File size: 1,739 Bytes
b317718 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 | import OAI.Probability.ProjectionMoments.Model
namespace OAI
/-!
Finite-state Gaussian regression with noiseless labels and angular accuracy.
-/
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace
noncomputable section
universe u
namespace NoiselessRegression
namespace Learner
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}
end Learner
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)
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)
end NoiselessRegression
end
end OAI
|