TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Finite-candidate Gaussian-process acquisition search terminates within its budget

Statement

For a nonempty finite candidate list of length $M \geq 1$ , a horizon $N \geq 0$ , and an initial-list length $s$ with $0 \leq s \leq N$ , the procedure in Finite-candidate Gaussian-process acquisition search with explicit statuses terminates after at most $N - s$ observation calls. It returns complete with the posterior state at history length $N$ , or one of its explicit inadmissible-model, posterior-solve-unavailable, acquisition-inconclusive, or observation-unavailable statuses. Assume every validation, positive-definiteness check, Cholesky factorization, triangular solve, acquisition-certificate check, and observation call finishes in finite time or reports failure.

Facts & Assumptions

Given: A finite ordered candidate list, horizon $N \geq 0$ , initial-list length $0 \leq s \leq N$ , and the algorithm's validation, posterior-update, acquisition, and status instructions.

[F1]

At each successful horizon step the method inspects at most $M$ acquisition values and appends exactly one observation; invalid inputs and every solve, acquisition, or observation failure branch return immediately. Finite-candidate Gaussian-process acquisition search with explicit statuses

Proof

1.1givenF1algebra

The input validation and finite scan of $M$ candidates terminate by the Given condition. Initializing the posterior, updating it after an observation, and checking an acquisition certificate each use finitely many stated operations; each positive-definiteness check, factorization, solve, and observation call returns or reports failure in finite time. Every failure branch returns an explicit status.

2.1step 1.1F1induction∎

Each successful stage appends one observation, so there are at most $N - s$ such stages. After the final observation the algorithm computes the posterior at history length $N$ and returns complete; if that update or any earlier validation, solve, acquisition-certificate, or observation step fails, it returns the corresponding explicit status. Thus every branch terminates after finitely many operations.

Depends on

Finite-candidate Gaussian-process acquisition search with explicit statuses

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Peter I. Frazier, A Tutorial on Bayesian Optimization, §2 (standard reference, not scraped)
Niranjan Srinivas et al., Gaussian Process Optimization in the Bandit Setting, §3 (standard reference, not scraped)