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