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Expected improvement acquisition

Mathematical Methods AIMech:ExpectedImprovementAcquisition DRAFT

An acquisition criterion for deterministic scalar minimization that scores a candidate evaluation by the predictive expected positive reduction below the best observed objective value, with a closed form in the predictive mean and standard deviation under Gaussian uncertainty.

Identity Notes & Concept Scope Expected improvement is the expected nonnegative gain relative to a stated threshold; it is a criterion rather than a full optimizer. TPE implements a particular density-model ranking and SMAC uses expected-improvement proposals inside a broader configuration procedure. Neither maintained record is the general EI criterion. Probability of improvement measures an event probability rather than gain magnitude; EGO uses EI within a complete original optimization procedure.

Claims & Grounded Evidence

5 Claims

Mechanism #expected-positive-improvement

For deterministic scalar minimization, let b be the smallest observed objective value and Y the predictive uncertainty about the value at a candidate input. Expected improvement is E[max(b - Y, 0)]. It weights each positive improvement by its predictive probability, retaining the amount of gain rather than only the probability P(Y < b). The random variable represents uncertainty about an unevaluated deterministic function, not observation noise.

supports Source: Efficient Global Optimization of Expensive Black-Box Functions · Locator: Section 4.1, PDF p.17 / printed p.471, improvement definition and Eq.14.

Mechanism #gaussian-standard-deviation-form

If Y is Gaussian with mean m and strictly positive standard deviation s, define z = (b - m)/s. Then EI = (b - m) Phi(z) + s phi(z), where Phi and phi are the standard normal distribution and density. The denominator and density multiplier use s, not the variance s squared. In the inspected EGO construction, m and s come from a fitted DACE predictor and its estimated prediction error.

supports Source: Efficient Global Optimization of Expensive Black-Box Functions · Locator: Section 4.1, PDF pp.17–18 / printed pp.471–472, Eq.15 and definitions of Y, s, Phi and phi.

Mechanism #fixed-incumbent-monotonicity

Holding the incumbent fixed with finite m and s > 0, the partial derivatives are dEI/dm = -Phi(z) and dEI/ds = phi(z). EI therefore decreases with predictive mean and increases with predictive standard deviation when the other argument is fixed. A lower bound on m and an upper bound on s give an EI upper bound over an input box. These are partial derivatives of the score, not assertions of monotonicity in the input coordinates.

supports Source: Efficient Global Optimization of Expensive Black-Box Functions · Locator: Section 4.1, PDF pp.18–19 / printed pp.472–473, displayed partial derivatives and rectangular-region bound.

Scope #degenerate-prediction-boundary

For a degenerate predictive law with s = 0, Eq.14 gives EI = max(b - m, 0); division by s in Eq.15 is not defined there. Under exact noiseless interpolation at an already observed input, m equals its observed value and is at least b, so EI is zero. Zero variance by itself does not imply zero improvement at an arbitrary input. The general degenerate-law statement is a direct curator derivation from the defining expectation.

supports Source: Efficient Global Optimization of Expensive Black-Box Functions · Locator: Section 2, PDF pp.7–9 / printed pp.461–463, Eqs.7–11 and interpolation; Section 4.1, PDF pp.17–18 / printed pp.471–472, Eq.14 and zero-at-samples statement; general s = 0 qualification follows from Eq.14.

Scope #one-evaluation-acquisition-scope

EI assesses the gain from one prospective evaluation relative to the current incumbent. Its maximization supplies a query-selection rule, but the score alone does not specify a surrogate-fitting procedure, initial design, stopping policy or complete sequential optimizer. The original EGO algorithm combines this criterion with those additional stages; EI is not an exact alias for EGO.

supports Source: Efficient Global Optimization of Expensive Black-Box Functions · Locator: Section 4.1, PDF pp.17–19 / printed pp.471–473; Section 4.2, PDF pp.19–21 / printed pp.473–475, algorithm and one-additional-point limitation.

⚠️ Curated Limitations & Negative Results

The inspected formulation uses noiseless observations and a fixed observed incumbent. With noisy data, the best observation need not be the best latent value, and expected improvement in a noisy measurement is not generally improvement in the underlying objective. Uncertainty in a latent incumbent would require an appropriate joint model. Section 6, PDF p.33 / printed p.487 explicitly leaves noisy-function optimization unresolved; no noisy, constrained, batch or multiobjective EI extension is specified here.
The DACE mean and prediction error use estimated correlation and variance parameters. Their uncertainty is not fully integrated, and the source notes that its standard error may be too small. Expected gain is conditional on that predictive approximation; it is neither a calibrated probability-of-success guarantee nor a bound on the true remaining optimization error (Section 2, PDF p.9 / printed p.463; Section 4.2, PDF pp.20–21 / printed pp.474–475).
Changing the objective by a nonlinear increasing transformation preserves its ordering but can change improvement magnitudes and acquisition rankings. The source actually fits transformed responses in EGO. EI on the transformed scale must not be identified with EI on the original scale or obtained by applying the transform to a predicted mean alone. The transformation distinction is supported by the inspected procedure and separate bounded curator examples, not by a universal invariance claim (Section 4.2, PDF pp.19–20 / printed pp.473–474).
EI depends on both predictive location and spread. It is distinct from probability of improvement, variance maximization and optimizer-information acquisitions. The original bounded curator checks separate exact rational interpolation from floating Gaussian CDF, quadrature, derivative and transformation examples; they do not implement EGO or establish numerical accuracy in extreme Gaussian tails.
Evidence is the 38-page journal-layout mirror of Jones, Schonlau and Welch (1998), not a verified byte-identical publisher download. The separate 43-page author manuscript was cover-inspected only, and the cited Mockus et al. (1978) chapter was not obtained. No earliest-invention claim is made. The source Appendix 1 derivative-sign discrepancy and fixed-correlation Hessian caveat concern separate surrogate derivations and do not change the defining EI expectation or its positive-standard-deviation formula.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:MathematicalFunction` Mathematical Function	related_match	EI assigns an expected gain to a candidate through a mathematical rule involving a predictive law and incumbent. This relates the specific acquisition rule to the pinned mathematical-function concept without equating it with that broad class, an activation function or a complete optimizer. It asserts no upstream subclass.

Evidence Sources & Primary Literature

1 Source

Source ID	Publication / Document	Type / Access	Reference & Checksum
`jones1998`	Efficient Global Optimization of Expensive Black-Box Functions Donald R. Jones, Matthias Schonlau, William J. Welch	primary paper full_text	ID: doi:10.1023/A:1008306431147 sha256:356260d0303a…

Curation Provenance

Curator: Codex

Method: ai_assisted

Model: GPT-6

Created: 2026-10-15T04:14:51Z

Updated: 2026-10-15T04:14:51Z

Source File: data/curated/ExpectedImprovementAcquisition.yaml