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Project-declaredLean 4.33.0-rc1 · mathlib@169c26b52a38

Is PACLearner For mono ε

Cslib.MachineLearning.PACLearning.IsPACLearnerFor.mono_ε

computer sciencecomputabilityprogram semanticsdefs

Plain-language statement

A PAC learner with accuracy ε₁ is also a PAC learner with any weaker accuracy ε₂ ≥ ε₁: the bad event {error > opt + ε} only shrinks.

Exact Lean statement

theorem IsPACLearnerFor.mono_ε {m : ℕ} {δ : Set.Ioo (0 : ℝ≥0) 1}
    {ε₁ ε₂ : Set.Ioo (0 : ℝ≥0) 1} (hε : ε₁.val ≤ ε₂.val)
    {C : ConceptClass α β} {𝒟 : Set (Measure (α × β))}
    (h : IsPACLearnerFor m ε₁ δ C 𝒟) :
    IsPACLearnerFor m ε₂ δ C 𝒟

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
theorem IsPACLearnerFor.mono_ε {m : ℕ} {δ : Set.Ioo (0 : ℝ≥0) 1}    {ε₁ ε₂ : Set.Ioo (0 : ℝ≥0) 1} (hε : ε₁.val ≤ ε₂.val)    {C : ConceptClass α β} {𝒟 : Set (Measure (α × β))}    (h : IsPACLearnerFor m ε₁ δ C 𝒟) :    IsPACLearnerFor m ε₂ 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 hε' : (↑ε₁.val : ℝ≥0∞) ≤ ↑ε₂.val := by exact_mod_cast hε  calc optimalError D C + (↑ε₁.val : ℝ≥0∞)      ≤ optimalError D C + ↑ε₂.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:228-239

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Related declarations

Project-declaredLean 4.33.0-rc1

Unique minimal

Cslib.Automata.DA.FinAcc.unique_minimal

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

Buchi Family cover

Cslib.Automata.NA.Buchi.buchiFamily_cover

Project documentation

na.buchiFamily is a cover if na has only finitely many states. This theorem uses the Ramsey theorem for infinite graphs and does not depend on any details of na.BuchiCongruence other than that it is of finite index.

computer sciencecomputabilityprogram semantics

Source project: Lean Computer Science Library

Person-level attribution pending.

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