| import OAI.Probability.ProjectionMoments.Model |
|
|
| namespace OAI |
|
|
| /-! |
| Finite-state Gaussian regression with noiseless labels and angular accuracy. |
| -/ |
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|
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace |
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| noncomputable section |
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| universe u |
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| namespace NoiselessRegression |
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| namespace Learner |
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| def success {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω] |
| (L : Learner d M T Ω) (ρ : Measure Ω) (ε : ℝ) (s : UnitSphere d) : ℝ≥0∞ := |
| (ρ.prod (sampleLaw d T)) |
| {z | angularError s (L.estimate s z.1 z.2) ≤ ε} |
|
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| end Learner |
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| def SubquadraticMemory : Prop := |
| ∃ c : ℝ, 0 < c ∧ ∀ M : ℕ → ℕ, |
| Asymptotics.IsLittleO atTop (fun d => (M d : ℝ)) (fun d => (d : ℝ) ^ 2) → |
| ∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d → ∀ T : ℕ, ∀ ε : ℝ, |
| 0 < ε → ε ≤ 1 / 10 → |
| ∀ (Ω : Type u) [MeasurableSpace Ω] (ρ : Measure Ω), IsProbabilityMeasure ρ → |
| ∀ L : Learner d (M d) T Ω, L.Admissible ρ → |
| (((2 : ℕ) / (3 : ℕ) : ℝ≥0∞) ≤ L.uniformSuccess ρ ε ∨ |
| ∀ s, ((2 : ℕ) / (3 : ℕ) : ℝ≥0∞) ≤ L.success ρ ε s) → |
| c * (d : ℝ) * Real.log (1 / ε) ≤ (T : ℝ) |
|
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| def FixedQuadraticMemory : Prop := |
| ∀ A₀ : ℝ, 0 < A₀ → ∃ c : ℝ, 0 < c ∧ ∃ d₀ : ℕ, |
| ∀ d : ℕ, d₀ ≤ d → ∀ M T : ℕ, (M : ℝ) ≤ A₀ * (d : ℝ) ^ 2 → |
| ∀ ε : ℝ, 0 < ε → ε ≤ 1 / 10 → |
| ∀ (Ω : Type u) [MeasurableSpace Ω] (ρ : Measure Ω), IsProbabilityMeasure ρ → |
| ∀ L : Learner d M T Ω, L.Admissible ρ → |
| ((2 : ℕ) / (3 : ℕ) : ℝ≥0∞) ≤ L.uniformSuccess ρ ε → |
| c * (d : ℝ) * Real.log (1 / ε) ≤ (T : ℝ) |
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| end NoiselessRegression |
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| end |
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| end OAI |
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|
