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upper · Normalized SampleWiki statement

Finite-sum RM-ULMC

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 (2026), §10.1.2primary theorem audit in progress
Complexity / rate
\[\widetilde O(m+\kappa^2+\kappa^{4/3}d^{1/3}m^{2/3}/\varepsilon^{2/3})\]

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 · Underdamped contraction and discretization route

This is a source-linked reading map for Chewi (2026), §10.1.2, not a transcription of the paper's proof. Exact source proof equations replace this map once theorem-level audit is complete.

01

Obtain continuous-time contraction for the underdamped Langevin dynamics under the source functional inequality.

02

Bound the randomized-midpoint/discretization error over one or several steps.

03

Convert the path or local numerical error into the requested terminal divergence/TV guarantee.

04

Balance contraction time against discretization error to choose step size and iteration count.

\[\widetilde O(m+\kappa^2+\kappa^{4/3}d^{1/3}m^{2/3}/\varepsilon^{2/3})\]
Assumptions and implicit prerequisites

What must be true before the rate can be read

Model and geometry

  • The oracle model is part of the theorem: stochastic-gradient noise and finite-sum access are not interchangeable.
  • Variance, component count, convexity/functional-inequality constants, and initialization are inherited from the cited result.
  • The displayed query complexity counts the oracle calls used by that source model.

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
Complexity / rate
\[\widetilde O(m+\kappa^2+\kappa^{4/3}d^{1/3}m^{2/3}/\varepsilon^{2/3})\]
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-STOCHASTIC-FINITE-SUM-UPPER-FINITE-SUM-RM-ULMC · source snapshot 40e4d538a2228b55