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Predictive Entropy Search

Algorithms AIMech:PredictiveEntropySearch DRAFT

A Bayesian global optimization algorithm that approximates information about the global maximizer by subtracting maximizer-conditioned predictive entropy from ordinary predictive entropy, using approximate maximizer samples and Gaussian constraint approximations.

Identity Notes & Concept Scope This record maintains the original PES approximate maximizer-sampling and predictive-entropy algorithm. Its ideal fixed-hyperparameter information target agrees with ES, but finite-feature maximizer draws and constraint/moment approximations do not identify the ES representer-point/EP-derivative procedure. Direction of optimization alone is not the distinction. General GP regression, derivative observations and generic entropy calculations are components or different methods. Constrained, parallel and max-value methods are not automatically aliases. No acronym alias is adopted; current canonical and ignored-inclusive evidence is refreshed before creation.

Claims & Grounded Evidence

6 Claims

Mechanism #optimizer-information-and-entropy-rearrangement

For a latent objective with a Gaussian-process prior and Gaussian observation noise, using a covariance sufficiently differentiable for the method’s derivative conditioning, fix the data, candidate location and hyperparameters. Let S be the global maximizer location and Y a prospective noisy observation. When the entropy expressions are well-defined, I(S;Y) equals both the expected reduction of optimizer entropy and H(Y)-E_S[H(Y|S)]. PES uses the second form. This is the same ideal information objective targeted by Entropy Search after a consistent minimum/maximum sign change; their computational approximations differ. Maximizing H(Y) alone omits the conditional-entropy term and is not this objective.

supports Source: Predictive Entropy Search for Efficient Global Optimization of Black-box Functions · Locator: Sections 1–2, PDF pp. 1–3, Equations 1–3; small original finite-channel illustrations in research/discovery/entropy-search-boundaries.py
supports Source: Entropy Search for Information-Efficient Global Optimization · Locator: Sections 2.3 and 2.5–2.6, PDF pp. 8–10, 12–14 / printed pp. 1816–1818, 1820–1822

Mechanism #finite-feature-maximizer-samples

For a shift-invariant real covariance kernel with a usable spectral distribution, the source constructs finite random Fourier features, samples Gaussian posterior feature weights, and numerically maximizes each resulting sample function. These locations approximate draws from the posterior maximizer law. For fixed features the linear model has the corresponding finite-rank Gaussian-process kernel. Averaging feature inner products to recover the original kernel does not make a finite-feature posterior or its maximizer samples exact for the original process. Feature count and numerical function maximization are additional approximation choices.

supports Source: Predictive Entropy Search for Efficient Global Optimization of Black-box Functions · Locator: Section 2.1, PDF p. 3, Equation 5
supports Source: Predictive Entropy Search for Efficient Global Optimization of Black-box Functions: Supplementary Material · Locator: Appendix A, supplement PDF pp. 1–2 / printed pp. 10–11; scalar kernel-conditioning boundary illustration in research/discovery/entropy-search-boundaries.py

Mechanism #approximate-maximizer-constraint-inference

For each sampled maximizer, the source conditions the Gaussian process on a zero gradient and selected Hessian off-diagonal values. It initially describes zero off-diagonals, then substitutes the values from the sample function used to obtain that maximizer. Negative Hessian-diagonal factors and a soft comparison with the largest noisy observation are incorporated by expectation propagation to give a Gaussian approximation. These are approximate conditioning factors, not a certificate of local or global maximality: negative diagonals do not imply negative definiteness when off-diagonals are nonzero, and Gaussian moment approximations do not retain the hard sign constraints exactly. The zero gradient assumption describes an interior stationary point and does not cover every bounded-domain maximum.

supports Source: Predictive Entropy Search for Efficient Global Optimization of Black-box Functions · Locator: Section 2.2, PDF pp. 4–5, C1–C2, Equations 6–7 and sampled off-diagonal substitution after Equation 9; original Hessian and boundary examples in research/discovery/entropy-search-boundaries.py
supports Source: Predictive Entropy Search for Efficient Global Optimization of Black-box Functions: Supplementary Material · Locator: Appendices B.1–B.2, supplement PDF pp. 2–3 / printed pp. 11–12

Mechanism #candidate-comparison-and-predictive-variance

At a candidate x, form a Gaussian approximation to the joint latent values f(x) and f(S), then impose f(x)<f(S). For a valid covariance and positive variance of f(S)-f(x), the source obtains the variance of the resulting truncated-Gaussian marginal. It then replaces that marginal by a Gaussian with the same moments. Observation-noise variance is added before evaluating the Gaussian predictive entropy. The truncated marginal itself is generally not Gaussian, and its moments do not make the true maximizer-conditioned predictive distribution exact. The Gaussian integral formulation requires consistent block ordering when using the supplement’s matrix notation.

supports Source: Predictive Entropy Search for Efficient Global Optimization of Black-box Functions · Locator: Section 2.2, PDF p. 5, Equations 8–9 and entropy approximation immediately following them
supports Source: Predictive Entropy Search for Efficient Global Optimization of Black-box Functions: Supplementary Material · Locator: Appendix B.3, supplement PDF pp. 3–4 / printed pp. 12–13, Equation 12 and following Gaussian prediction expressions; original block-order example in research/discovery/entropy-search-boundaries.py

Mechanism #hyperparameter-conditional-acquisition-average

The source samples Gaussian-process hyperparameters from their posterior and pairs each draw with an approximate maximizer draw. It averages the difference between one-half log ordinary predictive variance and one-half log maximizer-conditioned predictive variance, adding the corresponding draw’s observation-noise variance to both. This approximates an average of fixed-hyperparameter information acquisitions, E_psi[I(S;Y|psi,D,x)]. Such a conditional-information average is not generally equal to I(S;Y|D,x) under the fully marginalized model. The candidate-independent Gaussian and expectation-propagation terms can be computed once per draw and reused while evaluating the acquisition.

supports Source: Predictive Entropy Search for Efficient Global Optimization of Black-box Functions · Locator: Algorithm 2, PDF p. 2; Section 2.3, PDF p. 5, Equation 10; original finite XOR conditional/marginal information example in research/discovery/entropy-search-boundaries.py

Scope #sequential-query-and-final-recommendation

The sequential optimizer numerically maximizes the approximate acquisition to choose the next location, observes the noisy objective there, augments the data and repeats. Algorithm 1 returns a maximizer of the final Gaussian-process posterior mean as its recommendation. That recommendation is not defined as the mode of the optimizer-location distribution, and neither the approximation nor a finite evaluation budget certifies the true global optimum.

supports Source: Predictive Entropy Search for Efficient Global Optimization of Black-box Functions · Locator: Sections 1–2, PDF pp. 1–3; Algorithms 1–2, PDF p. 2

⚠️ Curated Limitations & Negative Results

This is the original PES approximation, not a generic name for all entropy acquisitions. Entropy Search directly approximates minimizer probabilities and prospective loss; PES instead uses approximate optimizer sampling and predictive-entropy inference. Their ideal fixed-hyperparameter information target is shared. Constrained, parallel and max-value entropy methods are not automatically the same identity or assessed variants here.
C3 on main PDF p. 4 reverses the maximum inequality in its all-domain prose, although its heading and Equation 8 on p. 5 use the intended candidate comparison. The stability paragraph after Equation 9 tests an undefined symbol s. This record retains the stated positive-variance domain and does not silently substitute a numerical stabilization rule. At x=S the strict comparison and variance require separate degenerate-case handling.
Supplement B.1 defines K on [z;c], and B.3 defines its cross-covariance in that order, but the displayed prediction uses [c;tilde_m] and diag(0,tilde_V), corresponding to the opposite order. Equation 12’s Gaussian integral avoids that ambiguity. With [z;c] maintained, the compatible blocks are [tilde_m;c] and diag(tilde_V,0); alternatively every matrix must be reordered consistently. This is a curator notation assessment, not an author-issued erratum or a tested PES software correction.
Differential-entropy identities require suitable densities and finite terms. A bounded-domain optimizer law can have boundary atoms, and interior derivative constraints omit maxima such as the endpoint of f(x)=x on [0,1]. General mutual information can be defined with appropriate measures, but that fact alone does not validate these numerical approximations or provide a universal boundary policy.
The original curator checks cover 27 finite positive binary channels and six constructed algebraic or logical boundaries. They illustrate information symmetry, conditional versus marginal information, Hessian and covariance limitations. They do not fit a Gaussian process, execute expectation propagation, sample random-feature functions, optimize an acquisition, run author code or reproduce benchmarks. No general accuracy, speed, regret or convergence advantage over Entropy Search follows from these checks.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:ActiveLearning` Active Learning	related_match	The pinned class describes interactive querying of an information source for outputs. This algorithm selects informative objective evaluations using a surrogate and optimizer-location uncertainty. The relation records that query-learning association without identifying the method with annotation or pool-based classification, asserting subclass membership, or making the broad AIO class an exact optimizer identity.

Evidence Sources & Primary Literature

3 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`hernandez2014`	Predictive Entropy Search for Efficient Global Optimization of Black-box Functions José Miguel Hernández-Lobato, Matthew W. Hoffman, Zoubin Ghahramani	primary paper full_text	sha256:82eb9e614937…
`hernandez2014-supplement`	Predictive Entropy Search for Efficient Global Optimization of Black-box Functions: Supplementary Material José Miguel Hernández-Lobato, Matthew W. Hoffman, Zoubin Ghahramani	primary paper full_text	sha256:c8cdec73f845…
`hennig2012`	Entropy Search for Information-Efficient Global Optimization Philipp Hennig, Christian J. Schuler	primary paper full_text	sha256:c1141315e0ce…

Curation Provenance

Curator: Codex

Method: ai_assisted

Model: GPT-6

Created: 2026-10-15T00:57:14Z

Updated: 2026-10-15T00:57:14Z

Source File: data/curated/PredictiveEntropySearch.yaml