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e320e82 | 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 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 | import OAI.Combinatorics.ExpectationThreshold.RoundingMasses
import OAI.Combinatorics.ExpectationThreshold.RoundingLowerBound
import OAI.Combinatorics.ExpectationThreshold.RoundingProbabilityContradiction
import OAI.Combinatorics.ExpectationThreshold.ColorMean
import OAI.Combinatorics.ExpectationThreshold.SelectorMain
namespace OAI
/-!
# Actual rounding of an unrestricted fractional cover
The concrete member masses feed the proved multiscale selector. The same
independent color law supplies the lower and upper bounds on the subset-product
score. No selector hypothesis or bound on support sizes appears in the result.
-/
open scoped BigOperators
noncomputable section
namespace TalagrandThreshold
variable {α : Type*} [Fintype α] [DecidableEq α]
omit [DecidableEq α] in
theorem fractionalCover_empty_not_mem {F : Family α} {g : Finset α → ℝ} {r : ℝ}
(hr : 0 ≤ r) (hcover : FractionalCover F g) (hcost : fracCost r g ≤ 1 / 2) :
(∅ : Finset α) ∉ F := by
intro hempty
have hcoverEmpty : (1 : ℝ) ≤ g ∅ := by
simpa only [Finset.powerset_empty, Finset.sum_singleton] using hcover.2 ∅ hempty
have hweight : g ∅ ≤ (1 / 2 : ℝ) :=
(empty_weight_le_fracCost hr hcover.nonneg).trans hcost
exact (by norm_num : ¬ ((1 : ℝ) ≤ 1 / 2)) (hcoverEmpty.trans hweight)
/-- The actual normalized probability masses attached to family members. -/
def fractionalMassFamily (F : Family α) (g : Finset α → ℝ) {r : ℝ}
(hr : 0 ≤ r) (hcover : FractionalCover F g) (hcost : fracCost r g ≤ 1 / 2) :
Selector.MassFamily F where
weight H := roundingMass g H.val
nonneg H x := roundingMass_nonneg g H.val hcover.nonneg
(memberMass_pos_of_fractionalCover hcover H.property hr hcost) x
total H := roundingMass_sum g H.val
(memberMass_pos_of_fractionalCover hcover H.property hr hcost)
supported H _ hx := roundingMass_eq_zero_of_not_mem g H.val hx
/-- Every fractional cover of cost at most one half rounds at the exact
parameter `r / (25 * 512^4)`. Its support may contain subsets of any size. -/
theorem fractionalCover_rounding (F : Family α) (g : Finset α → ℝ) {r : ℝ}
(hr0 : 0 < r) (hr1 : r ≤ 1) (hcover : FractionalCover F g)
(hcost : fracCost r g ≤ 1 / 2) :
Small (r / ((25 : ℝ) * (512 : ℝ) ^ 4)) F := by
classical
by_cases hF : F.Nonempty
· let p : ℝ := r / ((25 : ℝ) * (512 : ℝ) ^ 4)
have hp0 : 0 < p := div_pos hr0 rounding_constant_pos
have hp1 : p ≤ 1 := (div_le_one rounding_constant_pos).mpr
(hr1.trans (by norm_num : (1 : ℝ) ≤ (25 : ℝ) * (512 : ℝ) ^ 4))
have hnoempty := fractionalCover_empty_not_mem hr0.le hcover hcost
let mu := fractionalMassFamily F g hr0.le hcover hcost
obtain ⟨s, hs, hnext⟩ := exists_scale hp0
let μ := coloringLaw (α := α) p s hp0 hs
let Y : Color s → ℝ := fun c => (512 : ℝ) ^ 4 / (512 : ℝ) ^ (c.val + 1)
let Z : Coloring α s → ℝ := fun a => productScore g (fun x => Y (a x))
let P : Coloring α s → Prop := fun a => ∃ H : ↥F, Selector.meanColor mu H a ≤ 2
change Small p F
by_contra hnotsmall
have hprob : (9 / 10 : ℝ) ≤ μ.prob P :=
Selector.average_color F hF (fun H hH => members_nonempty hnoempty hH)
mu p hp0 hp1 hnotsmall s hs
have hY : ∀ c, 0 ≤ Y c := by
intro c
exact div_nonneg (pow_nonneg (by norm_num) _) (pow_nonneg (by norm_num) _)
have hZ : ∀ a, 0 ≤ Z a := fun a =>
productScore_nonneg hcover.nonneg (fun x => hY (a x))
have hlower : ∀ a, P a → (1 / 4 : ℝ) ≤ Z a := by
intro a ha
obtain ⟨H, hH⟩ := ha
apply productScore_lower_of_mean g H.val (fun x => (a x).val + 1) hcover.nonneg
(memberMass_ge_half hcover H.property hr0.le hcost)
simpa only [Selector.meanColor, mu, fractionalMassFamily, Nat.cast_add,
Nat.cast_one] using hH
have hmean : (colorLaw p s hp0 hs).expectation Y ≤ r / 5 := by
change (∑ c : Color s,
colorProb p s c * ((512 : ℝ) ^ 4 / (512 : ℝ) ^ (c.val + 1))) ≤ r / 5
exact color_mean_le_r_div_five hr0 hs hnext
have hupper : μ.expectation Z ≤ (1 / 10 : ℝ) :=
expectation_productScore_le_tenth (colorLaw p s hp0 hs) g Y
hcover.nonneg hY hr0.le hmean hcost
exact rounding_expectation_contradiction μ P Z hZ hprob hlower hupper
· have hFempty : F = ∅ := Finset.not_nonempty_iff_eq_empty.mp hF
rw [hFempty]
exact small_empty _
/-- Parameterwise rounding also includes density zero, with an actual cover. -/
theorem fractionalSmall_rounding (F : Family α) {r : ℝ}
(hr0 : 0 ≤ r) (hr1 : r ≤ 1) (hF : FractionalSmall r F) :
Small (r / ((25 : ℝ) * (512 : ℝ) ^ 4)) F := by
obtain ⟨g, hcover, hcost⟩ := hF
by_cases hr : r = 0
· subst r
rw [zero_div]
exact small_zero F (fractionalCover_empty_not_mem le_rfl hcover hcost)
· exact fractionalCover_rounding F g (lt_of_le_of_ne hr0 (Ne.symm hr)) hr1 hcover hcost
end TalagrandThreshold
end
end OAI
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