REVIEW 1 major objections 3 minor 12 references
Derivation of Closed Form of Expected Improvement for Gaussian Process Trained on Log-Transformed Objective
T0 review · 1 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read This paper supplies the step-by-step derivation behind the log-transformed expected-improvement formula used in Bayesian optimization.
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (1)
- [Section 1 and Section 2.2, Eq. (6)] 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.
minor comments (3)
- [Section 2.2, Eq. (6)] 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 3] 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 2.2, Eq. (8)] 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.
Circularity Check
No significant circularity: the derivation is self-contained and only reproduces an external published formula.
full rationale
The paper derives the closed form of expected improvement for a Gaussian process trained on log-transformed objectives directly from the definition of EI and the Gaussian predictive density. Equation (6) states the acquisition function as an integral, and Equations (7)-(9) transform it via substitution and completing the square; no fitted parameters, no hidden empirical inputs, and no load-bearing self-citations appear. The final formula is compared with Hutter et al. (2009) only as a check, not as a premise. The author's own prior work (Watanabe, 2023) is cited only for background on TPE, not to justify the derivation. The only substantive caveat is a domain restriction: Section 1 states y ∈ R while Section 2.2 defines g = log y, which requires y > 0; this is a correctness/scope issue, not circularity. The derivation is algebraically self-contained and independent of the cited result.
Assumptions & free parameters
assumptions (3)
- domain assumption The GP predictive distribution for the log-transformed objective is Gaussian: g ~ N(μ(x), σ(x)^2).
- domain assumption The objective y is strictly positive so that g = log y is well-defined.
- standard math Properties of the standard normal distribution: Φ'(z)=φ(z) and ∫_{-∞}^z t φ(t) dt = -φ(z).
Cite this review
Pith. "Pith review of Derivation of Closed Form of Expected Improvement for Gaussian Process Trained on Log-Transformed Objective." pith.science (2026). https://pith.science/paper/TEM5ZS35
@misc{pith2026241118095,
author = {Pith},
title = {Pith review of: Derivation of Closed Form of Expected Improvement for Gaussian Process Trained on Log-Transformed Objective},
year = {2026},
howpublished = {\url{https://pith.science/paper/TEM5ZS35}},
note = {Machine review of arXiv:2411.18095}
}
read the original abstract
Expected Improvement (EI) is arguably the most widely used acquisition function in Bayesian optimization. However, it is often challenging to enhance the performance with EI due to its sensitivity to numerical precision. Previously, Hutter et al. (2009) tackled this problem by using Gaussian process trained on the log-transformed objective function and it was reported that this trick improves the predictive accuracy of GP, leading to substantially better performance. Although Hutter et al. (2009) offered the closed form of their EI, its intermediate derivation has not been provided so far. In this paper, we give a friendly derivation of their proposition.
Reference graph
Works this paper leans on
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Ament, S., Daulton, S., Eriksson, D., Balandat, M., and Bakshy, E. (2024). Unexpected improvements to expected improvement for B ayesian optimization. In Advances in Neural Information Processing Systems
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Hutter, F., Hoos, H., Leyton-Brown, K., and Murphy, K. (2009). An experimental investigation of model-based parameter optimisation: SPO and beyond. In Conference on Genetic and Evolutionary Computation
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Reviewed August 12, 2026 · model on record in the stance chip above.
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