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dd14c26 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 | import OAI.Probability.MatroidProphet.GroupConditioning
import OAI.Probability.MatroidProphet.SafeStatistics
import OAI.Probability.MatroidProphet.Constants
namespace OAI
namespace MatroidProphet.MainAlgorithm
open Finset
variable {n : ℕ}
attribute [local instance] Classical.propDecidable
noncomputable def groupGround (M : Matroid (Fin n)) (d : MainMasks n)
(s : Fin n → Option ℤ) (j : ℕ) : Finset (Fin n) :=
(groupMask M d s univ j).toFinset
lemma mem_groupGround (M : Matroid (Fin n)) (d : MainMasks n)
(s : Fin n → Option ℤ) (j : ℕ) (e : Fin n) :
e ∈ groupGround M d s j ↔ candidate M d s e (s e) ∧ s e = (groups M d s)[j]? := by
rw [groupGround, Set.mem_toFinset]
simp [groupMask]
lemma groupGround_disjoint (M : Matroid (Fin n)) (d : MainMasks n)
(s : Fin n → Option ℤ) : Pairwise (fun i j => Disjoint (groupGround M d s i) (groupGround M d s j)) := by
intro i j hij
apply disjoint_left.mpr
intro e hei hej
obtain ⟨hc, hei⟩ := (mem_groupGround M d s i e).mp hei
obtain ⟨_, hej⟩ := (mem_groupGround M d s j e).mp hej
have hi : i < (groups M d s).length := by
by_contra hi
exact hc.1 (hei.trans (List.getElem?_eq_none (by omega)))
exact hij ((List.Nodup.getElem?_inj hi (groups_nodup M d s)).mp (hei.symm.trans hej))
lemma groupMask_eq_inter_ground (M : Matroid (Fin n)) (d : MainMasks n)
(s : Fin n → Option ℤ) (A : Finset (Fin n)) (j : ℕ) :
groupMask M d s A j = (A : Set (Fin n)) ∩ (groupGround M d s j : Set (Fin n)) := by
ext e
simp only [groupMask, Set.mem_ofPred_eq, Set.mem_inter_iff, Finset.mem_coe, mem_groupGround]
lemma listedSafeStatistic_guards (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(κ : ℕ) (d : MainMasks n) (s : Fin n → Option ℤ) (i : ℤ)
(Y : Set (Fin n)) (ε : Fin 2) (C T : Finset (Fin n))
(hi : i ∈ groups M d s) :
listedSafeStatistic M hE κ (withMasks d d.D C T) s i Y ε =
safeLayerStatistic M hE κ (groupMask M d s d.D)
(fun j => (C : Set (Fin n)) ∩ (groupGround M d s j : Set (Fin n)))
(fun j => (T : Set (Fin n)) ∩ (groupGround M d s j : Set (Fin n)))
((groups M d s).idxOf i) (trueGroup M d s i : Set (Fin n))
((d.D : Set (Fin n)) ∩ Y) ε := by
change (if i ∈ groups M d s then _ else _) = _
rw [ite_eq_left hi]
change safeLayerStatistic M hE κ (groupMask M d s d.D)
(groupMask M d s C) (groupMask M d s T) ((groups M d s).idxOf i)
(trueGroup M d s i : Set (Fin n)) ((d.D : Set (Fin n)) ∩ Y) ε = _
congr 1
· funext j
exact groupMask_eq_inter_ground M d s C j
· funext j
exact groupMask_eq_inter_ground M d s T j
lemma unlisted_density_inter_empty (M : Matroid (Fin n)) (d : MainMasks n)
(s : Fin n → Option ℤ) (i : ℤ) (hi : i ∉ groups M d s)
(Y : Set (Fin n)) (hY : Y ⊆ (trueGroup M d s i : Set (Fin n))) :
(d.D : Set (Fin n)) ∩ Y = ∅ := by
apply Set.eq_empty_iff_forall_notMem.mpr
rintro e ⟨heD, heY⟩
exact hi ((mem_groups M d s i).mpr ((mem_listed_iff M d s i).mpr ⟨e, heD, hY heY⟩))
theorem listedSafeStatistic_fixed_density (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(κ : ℕ) (hκ : 0 < κ) (d : MainMasks n) (s : Fin n → Option ℤ) (i : ℤ)
(Y : Set (Fin n)) (hY : Y ⊆ (trueGroup M d s i : Set (Fin n))) (hI : M.Indep Y)
(q : Fin n → ℝ) (hq0 : ∀ e, 0 ≤ q e) (hq1 : ∀ e, q e ≤ 1) :
(((2:ℝ)^12)⁻¹) / 4 * ((d.D : Set (Fin n)) ∩ Y).ncard -
(trueGroup M d s i).card / (κ:ℝ) ≤
bitsExpectation q univ (fun C => bitsExpectation q univ (fun T =>
fairParityExpectation (fun p =>
(listedSafeStatistic M hE κ (withMasks d d.D C T) s i Y (boolParity p) : ℝ)))) := by
by_cases hi : i ∈ groups M d s
· have hsafe := safeLayerStatistic_lower M hE κ hκ (groupMask M d s d.D)
(groupGround M d s) (groupGround_disjoint M d s) q hq0 hq1
((groups M d s).idxOf i) (trueGroup M d s i : Set (Fin n))
((d.D : Set (Fin n)) ∩ Y) (Set.inter_subset_right.trans hY) (hI.subset Set.inter_subset_right)
have hcard : (groupMask M d s d.D ((groups M d s).idxOf i)).ncard ≤ (trueGroup M d s i).card := by
rw [groupMask_at_key M d s d.D _ i (groups_get?_idxOf M d s i hi), Set.ncard_coe_finset]
exact card_le_card inter_subset_right
have hdiv : ((groupMask M d s d.D ((groups M d s).idxOf i)).ncard : ℝ) / (κ:ℝ) ≤
(trueGroup M d s i).card / (κ:ℝ) :=
div_le_div_of_nonneg_right (by exact_mod_cast hcard) (Nat.cast_nonneg κ)
have heq : bitsExpectation q univ (fun C => bitsExpectation q univ (fun T =>
fairParityExpectation (fun p =>
(listedSafeStatistic M hE κ (withMasks d d.D C T) s i Y (boolParity p) : ℝ)))) =
bitsExpectation q univ (fun C => bitsExpectation q univ (fun T =>
fairParityExpectation (fun p => (safeLayerStatistic M hE κ (groupMask M d s d.D)
(fun j => (C : Set (Fin n)) ∩ (groupGround M d s j : Set (Fin n)))
(fun j => (T : Set (Fin n)) ∩ (groupGround M d s j : Set (Fin n)))
((groups M d s).idxOf i) (trueGroup M d s i : Set (Fin n))
((d.D : Set (Fin n)) ∩ Y) (boolParity p) : ℝ)))) := by
apply bitsExpectation_congr
intro C hC
apply bitsExpectation_congr
intro T hT
congr 1
funext p
rw [listedSafeStatistic_guards M hE κ d s i Y (boolParity p) C T hi]
rw [heq]
exact (sub_le_sub_left hdiv _).trans hsafe
· have hempty := unlisted_density_inter_empty M d s i hi Y hY
have hzero (C T : Finset (Fin n)) (p : Bool) :
listedSafeStatistic M hE κ (withMasks d d.D C T) s i Y (boolParity p) = 0 := by
change (if i ∈ groups M d s then _ else 0) = 0
rw [ite_eq_right hi]
simp only [hempty, Set.ncard_empty, Nat.cast_zero, mul_zero, zero_sub, hzero,
fairParityExpectation, zero_add, zero_div, bitsExpectation_const]
exact neg_nonpos.mpr (div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _))
theorem listedSafeStatistic_lower (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(κ : ℕ) (hκ : 0 < κ) (d : MainMasks n) (s : Fin n → Option ℤ) (i : ℤ)
(Y : Set (Fin n)) (hY : Y ⊆ (trueGroup M d s i : Set (Fin n))) (hI : M.Indep Y)
(q : Fin n → ℝ) (hq0 : ∀ e, 0 ≤ q e) (hq1 : ∀ e, q e ≤ 1) :
retainedFraction * Y.ncard - (trueGroup M d s i).card / (κ:ℝ) ≤
tripleMaskExpectation (fun _ => (1:ℝ)/4) q q univ (fun D C T =>
fairParityExpectation (fun p =>
(listedSafeStatistic M hE κ (withMasks d D C T) s i Y (boolParity p) : ℝ))) := by
have hfixed (D : Finset (Fin n)) :
(((2:ℝ)^12)⁻¹) / 4 * ((D : Set (Fin n)) ∩ Y).ncard -
(trueGroup M d s i).card / (κ:ℝ) ≤
bitsExpectation q univ (fun C => bitsExpectation q univ (fun T =>
fairParityExpectation (fun p =>
(listedSafeStatistic M hE κ (withMasks d D C T) s i Y (boolParity p) : ℝ)))) :=
listedSafeStatistic_fixed_density M hE κ hκ (withMasks d D ∅ ∅) s i Y hY hI q hq0 hq1
have hmean := bitsExpectation_mono (fun _ : Fin n => (1:ℝ)/4) (by intro e; norm_num)
(by intro e; norm_num) univ (fun D _ => hfixed D)
have hcard : bitsExpectation (fun _ : Fin n => (1:ℝ)/4) univ
(fun D => (((D : Set (Fin n)) ∩ Y).ncard : ℝ)) = Y.ncard / 4 := by
have hset (D : Finset (Fin n)) : (D : Set (Fin n)) ∩ Y =
((Y.toFinset ∩ D : Finset (Fin n)) : Set (Fin n)) := by
ext e
simp only [Set.mem_inter_iff, Finset.mem_coe, mem_inter, Set.mem_toFinset]
exact and_comm
simp only [hset, Set.ncard_coe_finset]
rw [bitsExpectation_card_inter (fun _ : Fin n => (1:ℝ)/4) univ Y.toFinset (subset_univ _)]
simp only [sum_const, nsmul_eq_mul, ← Set.ncard_eq_toFinset_card']
ring
have hnum : (((2:ℝ)^12)⁻¹) / 4 / 4 = retainedFraction := by
norm_num [retainedFraction]
have heq : bitsExpectation (fun _ : Fin n => (1:ℝ)/4) univ
(fun D => (((2:ℝ)^12)⁻¹) / 4 * ((D : Set (Fin n)) ∩ Y).ncard -
(trueGroup M d s i).card / (κ:ℝ)) =
retainedFraction * Y.ncard - (trueGroup M d s i).card / (κ:ℝ) := by
simp only [sub_eq_add_neg, bitsExpectation_add, bitsExpectation_mul_const,
bitsExpectation_const, hcard]
congr 1
calc
(((2:ℝ)^12)⁻¹) / 4 * (Y.ncard / 4) =
((((2:ℝ)^12)⁻¹) / 4 / 4) * Y.ncard := by ring
_ = retainedFraction * Y.ncard := by rw [hnum]
rw [heq] at hmean
exact hmean
end MatroidProphet.MainAlgorithm
end OAI
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