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best lower · Normalized SampleWiki statement

General first-order Fisher lower bound

A paper-first mathematical case study: read the theorem, the derivation route, and the hidden prerequisites before opening formal infrastructure.

Formal topologyOpen this result's proof branch↗
Statement

Normalized SampleWiki statement

Primary theorem audit pending — this is not presented as a verbatim paper theorem

Chewi–Gerber–Lee–Lu (ALT 2023)primary theorem audit in progress
Guarantee / accuracy
\[\operatorname{FI}\le\varepsilon^2\]
Complexity / rate
\[\varepsilon^{-2+o(1)}\]

Reading. Read the accuracy guarantee together with the complexity/rate: the source assumptions and its notion of oracle cost are part of the result.

Primary theorem audit pending. The formulas below are the cleaned SampleWiki normalization of the cited result, not a claim of verbatim theorem transcription.

Proof / derivation

Reader derivation map · Hard-instance and oracle-separation route

This is a source-linked reading map for Chewi–Gerber–Lee–Lu (ALT 2023), not a transcription of the paper's proof. Exact source proof equations replace this map once theorem-level audit is complete.

01

Construct a family of targets satisfying the same model/regularity assumptions as the upper-bound problem.

02

Show that too few oracle queries cannot reliably distinguish the hard instances or locate the relevant geometry.

03

Translate indistinguishability into separation in the theorem's target accuracy metric.

\[\operatorname{FI}\le\varepsilon^2\]
04

Solve the resulting information/query inequality to obtain the displayed lower-bound scaling.

\[\varepsilon^{-2+o(1)}\]
Assumptions and implicit prerequisites

What must be true before the rate can be read

Model and geometry

  • No global log-concavity is assumed merely because Fisher information is the output criterion.
  • The potential/score smoothness and finite initial-information quantity are inherited from the cited theorem.
  • Relative Fisher information requires a legitimate density/score representative; ASTIS keeps this regularity obligation visible.

Analytic / proof prerequisites

  • Exact theorem-level parameter ranges and technical hypotheses remain controlled by the cited paper until primary-source audit is complete.
ASTIS rigorous LaTeX
Guarantee / accuracy
\[\operatorname{FI}\le\varepsilon^2\]
Complexity / rate
\[\varepsilon^{-2+o(1)}\]
Lean formalization

This fold is intentionally quiet while the source statement, proof route, and assumptions are being completed case by case. A source-facing Lean theorem will appear here only after it compiles and its statement has been matched to the audited source.

References and provenance

SampleWiki setting

ASTIS-SW-SETTING-NONLOGCONCAVE-FISHER-BEST-LOWER-GENERAL-FIRST-ORDER-FISHER-LOWER-BOUND · source snapshot f0b372fb2da1e3a6