{"id":"1e8288ec-72de-4909-9632-191369412cec","arxiv_id":"2411.18095","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":1.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper provides a correct step-by-step derivation of the log-transformed expected improvement acquisition function, confirming the closed form of Hutter et al. (2009).","lead":"This paper derives the closed-form formula for expected improvement when a Gaussian process is trained on the logarithm of the objective function. The derivation matches the formula first stated without proof by Hutter et al. (2009), so the paper is a reference for practitioners who want to verify their implementations.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The algebra is correct, but the derivation silently restricts the objective to y>0 whereas Section 1 states y∈R; Eq. (6) is ill-defined for non-positive objectives.","rationale":"Reader's assessment is fair: the derivation is elementary and correct, and the main weakness is the unstated positivity requirement. I re-derived the key identity ∫_{-∞}^z exp(tσ) φ(t) dt = exp(σ²/2) Φ(z−σ), and it matches. I also checked that the transformation from y-space to g-space preserves the measure (p_Y(y)dy=p_G(g)dg), so Eq. (6) is the correct EI for a log-normal posterior. The only substantive issue I find is the scope condition: with y∈R stated in §1, the use of log is unjustified. That is precisely the reader's weakest assumption, so my read agrees and the verdict stays CONDITIONAL/UNCHANGED. The condition should be an explicit y>0 restriction or a stated handling of non-positive objectives.","tokens_in":4634,"tokens_out":5338,"duration_ms":53075,"concrete_test":"Run the derivation on the simple maximization f(x)=-x² with observations at x=±1, so y<0. Attempting to form g=log y as required by §2.2 fails immediately, whereas standard EI in Eq. (2) is well defined; this demonstrates the claimed domain y∈R is too broad. To confirm the fix, re-state the result with the explicit hypothesis y_n∈R_{>0}, y*∈R_{>0}; all steps in Eqs. (6)-(9) then hold unchanged.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The derivation in §2.2 is internally consistent: with g=log y, the change of variables in Eq. (6), the completion of the square in Eq. (8), and the final expression Eq. (9) all check out, and the result matches Hutter et al.'s published formula. The load-bearing gap is a domain error in the theorem statement. Section 1 defines y∈R with 'better when larger', and no positivity assumption appears anywhere before §2.2 introduces g=log y. Since exp(g)>0 for every real g, the whole derivation only covers strictly positive objectives. If any observed y_n≤0 or if the incumbent y*≤0, the objects g_n=log y_n, log y*, and the upper limit in Eq. (6) are undefined, so the 'closed form' does not apply to the general real-valued setting claimed. The paper also does not discuss a shift or alternative transformation that would extend it. This is not an algebraic flaw in the formula itself; it is an unstated domain restriction that must be part of the theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper provides a step-by-step derivation of the closed-form expected improvement for a Gaussian process trained on log-transformed objective values, i.e., α_logEI(x|D) = y*Φ(z) − exp(μ + σ²/2)Φ(z − σ) with z = (log y* − μ)/σ. Section 2.1 derives the standard EI closed form, and Section 2.2 adapts it to the log-transformed setting by a change of variables and completing the square. The final result is compared with the formula stated by Hutter et al. (2009). The paper is expository: it contains no new algorithms or experiments, only the missing derivation of a known result.","tokens_in":4812,"tokens_out":4192,"duration_ms":38270,"significance":"If the derivation is correct, as it appears to be, the paper fills a small but real expository gap: practitioners using log-transformed EI in packages such as SMAC3 often rely on a closed form whose intermediate steps were not previously published in detail. The derivation is self-contained, does not depend on fitted parameters, and reproduces the cited formula exactly. These are strengths. The contribution is deliberately modest, however: the result itself is known, and the paper offers no empirical validation or algorithmic novelty. Its value is as a reference for verification and teaching.","major_comments":[{"comment":"The paper states in Section 1 that the objective y is real-valued (y ∈ R) and that larger is better, but Section 2.2 defines g = log y and uses the upper integration limit log y* in Eq. (6). The logarithmic transformation is defined only for strictly positive arguments, so the derivation silently restricts the objective to y > 0. If any observed objective value y_n ≤ 0, or if the incumbent y* ≤ 0, then g_n = log y_n, log y*, and the integral in Eq. (6) are all undefined. The paper should explicitly state the positivity assumption on the objective (or discuss a shift/alternative transformation) before Eq. (6), and the theorem statement should be corrected to apply only to that domain. This is not an algebraic error, but it is a load-bearing domain restriction that must be part of the statement.","section":"Section 1 and Section 2.2, Eq. (6)"}],"minor_comments":[{"comment":"The notation in Eq. (6) mixes the original objective y* and the transformed variable g = log y without a verbal reminder. Please state explicitly that y* is the maximum of the observed y-values and that the upper limit of integration is log y*, so that readers do not confuse the two scales.","section":"Section 2.2, Eq. (6)"},{"comment":"The sentence beginning \"For example, /l.VarogEIdefined in BoTorch ...\" appears garbled at the rendering level and reads as if the same name is used for both the log-transformed-objective EI of Eq. (9) and the logarithm-of-EI of Ament et al. (2024). Please clarify the naming convention used in BoTorch and Optuna, and distinguish the two acquisition functions explicitly.","section":"Section 3"},{"comment":"When completing the square, the paper writes ∫ exp(zσ)φ(z) dz = exp(σ²/2)Φ(z − σ); this is correct, but a short sentence noting that the substitution u = z − σ leaves the Gaussian density normalized would improve readability for the target audience.","section":"Section 2.2, Eq. (8)"}],"recommendation":"major_revision","confidential_remarks":"The paper is a short expository derivation of a known formula. Its novelty is intentionally minimal, and whether it fits the journal's scope depends on whether the editors value derivation papers as reference material. Technically, the algebra is sound; the only substantive issue is the missing positivity assumption for the objective, which is easily fixed by adding a hypothesis to the theorem statement. Once that is done, the paper is acceptable as a concise reference contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis is a correct, self-contained derivation of a known formula: the log-transformed expected improvement from Hutter et al. (2009). The paper does not claim a new result; it claims to fill in the algebra that Hutter left out. That is a modest but legitimate contribution, and the derivation is clean. Eqs. (3)–(9) check out: the change of variables, the completion of the square, and the final expression α_logEI = y*Φ(z) − exp(μ + σ²/2)Φ(z − σ) match the published formula. The related-work note distinguishing log-EI (GP on logs) from log EI (log of the acquisition function) is useful and prevents a real source of confusion.\n\nThe main soft spot is a domain error in the statement. Section 1 says y ∈ R, but the whole derivation depends on g = log y, which is only defined for y > 0. If any observed objective is zero or negative, or if the incumbent y* ≤ 0, then Eq. (6) is undefined. This is not an algebraic flaw; the theorem statement just needs a positivity assumption, or a note on how to handle non-positive objectives (e.g., a shift). The paper should also be careful with the claim that the intermediate derivation “has not been provided so far.” That is hard to verify and not essential; a weaker claim would be safer.\n\nI would accept this for peer review—it is short, correct, and useful as a reference for practitioners implementing log-EI. The reviewer should primarily check whether the positivity condition is stated and whether the derivation matches the cited source. The significance is low: no new result, no new experiments. But as an expository note, it deserves a fair review rather than a desk reject. I would not cite it as the primary source (Hutter et al. is the right cite), but I might point people to it as a readable walkthrough.","headline":"Correct and clean derivation of a known log-EI formula, but the statement silently assumes positive objectives; worth a quick review as an expository reference.","tokens_in":5258,"tokens_out":2373,"would_cite":false,"duration_ms":21913,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper supplies the step-by-step derivation behind the log-transformed expected-improvement formula used in Bayesian optimization.","keywords":["expected improvement","Bayesian optimization","log-transformed objective","Gaussian process","closed-form derivation","acquisition function","completing the square"],"falsifier":"Pick any GP posterior with mean $\\mu$, standard deviation $\\sigma$, and a positive incumbent $y^{\\star}$, and evaluate $\\int_{-\\infty}^{\\log y^{\\star}} (y^{\\star}-\\exp(g))\\varphi((g-\\mu)/\\sigma)/\\sigma\\,dg$ by numerical quadrature. If the result differs from $y^{\\star}\\Phi(z)-\\exp(\\mu+\\sigma^2/2)\\Phi(z-\\sigma)$ beyond quadrature tolerance, the derivation is wrong; additionally, the formula is undefined for $y^{\\star}\\le 0$, which would refute the paper's unstated domain assumption.","tokens_in":4432,"feed_emoji":"📐","tokens_out":8827,"duration_ms":72213,"temperature":0.7,"pith_summary":"Expected improvement is the standard acquisition function in Bayesian optimization, but its usual form is numerically fragile. A widely used remedy is to train the Gaussian process on the log of the objective and use a closed-form expression for expected improvement in that space; that expression was stated in 2009, but the derivation was never published. This paper fills in the missing steps. It shows, from the definition of expected improvement and the Gaussian posterior on the log-transformed objective, that the acquisition function reduces to $\\alpha_{\\mathrm{logEI}}(x|D) = y^{\\star}\\Phi(z) - \\exp(\\mu + \\sigma^2/2)\\Phi(z-\\sigma)$ with $z = (\\log y^{\\star} - \\mu)/\\sigma$, the same formula practitioners have been using. A sympathetic reader gains a verifiable basis for the method and a reference for investigating or extending it.","feed_headline":"Closed form of log-EI derived from first principles","feed_subtitle":"The exact expression behind a widely used Bayesian optimization acquisition function is now fully documented.","key_machinery":"The carrying object is the change of variables from objective space to standardized GP space. For ordinary EI, $z=(y^{\\star}-\\mu)/\\sigma$ reduces the integral to $z\\sigma\\Phi(z)+\\sigma\\varphi(z)$, using the identity $\\int t\\varphi(t)\\,dt=-\\varphi(z)$. For log-EI, $g=\\log y$ is modeled as $g\\sim N(\\mu,\\sigma^2)$ and $t=(g-\\mu)/\\sigma$ converts $\\int (y^{\\star}-\\exp(\\mu+\\sigma t))\\varphi(t)\\,dt$ into two terms; the exponential term is handled by completing the square, $\\exp(\\sigma t - t^2/2) = \\exp(-(t-\\sigma)^2/2 + \\sigma^2/2)$, then a shift $u=t-\\sigma$ turns the remaining integral into $\\exp(\\mu+\\sigma^2/2)\\Phi(z-\\sigma)$. Those two substitutions plus the normal CDF identity are the whole mechanism.","core_discovery":"The paper's central claim is that the log-transformed expected-improvement formula is not an approximation or a heuristic: it is the exact integral $\\int_{-\\infty}^{\\log y^{\\star}} (y^{\\star} - \\exp(g)) p(g|x,D)\\,dg$ evaluated under a Gaussian posterior on $g = \\log y$. The derivation first recovers the classical expected-improvement closed form $z\\sigma\\Phi(z)+\\sigma\\varphi(z)$ with $z=(y^{\\star}-\\mu)/\\sigma$, then applies the same substitution to the log-transformed objective. After substituting $t=(g-\\mu)/\\sigma$, the remaining integral is evaluated by completing the square in the exponent, which produces the factor $\\exp(\\sigma^2/2)$ and the shifted normal CDF $\\Phi(z-\\sigma)$. The final expression is exactly the one proposed in the 2009 prior work, which the paper thereby places on fully documented footing.","pith_inferences":["The paper states the objective values as real numbers, but the formula requires $y>0$; extending the method to zero or negative objectives would need a shift or alternative transformation, a constraint the paper leaves implicit.","The same substitution-and-complete-the-square pattern should carry over to any location-scale model on the transformed objective, not only Gaussian processes, giving a template for other acquisition derivations.","A direct numerical check in one dimension (quadrature versus the closed form) would confirm the identity to machine precision and could be added as a unit test in libraries that implement log-EI.","The derivation establishes the formula's correctness, not the empirical claim that log transformation improves predictive accuracy, so the performance advantage still rests on the earlier experimental evidence."],"forward_implications":["The log-EI acquisition used in practice is exactly expected improvement for a GP trained on log-transformed observations, with no hidden approximation in the closed form.","Implementations can be checked against the derived expression, so a mismatch in a codebase is a bug, not an ambiguity in the formula.","The derivation supplies a starting point for further enhancements, such as differentiating the acquisition function or adapting it to other transformations.","Because the derivation is self-contained, it can serve as a pedagogical reference for why the correction factor $\\exp(\\sigma^2/2)$ appears."],"supporting_citations":[{"why":"It states the closed-form log-EI expression that this paper derives, so the proposition is the target of the proof.","marker":"Hutter et al. (2009)"},{"why":"It defines expected improvement as the integral the derivation starts from.","marker":"Jones et al. (1998)"},{"why":"It presents the log-EI closed form in the sequential model-based optimization setting alongside the 2009 paper.","marker":"Hutter et al. (2011)"},{"why":"It describes a widely used Bayesian optimization package that implements the log-EI acquisition, showing the formula's practical role.","marker":"Lindauer et al. (2022)"}],"fun_headline_variants":["Log-EI closed form derived, not just assumed","Exact derivation of EI for log-transformed GP","First-principles proof for log-EI acquisition","Full derivation of log-EI closed form provided","Log-transformed EI formula derived rigorously"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that every objective value is strictly positive, because the log transformation that defines $g=\\log y$ is undefined otherwise; if zeros or negative values can occur, the derivation and the formula do not apply.","fun_headline_variants_meta":{"raw":{"variants":["Log-EI closed form derived, not just assumed","Exact derivation of EI for log-transformed GP","First-principles proof for log-EI acquisition","Full derivation of log-EI closed form provided","Log-transformed EI formula derived rigorously"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000176,"raw_usage":{"total_tokens":1231,"prompt_tokens":825,"completion_tokens":406,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":441,"completion_tokens_details":{"reasoning_tokens":335}},"tokens_in":441,"tokens_out":406,"duration_ms":4058,"temperature":1.0,"reasoning_tokens":335,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:30:38.990005+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Pick any GP posterior with mean $\\mu$, standard deviation $\\sigma$, and a positive incumbent $y^{\\star}$, and evaluate $\\int_{-\\infty}^{\\log y^{\\star}} (y^{\\star}-\\exp(g))\\varphi((g-\\mu)/\\sigma)/\\sigma\\,dg$ by numerical quadrature. If the result differs from $y^{\\star}\\Phi(z)-\\exp(\\mu+\\sigma^2/2)\\Phi(z-\\sigma)$ beyond quadrature tolerance, the derivation is wrong; additionally, the formula is undefined for $y^{\\star}\\le 0$, which would refute the paper's unstated domain assumption.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It states the closed-form log-EI expression that this paper derives, so the proposition is the target of the proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It defines expected improvement as the integral the derivation starts from."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It presents the log-EI closed form in the sequential model-based optimization setting alongside the 2009 paper."}],"review_version":1}