lean/OAI/Computability/VertexCover/Analysis/CanonicalGradientValid.lean

import Mathlib
import OAI.Computability.VertexCover.Analysis.ZeroMemIncidenceHull
import OAI.Computability.VertexCover.Analysis.DistanceMinusSelect
import OAI.Computability.VertexCover.Analysis.AeRegular

namespace OAI

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namespace VertexCover.LabelCover
open MeasureTheory

theorem separator_continuous (Φ : LabelCover) {d : ℕ}
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) :
    Continuous (Φ.separator A hA) :=
  ((Φ.distanceTo_continuous A hA).comp continuous_neg |>.sub
    (Φ.distanceTo_continuous A hA)).div_const 2

noncomputable def batchFunction (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) (s : Φ.BatchWeights J) : ℝ :=
  Φ.separator A hA (c0 + Φ.batchSum seed J s)

theorem batchFunction_continuous (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) :
    Continuous (Φ.batchFunction seed J c0 A hA) :=
  (Φ.separator_continuous A hA).comp
    (continuous_const.add (Φ.batchSumCLM seed J).continuous)

theorem batchFunction_hasFDerivAt (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) (s : Φ.BatchWeights J)
    (hs : VertexCover.AffineSelector.Regular (Φ.affineSlope seed J)
      (Φ.affineOffset c0 (A := A)) s) :
    ∃ im ip : Φ.AffineIndex A, HasFDerivAt (Φ.batchFunction seed J c0 A hA)
      ((1/2 : ℝ) • (Φ.affineSlope seed J im - Φ.affineSlope seed J ip)) s := by
  have hc : Continuous (fun s : Φ.BatchWeights J => c0 + Φ.batchSum seed J s) :=
    continuous_const.add (Φ.batchSumCLM seed J).continuous
  obtain ⟨ip, hp⟩ := VertexCover.AffineSelector.hasFDerivAt _ _
    ((Φ.distanceTo_continuous A hA).comp hc)
    (Φ.distance_plus_select seed J c0 A hA) hs
  obtain ⟨im, hm⟩ := VertexCover.AffineSelector.hasFDerivAt _ _
    ((Φ.distanceTo_continuous A hA).comp hc.neg)
    (Φ.distance_minus_select seed J c0 A hA) hs
  refine ⟨im, ip, ?_⟩
  have heq : Φ.batchFunction seed J c0 A hA = (fun y => (1/2 : ℝ) *
      (Φ.distanceTo A hA (-(c0 + Φ.batchSum seed J y)) -
        Φ.distanceTo A hA (c0 + Φ.batchSum seed J y))) := by
    funext y
    unfold batchFunction separator
    ring
  rw [heq]
  exact (hm.sub hp).const_mul (1/2 : ℝ)

def BatchGradientValid (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
    (J : Finset (Fin d)) (g : Φ.BatchWeights J) : Prop :=
  ∃ G ∈ Φ.incidenceHull seed, ∀ (j : J) k, g j k = G j k

theorem signed_incidence_mem (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
    {I : Finset (Φ.Coordinate d)} (hI : Φ.Compatible I) (e : Bool) :
    affineSign e • Φ.incidence seed I ∈ Φ.incidenceHull seed := by
  apply subset_convexHull ℝ
  cases e
  · exact ⟨I, hI, Or.inr (by simp)⟩
  · exact ⟨I, hI, Or.inl (by simp)⟩

theorem neg_signed_incidence_mem (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
    {I : Finset (Φ.Coordinate d)} (hI : Φ.Compatible I) (e : Bool) :
    -(affineSign e • Φ.incidence seed I) ∈ Φ.incidenceHull seed := by
  apply subset_convexHull ℝ
  cases e
  · exact ⟨I, hI, Or.inl (by simp)⟩
  · exact ⟨I, hI, Or.inr (by simp)⟩

theorem batchGradient_affineSlope (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
    (J : Finset (Fin d)) {A : Finset (Φ.Coordinate d → ℝ)}
    (i : Φ.AffineIndex A) (j : J) (k : Fin (Φ.WeightDimension d)) :
    Φ.batchGradient J (Φ.affineSlope seed J i) j k =
      affineSign i.2.2.2 * Φ.incidence seed i.2.1.val j k := by
  change affineSign i.2.2.2 *
    Φ.batchGradient J ((Φ.selectionForm i.2.1.val).comp (Φ.batchSumCLM seed J)) j k = _
  rw [Φ.batchGradient_form seed J ((Φ.mem_compatibleSelections _).mp i.2.1.property)]

theorem batchGradient_half_difference_valid (Φ : LabelCover) {d : ℕ}
    (seed : Φ.Seeds d) (J : Finset (Fin d)) {A : Finset (Φ.Coordinate d → ℝ)}
    (im ip : Φ.AffineIndex A) :
    Φ.BatchGradientValid seed J (Φ.batchGradient J
      ((1/2 : ℝ) • (Φ.affineSlope seed J im - Φ.affineSlope seed J ip))) := by
  let gm := affineSign im.2.2.2 • Φ.incidence seed im.2.1.val
  let gp := -(affineSign ip.2.2.2 • Φ.incidence seed ip.2.1.val)
  have hm := Φ.signed_incidence_mem seed
    ((Φ.mem_compatibleSelections _).mp im.2.1.property) im.2.2.2
  have hp := Φ.neg_signed_incidence_mem seed
    ((Φ.mem_compatibleSelections _).mp ip.2.1.property) ip.2.2.2
  refine ⟨(1/2 : ℝ) • gm + (1/2 : ℝ) • gp,
    (convex_convexHull ℝ _) hm hp (by norm_num) (by norm_num) (by norm_num), ?_⟩
  intro j k
  change (1/2 : ℝ) * (Φ.batchGradient J (Φ.affineSlope seed J im) j k -
    Φ.batchGradient J (Φ.affineSlope seed J ip) j k) = _
  rw [Φ.batchGradient_affineSlope, Φ.batchGradient_affineSlope]
  simp only [gm, gp, Pi.add_apply, Pi.smul_apply, Pi.neg_apply, smul_eq_mul]
  ring

noncomputable def canonicalGradient (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) (s : Φ.BatchWeights J) :
    Φ.BatchWeights J := by
  classical
  exact if VertexCover.AffineSelector.Regular (Φ.affineSlope seed J)
    (Φ.affineOffset c0 (A := A)) s then
      Φ.batchGradient J (fderiv ℝ (Φ.batchFunction seed J c0 A hA) s) else 0

theorem canonicalGradient_valid (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) (s : Φ.BatchWeights J) :
    Φ.BatchGradientValid seed J (Φ.canonicalGradient seed J c0 A hA s) := by
  classical
  unfold canonicalGradient
  split_ifs with hs
  · obtain ⟨im, ip, hder⟩ := Φ.batchFunction_hasFDerivAt seed J c0 A hA s hs
    rw [hder.fderiv]
    exact Φ.batchGradient_half_difference_valid seed J im ip
  · exact ⟨0, Φ.zero_mem_incidenceHull seed, fun _ _ => rfl⟩

end VertexCover.LabelCover


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end OAI