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Project-declaredLean 4.33.0-rc1 · mathlib@169c26b52a38

Is PACLearner For antitone C

Cslib.MachineLearning.PACLearning.IsPACLearnerFor.antitone_C

computer sciencecomputabilityprogram semanticsdefs

Plain-language statement

The deterministic PAC learner predicate is antitone in the concept class: a learner for a larger class C' is also a learner for any subclass C ⊆ C', since the agnostic benchmark optimalError _ C ≥ optimalError _ C' makes the error requirement easier.

Exact Lean statement

theorem IsPACLearnerFor.antitone_C {m : ℕ} {ε δ : Set.Ioo (0 : ℝ≥0) 1}
    {C C' : ConceptClass α β} (hC : C ⊆ C')
    {𝒟 : Set (Measure (α × β))} (h : IsPACLearnerFor m ε δ C' 𝒟) :
    IsPACLearnerFor m ε δ C 𝒟

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
theorem IsPACLearnerFor.antitone_C {m : ℕ} {ε δ : Set.Ioo (0 : ℝ≥0) 1}    {C C' : ConceptClass α β} (hC : C ⊆ C')    {𝒟 : Set (Measure (α × β))} (h : IsPACLearnerFor m ε δ C' 𝒟) :    IsPACLearnerFor m ε δ C 𝒟 := by  obtain ⟨A, hA⟩ := h  refine ⟨A, fun D inst hD => le_trans (measure_mono ?_) (@hA D inst hD)⟩  intro S hS  have h_opt : optimalError D C' ≤ optimalError D C := iInf_le_iInf_of_subset hC  calc optimalError D C' + (↑ε.val : ℝ≥0∞)      ≤ optimalError D C + ↑ε.val := by gcongr    _ < error D (A S) := hS
Project
Lean Computer Science Library
License
Apache-2.0
Commit
f36649cff2c9
Source
Cslib/MachineLearning/PACLearning/Defs.lean:244-254

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Plain-language statement

The minimal DFA M accepting the language l is unique up to unique isomorphism.

computer sciencecomputabilityprogram semantics

Source project: Lean Computer Science Library

Person-level attribution pending.

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Project-declaredLean 4.33.0-rc1

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computer sciencecomputabilityprogram semantics

Source project: Lean Computer Science Library

Person-level attribution pending.

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