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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 | import OAI.Probability.MatroidProphet.Algorithm.Cost
import OAI.Probability.MatroidProphet.Algorithm.Mean
import OAI.Probability.MatroidProphet.GroupTransfer
namespace OAI
namespace MatroidProphet.MainAlgorithm
open Finset
variable {n : ℕ}
attribute [local instance] Classical.propDecidable
lemma trueGroup_eq_candidate_filter (M : Matroid (Fin n)) (d : MainMasks n)
(w : Fin n → Option ℤ) (i : ℤ) :
trueGroup M d w i = (candidateLabels M w d.H).filter (fun e => w e = some i) := by
unfold trueGroup candidateLabels candidateSet
ext e
simp only [mem_filter, mem_univ, true_and]
rfl
noncomputable def higherTrueGroups (M : Matroid (Fin n)) (d : MainMasks n)
(w : Fin n → Option ℤ) (i : ℤ) : Finset (Fin n) :=
(candidateLabels M w d.H).filter (fun e => ∃ k, w e = some k ∧ i < k)
lemma trueZ_eq_listedRankStatistic (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (w : Fin n → Option ℤ) (i : ℤ) :
trueZ M hE d w i = listedRankStatistic M hE (2^100) d w i (boolParity d.odd) := by
unfold trueZ listedRankStatistic
split_ifs with hi
· unfold listedZ
rw [groupMask_at_key M d w univ _ i (groups_get?_idxOf M d w i hi), univ_inter]
· rfl
lemma mainMean_trueZ (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (w : Fin n → Option ℤ) (i : ℤ) :
mainMean d (fun d' => (trueZ M hE d' w i : ℝ)) =
tripleMaskExpectation (fun _ => (1:ℝ)/4) (fun _ => thinningRate) (fun _ => thinningRate)
univ (fun D C T => fairParityExpectation (fun p =>
(listedRankStatistic M hE (2^100) (withMasks d D C T) w i (boolParity p) : ℝ))) := by
unfold mainMean
apply tripleMaskExpectation_congr
intro D hD C hC T hT
congr 1
funext p
dsimp only
rw [trueZ_eq_listedRankStatistic]
rfl
theorem mainMean_trueZ_lower (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (w : Fin n → Option ℤ) (i : ℤ)
(Y : Set (Fin n)) (hY : Y ⊆ (trueGroup M d w i : Set (Fin n))) (hI : M.Indep Y)
(hn : densityThreshold ≤ ((trueGroup M d w i).card : ℝ)) :
(retainedFraction * Y.ncard - ((2:ℝ)^23)⁻¹ * (trueGroup M d w i).card) / densityThreshold ≤
mainMean d (fun d' => (trueZ M hE d' w i : ℝ)) := by
rw [mainMean_trueZ]
exact listedRankStatistic_lower M hE d w i Y hY hI hn
lemma true_expected_nominal_to_final (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (w : Fin n → Option ℤ) (i : ℤ)
(t : ℝ) (ht0 : 0 ≤ t) (ht1 : t ≤ 1) :
densityThreshold * t * bitsExpectation (fun _ : Fin n => t) univ (fun T =>
(trueZ M hE {d with T := T} w i : ℝ)) ≤
densityThreshold * bitsExpectation (fun _ : Fin n => t) univ (fun T =>
(trueLambda M hE {d with T := T} w i : ℝ)) +
t * (densityThreshold * (d.C ∩ trueGroup M d w i).card +
densityThreshold * (d.C ∩ higherTrueGroups M d w i).card +
(higherTrueGroups M d w i).card) := by
by_cases hi : i ∈ groups M d w
· have hh : (groups M d w).idxOf i < (groups M d w).length := List.idxOf_lt_length_of_mem hi
have hl := listed_expected_nominal_to_final M hE d w _ hh t ht0 ht1
have hc := listed_cost_le_true_counts M d w (groups_get?_idxOf M d w i hi) (2^100)
rw [← trueGroup_eq_candidate_filter M d w i] at hc
have hc' : (((2^100) * (groupMask M d w d.C ((groups M d w).idxOf i)).ncard +
∑ j ∈ range ((groups M d w).length-((groups M d w).idxOf i+1)),
((2^100) * (groupMask M d w d.C ((groups M d w).idxOf i+1+j)).ncard +
(groupMask M d w d.D ((groups M d w).idxOf i+1+j)).ncard) : ℕ) : ℝ) ≤
densityThreshold * (d.C ∩ trueGroup M d w i).card +
densityThreshold * (d.C ∩ higherTrueGroups M d w i).card +
(higherTrueGroups M d w i).card := by
calc
_ ≤ (((2^100) * (d.C ∩ trueGroup M d w i).card +
(2^100) * (d.C ∩ higherTrueGroups M d w i).card +
(higherTrueGroups M d w i).card : ℕ) : ℝ) := by exact_mod_cast hc
_ = _ := by norm_num [densityThreshold]
have hz (T : Finset (Fin n)) : trueZ M hE {d with T := T} w i =
listedZ M hE {d with T := T} w ((groups M d w).idxOf i) := by
change (if i ∈ groups M d w then _ else 0) = _
rw [ite_eq_left hi]
rfl
have hLam (T : Finset (Fin n)) : trueLambda M hE {d with T := T} w i =
listedLambda M hE {d with T := T} w ((groups M d w).idxOf i) := by
change (if i ∈ groups M d w then _ else 0) = _
rw [ite_eq_left hi]
rfl
simp only [hz, hLam]
exact hl.trans (add_le_add le_rfl (mul_le_mul_of_nonneg_left hc' ht0))
· have hz (T : Finset (Fin n)) : trueZ M hE {d with T := T} w i = 0 := by
change (if i ∈ groups M d w then _ else 0) = _
rw [ite_eq_right hi]
have hLam (T : Finset (Fin n)) : trueLambda M hE {d with T := T} w i = 0 := by
change (if i ∈ groups M d w then _ else 0) = _
rw [ite_eq_right hi]
simp only [hz, hLam, Nat.cast_zero, bitsExpectation_const, mul_zero, zero_add]
apply mul_nonneg ht0
exact add_nonneg (add_nonneg
(mul_nonneg constants_positive.2.1.le (Nat.cast_nonneg _))
(mul_nonneg constants_positive.2.1.le (Nat.cast_nonneg _))) (Nat.cast_nonneg _)
lemma trueGroup_eq_of_H_eq (M : Matroid (Fin n)) (d d' : MainMasks n)
(w : Fin n → Option ℤ) (i : ℤ) (hH : d'.H = d.H) :
trueGroup M d' w i = trueGroup M d w i := by
rw [trueGroup_eq_candidate_filter, trueGroup_eq_candidate_filter, hH]
lemma higherTrueGroups_eq_of_H_eq (M : Matroid (Fin n)) (d d' : MainMasks n)
(w : Fin n → Option ℤ) (i : ℤ) (hH : d'.H = d.H) :
higherTrueGroups M d' w i = higherTrueGroups M d w i := by
simp only [higherTrueGroups, hH]
lemma mainMean_resample_test (d : MainMasks n) (f : MainMasks n → ℝ) :
mainMean d (fun d' => bitsExpectation (fun _ : Fin n => thinningRate) univ
(fun T => f {d' with T := T})) = mainMean d f := by
unfold mainMean tripleMaskExpectation
apply bitsExpectation_congr
intro D hD
apply bitsExpectation_congr
intro C hC
change bitsExpectation (fun _ : Fin n => thinningRate) univ (fun _ =>
fairParityExpectation (fun p => bitsExpectation (fun _ : Fin n => thinningRate) univ
(fun T => f {withMasks d D C T with odd := p}))) = _
rw [bitsExpectation_const, ← bitsExpectation_fairParity]
theorem mainMean_nominal_to_final (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (w : Fin n → Option ℤ) (i : ℤ) :
densityThreshold * thinningRate * mainMean d (fun d' => (trueZ M hE d' w i : ℝ)) ≤
densityThreshold * mainMean d (fun d' => (trueLambda M hE d' w i : ℝ)) +
thinningRate * (densityThreshold * thinningRate * (trueGroup M d w i).card +
densityThreshold * thinningRate * (higherTrueGroups M d w i).card +
(higherTrueGroups M d w i).card) := by
have hmean : mainMean d (fun d' => densityThreshold * thinningRate *
bitsExpectation (fun _ : Fin n => thinningRate) univ (fun T =>
(trueZ M hE {d' with T := T} w i : ℝ))) ≤
mainMean d (fun d' => densityThreshold *
bitsExpectation (fun _ : Fin n => thinningRate) univ (fun T =>
(trueLambda M hE {d' with T := T} w i : ℝ)) +
thinningRate * (densityThreshold * (d'.C ∩ trueGroup M d w i).card +
densityThreshold * (d'.C ∩ higherTrueGroups M d w i).card +
(higherTrueGroups M d w i).card)) := by
apply mainMean_mono
intro d' hH
have hp := true_expected_nominal_to_final M hE d' w i thinningRate
constants_positive.2.2.1.le (by norm_num [thinningRate])
rwa [trueGroup_eq_of_H_eq M d d' w i hH,
higherTrueGroups_eq_of_H_eq M d d' w i hH] at hp
simp only [mainMean_add, mainMean_mul, mainMean_const, mainMean_guard_count] at hmean
rw [mainMean_resample_test d (fun d' => (trueZ M hE d' w i : ℝ)),
mainMean_resample_test d (fun d' => (trueLambda M hE d' w i : ℝ))] at hmean
simpa only [mul_assoc] using hmean
theorem true_group_payoff (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (w : Fin n → Option ℤ) (i : ℤ)
(Y : Set (Fin n)) (hY : Y ⊆ (trueGroup M d w i : Set (Fin n))) (hI : M.Indep Y)
(hn : densityThreshold ≤ ((trueGroup M d w i).card : ℝ)) :
thinningRate / densityThreshold *
(retainedFraction * Y.ncard -
(((2:ℝ)^23)⁻¹ + thinningRate * densityThreshold) * (trueGroup M d w i).card -
(1 + thinningRate * densityThreshold) * (higherTrueGroups M d w i).card) ≤
mainMean d (fun d' => (trueLambda M hE d' w i : ℝ)) := by
have hZ := mainMean_trueZ_lower M hE d w i Y hY hI hn
have hcost := mainMean_nominal_to_final M hE d w i
have hk : 0 < densityThreshold := constants_positive.2.1
have ht : 0 ≤ thinningRate := constants_positive.2.2.1.le
have hmul := mul_le_mul_of_nonneg_left hZ (mul_nonneg hk.le ht)
have hcancel : densityThreshold * thinningRate *
((retainedFraction * Y.ncard - ((2:ℝ)^23)⁻¹ * (trueGroup M d w i).card) / densityThreshold) =
thinningRate * (retainedFraction * Y.ncard - ((2:ℝ)^23)⁻¹ * (trueGroup M d w i).card) := by
field_simp
rw [hcancel] at hmul
have hboth := hmul.trans hcost
rw [div_mul_eq_mul_div]
apply (div_le_iff₀ hk).mpr
nlinarith [hboth]
end MatroidProphet.MainAlgorithm
end OAI
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