lean/OAI/Probability/GaussianRegression/Main.lean

import OAI.Probability.GaussianRegression.Precision

namespace OAI

/-!
Quadratic and subquadratic memory lower bounds for Gaussian regression.
-/

open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
open InnerProductSpace
open scoped Pointwise

noncomputable section

universe u

namespace NoiselessRegression

theorem fixed_quadratic_memory : FixedQuadraticMemory.{u} := by
  intro A hA
  refine ⟨1/(4*(400000+100*A)), by positivity, 32, ?_⟩
  intro d hd M T hM ε hε hε10 Ω _ ρ hρ L hL hs
  let : IsProbabilityMeasure ρ := hρ
  have hT : d ≤ 4*T := by
    by_contra h
    have hb := hs.trans (L.linear_success_bound ρ (by omega) (by omega) hε10 hL)
    have hr := ENNReal.toReal_mono ENNReal.ofReal_ne_top hb
    norm_num [ENNReal.toReal_div] at hr
  let q := (d-1)/4
  let a := 2*q
  have hq : 2 ≤ q := by dsimp [q]; omega
  have hqa : 2*q ≤ a := le_rfl
  have ha : 2*a ≤ d-1 := by dsimp [a, q]; omega
  have hdq : d ≤ 8*q := by dsimp [q]; omega
  have hda : d ≤ 4*a := by dsimp [a]; omega
  exact precision_from_stream_bound hd hT (by omega) hdq hda hM hA hε (by linarith)
    (L.streaming_success_bound_real ρ hL (by omega) hε ha hq hqa hs)

section LowerFubini
variable {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
  {μ : Measure X} {ν : Measure Y} [SFinite μ] [SFinite ν]
lemma prod_null_set_lower_bound {S : Set (X × Y)} (hS : NullMeasurableSet S (μ.prod ν))
    {c : ℝ≥0∞} (h : ∀ᵐ x ∂μ, c ≤ ν {y | (x,y) ∈ S}) :
    c * μ univ ≤ (μ.prod ν) S := by
  have hm : AEMeasurable (S.indicator (fun _ => (1 : ℝ≥0∞))) (μ.prod ν) :=
    aemeasurable_const.indicator₀ hS
  rw [← one_mul ((μ.prod ν) S), ← lintegral_indicator_const₀ hS 1, lintegral_prod _ hm]
  have hc : c * μ univ = ∫⁻ _ : X, c ∂μ := (lintegral_const c).symm
  rw [hc]
  apply lintegral_mono_ae
  filter_upwards [h, ae_fiber_aemeasurable hm.prod_swap] with x hx hmx
  have hs : NullMeasurableSet {y | (x,y) ∈ S} ν := by
    have heq : {y | (x,y) ∈ S} = {y | S.indicator (fun _ => (1 : ℝ≥0∞)) (x,y) = 1} := by
      ext y
      by_cases hy : (x,y) ∈ S <;> simp [hy]
    rw [heq]
    exact nullMeasurableSet_eq_fun hmx aemeasurable_const
  change c ≤ ∫⁻ y, ({y | (x,y) ∈ S}).indicator (fun _ => (1 : ℝ≥0∞)) y ∂ν
  simpa [lintegral_indicator_const₀ hs] using hx
end LowerFubini

namespace Learner
lemma uniformSuccess_of_pointwise {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω]
    (L : Learner d M T Ω) (ρ : Measure Ω) [IsProbabilityMeasure ρ]
    (hL : L.Admissible ρ) (hd : 0 < d) (ε : ℝ) {c : ℝ≥0∞}
    (hs : ∀ s, c ≤ L.success ρ ε s) : c ≤ L.uniformSuccess ρ ε := by
  let : NeZero d := ⟨hd.ne'⟩
  have hb := prod_null_set_lower_bound (L.angularSuccess_nullMeasurable ρ ε hL)
    (c := c) (Filter.Eventually.of_forall hs)
  simpa only [measure_univ, mul_one, uniformSuccess] using hb
end Learner

theorem subquadratic_memory : SubquadraticMemory.{u} := by
  obtain ⟨c, hc, d₁, hfixed⟩ := fixed_quadratic_memory (1 : ℝ) (by norm_num)
  refine ⟨c, hc, ?_⟩
  intro M hM
  obtain ⟨d₂, hd₂⟩ := eventually_atTop.1 hM.eventuallyLE
  refine ⟨max d₁ (max d₂ 1), ?_⟩
  intro d hd T ε hε hε10 Ω _ ρ hρ L hL hs
  let : IsProbabilityMeasure ρ := hρ
  have hmem : (M d : ℝ) ≤ 1 * (d : ℝ)^2 := by
    have hm := hd₂ d (by omega)
    simpa only [Real.norm_eq_abs, abs_of_nonneg (Nat.cast_nonneg (M d) : (0 : ℝ) ≤ M d),
      abs_of_nonneg (sq_nonneg (d : ℝ)), one_mul] using hm
  apply hfixed d (by omega) (M d) T hmem ε hε hε10 Ω ρ hρ L hL
  rcases hs with hs | hs
  · exact hs
  · exact L.uniformSuccess_of_pointwise ρ hL (by omega) ε hs

theorem main : SubquadraticMemory.{u} ∧ FixedQuadraticMemory.{u} :=
  ⟨subquadratic_memory, fixed_quadratic_memory⟩

end NoiselessRegression

end

end OAI