall-tau rank correlation between approximate and exact acquisitions. If the rank correlation is low for small α, the approximation is not a faithful proxy and the ensemble's weighting is unreliable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The AES acquisition in Eq. (10) requires the conditional predictive p(y|D,x,{y*,x*}), which Section 3.2 approximates by the Gaussian N(m_tr, v_tr+σ^2) with moments from the truncated unconditional GP (Eqs. 11-12). This is the same heuristic as JES, but the alpha-divergence for small α is sensitive to the distribution's shape, not just its mean and variance. Figure 4 shows the approximation underestimates the exact acquisition for α=0.001 and α=0.1 in 1D, and the error is larger for smaller α. Since the ensemble Eq. (16) includes α=0.001 and α=0.1, the normalization weights w_α are based on underestimated maxima; if the underestimation is α-dependent, the ensemble may not reflect the intended weighted alpha-divergence. Crucially, the paper does not quantify the approximation error in higher dimensions or check whether the maximizer of the approximate acquisition (which determines the next query) coincides with the maximizer of the exact acquisition. If the approximation changes the ranking of points, the method optimizes a different function than claimed, and the strong noiseless performance in Sections 5.2-5.4 may not be attributable to the alpha-divergence objective.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":30387,"tokens_out":7116,"duration_ms":82326,"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":[{"comment":"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.","section":"Section 5.1, Figure 4, Eq. (15)"},{"comment":"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.","section":"Section 3.3, Eq. (16)"}],"minor_comments":[{"comment":"In the bullet list near Eq. (9), 'DKL(p(x) ∥ x(x))' should read 'DKL(p(x) ∥ q(x))'.","section":"Section 3.1"},{"comment":"In the conclusion, 'arxvic' appears to be a typo for 'arXiv' or 'also'.","section":"Section 6"},{"comment":"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.","section":"Figure 8 caption"},{"comment":"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.","section":"Table 1 and Section 3.3"},{"comment":"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.","section":"Section 5.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid extension of JES, with transparent derivations and honest reporting of limitations. The main risk is the unverified fidelity of the Gaussian approximation in higher dimensions, which is load-bearing for the empirical claims. The alpha-divergence generalization is natural but not surprising; the ensemble mechanism is simple. The manuscript fits the journal's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know: this is a solid, incremental but genuinely new acquisition function for BO. The alpha-divergence generalization of JES is real, the closed-form integral is neat, and the ensemble over alpha is a practical workaround. The paper deserves a serious referee.\n\nThe derivation from Eq. 10 to 15 is correct, and the appendices back it up. I checked the exponential-family step and the limit claims; they hold. The 1D comparison against a more exact Monte Carlo shows the approximation underestimates the exact acquisition for small alpha, but the ensemble matches well. The experiments are extensive: synthetic GP, benchmarks, and a neural network tuning task. In noiseless settings, the ensemble beats JES and other information-based methods on several problems. The authors are also honest that no single alpha dominates and that the ensemble is the safe choice.\n\nThe main concern is that the truncated-Gaussian approximation is only validated in 1D. For small alpha, the acquisition is underestimated, and in higher dimensions the approximation could mis-rank points. The stress-test note raises this, and it is a fair concern. It is not fatal: the ensemble uses the same approximation, and the empirical results are competitive across many problems, which suggests the approximation is not grossly wrong. Still, the paper would be stronger if it included a higher-dimensional check, e.g., comparing the maximizers of the approximate and exact acquisition on a 2D problem.\n\nA minor point: the ensemble's 11x cost is brushed aside as negligible. That is fine in BO, but the paper could be more precise about when that assumption breaks. Also, the paper does not compare to the latest non-information-based methods, but that is out of scope.\n\nOverall, this is a paper that BO practitioners will want to read. It deserves peer review. I would recommend conditional acceptance, asking for a higher-dimensional approximation check before publishing.","headline":"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.","tokens_in":30920,"tokens_out":1588,"would_cite":true,"duration_ms":15968,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Alpha Entropy Search","Bayesian optimization","alpha-divergence","acquisition function","Joint Entropy Search","Gaussian process","truncated Gaussian approximation","ensemble of acquisition functions"],"falsifier":"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.","tokens_in":29922,"feed_emoji":"🎯","tokens_out":9107,"duration_ms":77781,"temperature":0.7,"pith_summary":"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.","feed_headline":"Alpha-divergence acquisition rivals top Bayesian optimizers","feed_subtitle":"Eleven alpha values are averaged into one acquisition that beats Joint Entropy Search on noiseless benchmarks.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the Joint Entropy Search acquisition and the truncated-Gaussian approximation of the conditional predictive distribution that AES adapts.","marker":"[15]"},{"why":"Establishes the mutual-information swap and the random-feature sampling of optimizer locations that AES uses for Monte Carlo expectation.","marker":"[12]"},{"why":"Defines Amari's alpha-divergence, the divergence family underlying the AES acquisition.","marker":"[17]"},{"why":"Describes the mode-seeking versus mode-covering behavior of alpha-divergences and motivates the (0,1) range of alpha values.","marker":"[18]"},{"why":"Provides random Fourier features for sampling functions from the GP posterior and extracting samples of {x*, y*}.","marker":"[25]"},{"why":"Introduces Max-value Entropy Search, one of the principal baselines AES is compared against.","marker":"[20]"}],"fun_headline_variants":["Alpha Entropy Search: a new information-based BO acquisition family","Replace KL with alpha-divergence in Bayesian optimization","Alpha-divergence powers new Bayesian optimization acquisition","Ensembling alpha values boosts Bayesian optimization search"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Alpha Entropy Search: a new information-based BO acquisition family","Replace KL with alpha-divergence in Bayesian optimization","Alpha-divergence powers new Bayesian optimization acquisition","Ensembling alpha values boosts Bayesian optimization search"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000606,"raw_usage":{"total_tokens":2942,"prompt_tokens":1178,"completion_tokens":1764,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":794,"completion_tokens_details":{"reasoning_tokens":1701}},"tokens_in":794,"tokens_out":1764,"duration_ms":13168,"temperature":1.0,"reasoning_tokens":1701,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:57:29.358963+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Hvarfner, F","cited_arxiv_id":null,"evidence_quote":"Supplies the Joint Entropy Search acquisition and the truncated-Gaussian approximation of the conditional predictive distribution that AES adapts."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the mutual-information swap and the random-feature sampling of optimizer locations that AES uses for Monte Carlo expectation."},{"cited_title":"Amari, Differential-geometrical methods in statistics, Vol","cited_arxiv_id":null,"evidence_quote":"Defines Amari's alpha-divergence, the divergence family underlying the AES acquisition."},{"cited_title":"Minka, et al., Divergence measures and message passing, Tech","cited_arxiv_id":null,"evidence_quote":"Describes the mode-seeking versus mode-covering behavior of alpha-divergences and motivates the (0,1) range of alpha values."},{"cited_title":"Rahimi, B","cited_arxiv_id":null,"evidence_quote":"Provides random Fourier features for sampling functions from the GP posterior and extracting samples of {x*, y*}."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces Max-value Entropy Search, one of the principal baselines AES is compared against."}],"review_version":1}