lean/OAI/Computability/VertexCover/Analysis/OwnGradientUnsupported.lean

import Mathlib
import OAI.Computability.VertexCover.Analysis.BatchFunctionOwnLipschitz

namespace OAI

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namespace VertexCover.Average
open MeasureTheory

theorem integral_l1_le {α ι : Type*} [MeasurableSpace α] [Fintype ι]
    (μ : Measure α) [IsProbabilityMeasure μ] (g : α → ι → ℝ)
    (hg : ∀ i, AEStronglyMeasurable (fun a => g a i) μ)
    (hb : ∀ a, ∑ i, |g a i| ≤ 1) :
    ∑ i, |∫ a, g a i ∂μ| ≤ 1 := by
  have hbi : ∀ a i, |g a i| ≤ 1 := by
    intro a i
    exact (Finset.single_le_sum (fun j _ => abs_nonneg (g a j)) (Finset.mem_univ i)).trans (hb a)
  have hi : ∀ i, Integrable (fun a => |g a i|) μ := by
    intro i
    exact (Integrable.of_bound (hg i) 1
      (Filter.Eventually.of_forall (fun a => by simpa only [Real.norm_eq_abs] using hbi a i))).abs
  calc
    (∑ i, |∫ a, g a i ∂μ|) ≤ ∑ i, ∫ a, |g a i| ∂μ := by
      apply Finset.sum_le_sum
      intro i _
      simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm (fun a => g a i)
    _ = ∫ a, ∑ i, |g a i| ∂μ := (integral_finsetSum _ (fun i _ => hi i)).symm
    _ ≤ ∫ _a, (1 : ℝ) ∂μ := integral_mono
      (integrable_finsetSum _ (fun i _ => hi i)) (integrable_const 1) hb
    _ = 1 := by simp

theorem finite_l1_le {α ι : Type*} [Fintype α] [Fintype ι]
    (g : α → ι → ℝ) (hb : ∀ a, ∑ i, |g a i| ≤ 1) :
    ∑ i, |(∑ a, g a i) / Fintype.card α| ≤ 1 := by
  classical
  by_cases he : IsEmpty α
  · let := he
    simp
  · have : Nonempty α := not_isEmpty_iff.mp he
    have hc : (0 : ℝ) < Fintype.card α := Nat.cast_pos.mpr Fintype.card_pos
    simp only [abs_div, abs_of_nonneg hc.le, ← Finset.sum_div]
    apply (div_le_iff₀ hc).mpr
    calc
      (∑ i, |∑ a, g a i|) ≤ ∑ i, ∑ a, |g a i| := by
        exact Finset.sum_le_sum (fun i _ => Finset.abs_sum_le_sum_abs _ _)
      _ = ∑ a, ∑ i, |g a i| := Finset.sum_comm
      _ ≤ ∑ _a : α, (1 : ℝ) := Finset.sum_le_sum (fun a _ => hb a)
      _ = 1 * Fintype.card α := by simp

end VertexCover.Average

namespace VertexCover.LabelCover
open MeasureTheory ProbabilityTheory

noncomputable def ownFiber (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
    (frozen : Φ.Seeds d) (j : J) (i : Φ.Query d) : Finset (Φ.HiddenSeeds J) := by
  classical
  exact Finset.univ.filter (fun hidden => Φ.query (Φ.spliceSeeds J frozen hidden) j = i)

@[simp] theorem mem_ownFiber (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
    (frozen : Φ.Seeds d) (j : J) (i : Φ.Query d) (hidden : Φ.HiddenSeeds J) :
    hidden ∈ Φ.ownFiber J frozen j i ↔ Φ.query (Φ.spliceSeeds J frozen hidden) j = i := by
  classical
  simp [ownFiber]

noncomputable def ownMean (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
    (frozen : Φ.Seeds d) (c0 : Φ.Coordinate d → ℝ)
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty)
    (j : J) (i : Φ.Query d) (s : Fin (Φ.WeightDimension d) → ℝ) : ℝ := by
  classical
  exact VertexCover.finiteMean (fun hidden : Φ.ownFiber J frozen j i =>
    ∫ other, Φ.batchFunction (Φ.spliceSeeds J frozen hidden) J c0 A hA
      (Function.update other j s) ∂Φ.batchLaw J)

noncomputable def ownGradient (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
    (frozen : Φ.Seeds d) (c0 : Φ.Coordinate d → ℝ)
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty)
    (j : J) (i : Φ.Query d) (s : Fin (Φ.WeightDimension d) → ℝ)
    (k : Fin (Φ.WeightDimension d)) : ℝ := by
  classical
  exact VertexCover.finiteMean (fun hidden : Φ.ownFiber J frozen j i =>
    ∫ other, Φ.canonicalGradient (Φ.spliceSeeds J frozen hidden) J c0 A hA
      (Function.update other j s) j k ∂Φ.batchLaw J)

theorem canonicalGradient_update_measurable (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
    (J : Finset (Fin d)) (c0 : Φ.Coordinate d → ℝ)
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty)
    (j : J) (s : Fin (Φ.WeightDimension d) → ℝ) (k : Fin (Φ.WeightDimension d)) :
    Measurable (fun other : Φ.BatchWeights J =>
      Φ.canonicalGradient seed J c0 A hA (Function.update other j s) j k) := by
  classical
  exact (measurable_pi_apply k).comp ((measurable_pi_apply j).comp
    ((Φ.canonicalGradient_measurable seed J c0 A hA).comp
      (continuous_id.update j continuous_const).measurable))

theorem ownGradient_block_bound (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
    (frozen : Φ.Seeds d) (c0 : Φ.Coordinate d → ℝ)
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty)
    (j : J) (i : Φ.Query d) (s : Fin (Φ.WeightDimension d) → ℝ) :
    ∑ k, |Φ.ownGradient J frozen c0 A hA j i s k| ≤ 1 := by
  classical
  apply VertexCover.Average.finite_l1_le
  intro hidden
  apply VertexCover.Average.integral_l1_le
  · intro k
    exact (Φ.canonicalGradient_update_measurable _ J c0 A hA j s k).aestronglyMeasurable
  · intro other
    exact Φ.canonicalGradient_block_bound _ J c0 A hA (Function.update other j s) j

theorem ownGradient_unused (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
    (frozen : Φ.Seeds d) (c0 : Φ.Coordinate d → ℝ)
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty)
    (j : J) (i : Φ.Query d) (s : Fin (Φ.WeightDimension d) → ℝ)
    (k : Fin (Φ.WeightDimension d)) (hk : ∀ a : i.LocalLabel, i.slot a ≠ k) :
    Φ.ownGradient J frozen c0 A hA j i s k = 0 := by
  classical
  unfold ownGradient VertexCover.finiteMean
  have hz : ∀ hidden : Φ.ownFiber J frozen j i, ∀ other : Φ.BatchWeights J,
      Φ.canonicalGradient (Φ.spliceSeeds J frozen hidden) J c0 A hA
        (Function.update other j s) j k = 0 := by
    intro hidden other
    apply Φ.canonicalGradient_unused
    have hq := (Φ.mem_ownFiber J frozen j i hidden).mp hidden.property
    rw [hq]
    exact hk
  simp only [hz, integral_zero, Finset.sum_const_zero, zero_div]

theorem ownGradient_unsupported (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
    (frozen : Φ.Seeds d) (c0 : Φ.Coordinate d → ℝ)
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty)
    (j : J) (i : Φ.Query d) (hi : Φ.ownFiber J frozen j i = ∅)
    (s : Fin (Φ.WeightDimension d) → ℝ) : Φ.ownGradient J frozen c0 A hA j i s = 0 := by
  classical
  funext k
  unfold ownGradient VertexCover.finiteMean
  simp only [Fintype.card_coe, hi, Finset.card_empty, Nat.cast_zero, div_zero, Pi.zero_apply]

end VertexCover.LabelCover


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end OAI