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Lean Formalization of Generalization Error Bound by Rademacher Complexity and Dudley's Entropy Integral

Sho Sonoda, Kazumi Kasaura, Yuma Mizuno, Kei Tsukamoto, Naoto Onda · Mar 25, 2025 · Citations: 0

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Coverage: Stale

Use this page to decide whether the paper is strong enough to influence an eval design. It summarizes the abstract plus available structured metadata. If the signal is thin, use it as background context and compare it against stronger hub pages before making protocol choices.

Best use

Background context only

Metadata: Stale

Trust level

Low

Signals: Stale

What still needs checking

Extraction flags indicate low-signal or possible false-positive protocol mapping.

Signal confidence: 0.15

Abstract

Understanding and certifying the generalization performance of machine learning algorithms -- i.e. obtaining theoretical estimates of the test error from a finite training sample -- is a central theme of statistical learning theory. Among the many complexity measures used to derive such guarantees, Rademacher complexity yields sharp, data-dependent bounds that apply well beyond classical $0$--$1$ classification. In this study, we formalize the generalization error bound by Rademacher complexity in Lean 4, building on measure-theoretic probability theory available in the Mathlib library. Our development provides a mechanically-checked pipeline from the definitions of empirical and expected Rademacher complexity, through a formal symmetrization argument and a bounded-differences analysis, to high-probability uniform deviation bounds via a formally proved McDiarmid inequality. A key technical contribution is a reusable mechanism for lifting results from countable hypothesis classes (where measurability of suprema is straightforward in Mathlib) to separable topological index sets via a reduction to a countable dense subset. As worked applications of the abstract theorem, we mechanize standard empirical Rademacher bounds for linear predictors under $\ell_2$ and $\ell_1$ regularization, and we also formalize a Dudley-type entropy integral bound based on covering numbers and a chaining construction.

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  • Extraction flags indicate low-signal or possible false-positive protocol mapping.
  • Extraction confidence is 0.15 (below strong-reference threshold).
  • No explicit evaluation mode was extracted from available metadata.
  • No benchmark/dataset or metric anchors were extracted.
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HFEPX Relevance Assessment

This paper is adjacent to HFEPX scope and is best used for background context, not as a primary protocol reference.

Best use

Background context only

Use if you need

Background context only.

Main weakness

Extraction flags indicate low-signal or possible false-positive protocol mapping.

Trust level

Low

Eval-Fit Score

0/100 • Low

Treat as adjacent context, not a core eval-method reference.

Human Feedback Signal

Not explicit in abstract metadata

Evaluation Signal

Weak / implicit signal

HFEPX Fit

Adjacent candidate

Extraction confidence: Low

If you are doing eval pipeline work, start here:

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What This Page Found In The Paper

Each field below shows whether the signal looked explicit, partial, or missing in the available metadata. Use this to judge what is safe to trust directly and what still needs full-paper validation.

Human Feedback Types

missing

None explicit

Confidence: Low Not found

No explicit feedback protocol extracted.

Evidence snippet: Understanding and certifying the generalization performance of machine learning algorithms -- i.e.

Evaluation Modes

missing

None explicit

Confidence: Low Not found

Validate eval design from full paper text.

Evidence snippet: Understanding and certifying the generalization performance of machine learning algorithms -- i.e.

Quality Controls

missing

Not reported

Confidence: Low Not found

No explicit QC controls found.

Evidence snippet: Understanding and certifying the generalization performance of machine learning algorithms -- i.e.

Benchmarks / Datasets

missing

Not extracted

Confidence: Low Not found

No benchmark anchors detected.

Evidence snippet: Understanding and certifying the generalization performance of machine learning algorithms -- i.e.

Reported Metrics

missing

Not extracted

Confidence: Low Not found

No metric anchors detected.

Evidence snippet: Understanding and certifying the generalization performance of machine learning algorithms -- i.e.

Rater Population

missing

Unknown

Confidence: Low Not found

Rater source not explicitly reported.

Evidence snippet: Understanding and certifying the generalization performance of machine learning algorithms -- i.e.

Human Data Lens

  • Uses human feedback: No
  • Feedback types: None
  • Rater population: Unknown
  • Unit of annotation: Unknown
  • Expertise required: Math
  • Signal basis: Structured extraction plus abstract evidence.

Evaluation Lens

  • Evaluation modes:
  • Agentic eval: None
  • Quality controls: Not reported
  • Signal confidence: 0.15
  • Known cautions: low_signal, possible_false_positive

Protocol And Measurement Signals

Benchmarks / Datasets

No benchmark or dataset names were extracted from the available abstract.

Reported Metrics

No metric terms were extracted from the available abstract.

Research Brief

Metadata summary

Understanding and certifying the generalization performance of machine learning algorithms -- i.e.

Based on abstract + metadata only. Check the source paper before making high-confidence protocol decisions.

Key Takeaways

  • Understanding and certifying the generalization performance of machine learning algorithms -- i.e.
  • obtaining theoretical estimates of the test error from a finite training sample -- is a central theme of statistical learning theory.
  • Among the many complexity measures used to derive such guarantees, Rademacher complexity yields sharp, data-dependent bounds that apply well beyond classical $0$--$1$ classification.

Researcher Actions

  • Compare this paper against nearby papers in the same arXiv category before using it for protocol decisions.
  • Check the full text for explicit evaluation design choices (raters, protocol, and metrics).
  • Use related-paper links to find stronger protocol-specific references.

Caveats

  • Generated from abstract + metadata only; no PDF parsing.
  • Signals below are heuristic and may miss details reported outside the abstract.

Recommended Queries

Research Summary

Contribution Summary

  • Understanding and certifying the generalization performance of machine learning algorithms -- i.e.
  • obtaining theoretical estimates of the test error from a finite training sample -- is a central theme of statistical learning theory.
  • Among the many complexity measures used to derive such guarantees, Rademacher complexity yields sharp, data-dependent bounds that apply well beyond classical 0--1 classification.

Why It Matters For Eval

  • Abstract shows limited direct human-feedback or evaluation-protocol detail; use as adjacent methodological context.

Researcher Checklist

  • Gap: Human feedback protocol is explicit

    No explicit human feedback protocol detected.

  • Gap: Evaluation mode is explicit

    No clear evaluation mode extracted.

  • Gap: Quality control reporting appears

    No calibration/adjudication/IAA control explicitly detected.

  • Gap: Benchmark or dataset anchors are present

    No benchmark/dataset anchor extracted from abstract.

  • Gap: Metric reporting is present

    No metric terms extracted.

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