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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 | import OAI.Probability.MatroidProphet.Algorithm.Worst
import OAI.Probability.MatroidProphet.GroupPayoff
import OAI.Probability.MatroidProphet.AnalyzedMass
namespace OAI
namespace MatroidProphet.MainAlgorithm
open Finset
variable {n : ℕ}
lemma higherTrueGroups_card_eq_sum (M : Matroid (Fin n)) (d : MainMasks n)
(w : Fin n → Option ℤ) (i : ℤ) :
((higherTrueGroups M d w i).card : ℝ) =
∑ j ∈ trueLevels M w d.H with i < j, ((trueGroup M d w j).card : ℝ) := by
simp only [trueGroup_eq_candidate_filter, higherTrueGroups]
exact highGroup_card w (candidateLabels M w d.H) (trueLevels M w d.H) Subset.rfl i
theorem conditional_main_payoff (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (w : Fin n → Option ℤ) :
thinningRate / (2 * densityThreshold) *
(retainedFraction * analyzedComparisonMass M w d.H - ((2:ℝ)^21)⁻¹ *
∑ e ∈ candidateLabels M w d.H, levelWeight weightBase (w e)) ≤
mainMean d (fun d' => mainWorstRounded M hE d' w) := by
classical
let J := positiveGreedy M w \ d.H
let Y : ℤ → Finset (Fin n) := fun i => J.filter (fun e => w e = some i)
have hY (i : ℤ) : (Y i : Set (Fin n)) ⊆ (trueGroup M d w i : Set (Fin n)) := by
rw [trueGroup_eq_candidate_filter]
intro e he
obtain ⟨heJ, hew⟩ := mem_filter.mp he
exact mem_filter.mpr ⟨positiveGreedy_survivors_subset M w d.H heJ, hew⟩
have hI (i : ℤ) : M.Indep (Y i : Set (Fin n)) :=
(positiveGreedy_survivors_indep M hE w d.H).subset (filter_subset _ _)
have hgroup (i : ℤ) (hi : i ∈ analyzedLevels M w d.H) :
thinningRate / densityThreshold *
(retainedFraction * (Y i).card -
(((2:ℝ)^23)⁻¹ + thinningRate * densityThreshold) * (trueGroup M d w i).card -
(1 + thinningRate * densityThreshold) *
∑ l ∈ trueLevels M w d.H with i < l, ((trueGroup M d w l).card : ℝ)) ≤
mainMean d (fun d' => (trueLambda M hE d' w i : ℝ)) := by
have hn : densityThreshold ≤ ((trueGroup M d w i).card : ℝ) := by
rw [trueGroup_eq_candidate_filter]
exact (mem_filter.mp hi).2
have hp := true_group_payoff M hE d w i (Y i : Set (Fin n)) (hY i) (hI i) hn
simpa only [Set.ncard_coe_finset, higherTrueGroups_card_eq_sum] using hp
have hpaid := source_weighted_group_payoff (trueLevels M w d.H) (analyzedLevels M w d.H)
(filter_subset _ _) (fun i => ((trueGroup M d w i).card : ℝ))
(fun i => ((Y i).card : ℝ)) (fun i => mainMean d (fun d' => (trueLambda M hE d' w i : ℝ)))
(fun i hi => Nat.cast_nonneg _) hgroup
have hmass : (∑ i ∈ trueLevels M w d.H, weightBase ^ i * ((trueGroup M d w i).card : ℝ)) =
∑ e ∈ candidateLabels M w d.H, levelWeight weightBase (w e) := by
simp only [trueGroup_eq_candidate_filter]
exact sum_weightGroups_mass w (candidateLabels M w d.H) (trueLevels M w d.H) Subset.rfl weightBase
rw [hmass] at hpaid
exact hpaid.trans (weighted_mean_trueLambda_le_mainWorstRounded M hE d w (analyzedLevels M w d.H))
end MatroidProphet.MainAlgorithm
end OAI
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