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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 | import OAI.Probability.MatroidProphet.Residual.Gap
import OAI.Probability.MatroidProphet.GroupConditioning
import OAI.Probability.MatroidProphet.SafeParity
namespace OAI
namespace MatroidProphet.MainAlgorithm
open Finset Pivots
variable {n : ℕ}
noncomputable def listedTransferGap (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) : ℝ :=
max ((listedSafeStatistic M hE κ d s i Y ε : ℝ) -
(κ : ℝ) * listedRankStatistic M hE κ d s i ε) 0
lemma trueGroup_nonloop (M : Matroid (Fin n)) (d : MainMasks n)
(s : Fin n → Option ℤ) (i : ℤ) :
∀ e ∈ trueGroup M d s i, e ∉ M.closure ∅ := by
intro e he hl
have hc := (mem_trueGroup M d s i e).mp he
exact hc.1.2.2 (M.closure_mono (Set.empty_subset _) hl)
noncomputable def focalResidualFamily (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(κ : ℕ) (d : MainMasks n) (s : Fin n → Option ℤ) (i : ℤ) (ε : Fin 2) :
Finset (Finset (Fin n)) :=
let r := focalReference M d s i
labeledResidualFamily M (trueGroup M d s i) κ
(guardedPath M hE κ (groupMask M r s r.D) (groupMask M r s r.C) (focalIndex M d s i))
(activation (focalIndex M d s i)) ε
(fun b => lowerCompetition M hE κ (groupMask M r s r.D) (groupMask M r s r.C)
(groupMask M r s r.T) b.val.val (focalIndex M d s i))
lemma listedTransferGap_focal_le (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)))
(ε : Fin 2) (A B C : Finset (Fin n))
(hA : A ⊆ trueGroup M d s i) (hB : B ⊆ trueGroup M d s i)
(hC : C ⊆ trueGroup M d s i) :
listedTransferGap M hE κ (focalMasks d (trueGroup M d s i) A B C) s i Y ε ≤
residualDelta * (trueGroup M d s i).card + ((trueGroup M d s i).card : ℝ) *
residualBadIndicator (focalResidualFamily M hE κ d s i ε)
(residualDelta * (trueGroup M d s i).card) (A ∪ B) := by
classical
by_cases hne : A.Nonempty
· let f := focalMasks d (trueGroup M d s i) A B C
have hg : groupMask M f s f.D (focalIndex M d s i) = (A : Set (Fin n)) :=
focal_group_mask M d s i A B C d.D A hA hne hA
have hDU : groupMask M f s f.D (focalIndex M d s i) ⊆ (trueGroup M d s i : Set (Fin n)) := by
rw [hg]
exact hA
have hI : (f.D : Set (Fin n)) ∩ Y ⊆ groupMask M f s f.D (focalIndex M d s i) := by
rw [hg, focal_density_inter M d s i A B C Y hY]
exact Set.inter_subset_left
have hOS : ((f.H ∪ f.D ∪ f.C : Finset (Fin n)) : Set (Fin n)) ∩
(trueGroup M d s i : Set (Fin n)) ⊆ ((A ∪ B : Finset (Fin n)) : Set (Fin n)) := by
have he := congrArg (fun X : Finset (Fin n) => (X : Set (Fin n)))
(focal_observed_inter M d s i A B C hA hB)
simpa only [coe_inter] using he.subset
have h := nominalTransferGap_le_bad M hE κ hκ (groupMask M f s f.D)
(groupMask M f s f.C) (groupMask M f s f.T) (focalIndex M d s i)
(trueGroup M d s i) (A ∪ B) ((f.H ∪ f.D ∪ f.C : Finset (Fin n)) : Set (Fin n))
((f.D : Set (Fin n)) ∩ Y) hDU hI (union_subset hA hB) hOS
(trueGroup_nonloop M d s i) ε
have hF := funext (focal_incoming_path M hE κ d s i A B C hA hB hC hne)
have hK := funext (fun b : ParityWindow (activation (focalIndex M d s i)) ε =>
focal_lowerCompetition M hE κ d s i A B C hA hB hC hne b.val.val)
rw [hF, hK] at h
simpa only [listedTransferGap, listedSafeStatistic_focal_nonempty M hE κ d s i A B C Y ε hA hne,
listedRankStatistic_focal_nonempty M hE κ d s i A B C ε hA hne,
nominalTransferGap, focalResidualFamily] using h
· have hAe : A = ∅ := not_nonempty_iff_eq_empty.mp hne
subst A
rw [listedTransferGap, listedSafeStatistic_focal_empty, listedRankStatistic_focal_empty]
simp only [Nat.cast_zero, mul_zero, sub_zero, max_self]
apply add_nonneg (mul_nonneg residualDelta_nonneg (Nat.cast_nonneg _))
apply mul_nonneg (Nat.cast_nonneg _)
unfold residualBadIndicator
split_ifs <;> norm_num
lemma focalTransferGap_expectation_le (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (s : Fin n → Option ℤ) (i : ℤ) (Y : Set (Fin n))
(hY : Y ⊆ (trueGroup M d s i : Set (Fin n)))
(hn : densityThreshold ≤ ((trueGroup M d s i).card : ℝ)) (ε : Fin 2) :
tripleMaskExpectation (fun _ => (1:ℝ)/4) (fun _ => thinningRate) (fun _ => thinningRate)
(trueGroup M d s i) (fun A B C =>
listedTransferGap M hE (2^100) (focalMasks d (trueGroup M d s i) A B C) s i Y ε) ≤
2 * residualDelta * (trueGroup M d s i).card := by
have ht0 : 0 ≤ thinningRate := constants_positive.2.2.1.le
have ht1 : thinningRate ≤ 1 := by norm_num [thinningRate]
have hu : (1:ℝ)/4 + (1-(1:ℝ)/4)*thinningRate ≤ 1/2 := by norm_num [thinningRate]
have hne : (trueGroup M d s i).Nonempty := by
apply card_pos.mp
have hp := lt_of_lt_of_le constants_positive.2.1 hn
exact_mod_cast hp
apply tripleMask_residual_error_le (fun _ => (1:ℝ)/4) (fun _ => thinningRate)
(fun _ => thinningRate) (fun _ => by norm_num) (fun _ => by norm_num)
(fun _ => ht0) (fun _ => ht1) (fun _ => ht0) (fun _ => ht1) (fun _ => hu)
(trueGroup M d s i) (focalResidualFamily M hE (2^100) d s i ε)
· exact labeledResidualFamily_subsets M _ _ _ _ ε _
· have hc := labeledResidualFamily_card M hE (trueGroup M d s i) hne (2^100)
(by norm_num) (guardedPath M hE (2^100)
(groupMask M (focalReference M d s i) s (focalReference M d s i).D)
(groupMask M (focalReference M d s i) s (focalReference M d s i).C) (focalIndex M d s i))
(activation (focalIndex M d s i)) ε
(fun b => lowerCompetition M hE (2^100)
(groupMask M (focalReference M d s i) s (focalReference M d s i).D)
(groupMask M (focalReference M d s i) s (focalReference M d s i).C)
(groupMask M (focalReference M d s i) s (focalReference M d s i).T) b.val.val (focalIndex M d s i))
simpa only [Nat.cast_pow, Nat.cast_ofNat, densityThreshold, focalResidualFamily] using hc
· exact hn
· intro A hA B hB C hC
exact listedTransferGap_focal_le M hE (2^100) (by positivity) d s i Y hY ε A B C hA hB hC
end MatroidProphet.MainAlgorithm
end OAI
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