| import OAI.Probability.MatroidProphet.WeightGroups |
| import OAI.Probability.MatroidProphet.WeightedPayoff |
| import OAI.Probability.MatroidProphet.Maximum.Score |
|
|
| namespace OAI |
|
|
| namespace MatroidProphet.MainAlgorithm |
|
|
| open Finset |
|
|
| variable {n : ℕ} |
|
|
| noncomputable def trueLevels (M : Matroid (Fin n)) (a : Fin n → Option ℤ) |
| (H : Finset (Fin n)) : Finset ℤ := weightLevels a (candidateLabels M a H) |
|
|
| noncomputable def analyzedLevels (M : Matroid (Fin n)) (a : Fin n → Option ℤ) |
| (H : Finset (Fin n)) : Finset ℤ := by |
| classical |
| exact (trueLevels M a H).filter fun i => |
| densityThreshold ≤ ((candidateLabels M a H).filter (fun e => a e = some i)).card |
|
|
| noncomputable def comparisonMass (M : Matroid (Fin n)) (a : Fin n → Option ℤ) |
| (H : Finset (Fin n)) : ℝ := |
| ∑ e ∈ positiveGreedy M a \ H, levelWeight weightBase (a e) |
|
|
| noncomputable def analyzedComparisonMass (M : Matroid (Fin n)) (a : Fin n → Option ℤ) |
| (H : Finset (Fin n)) : ℝ := by |
| classical |
| exact ∑ i ∈ analyzedLevels M a H, weightBase ^ i * |
| (((positiveGreedy M a \ H).filter (fun e => a e = some i)).card : ℝ) |
|
|
| lemma candidate_level_le_maximumRounded (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (w : Weights n) (H : Finset (Fin n)) (i : ℤ) |
| (hi : i ∈ trueLevels M (fun e => roundedLevel weightBase (w e)) H) : |
| weightBase ^ i ≤ maximumRounded M w := by |
| classical |
| obtain ⟨e, he, hei⟩ := (mem_weightLevels _ _ i).1 hi |
| have hc : candidate M ⟨H, ∅, ∅, ∅, false⟩ (fun e => roundedLevel weightBase (w e)) e |
| (roundedLevel weightBase (w e)) := (Finset.mem_filter.mp he).2 |
| have heI := candidate_singleton_indep M hE _ _ e _ hc |
| have h := roundedWeight_le_maximumRounded M w e heI |
| simpa only [roundedWeight, hei, levelWeight] using h |
|
|
| lemma comparisonMass_partition (M : Matroid (Fin n)) (a : Fin n → Option ℤ) |
| (H : Finset (Fin n)) : |
| comparisonMass M a H = |
| ∑ i ∈ trueLevels M a H, weightBase ^ i * |
| (((positiveGreedy M a \ H).filter (fun e => a e = some i)).card : ℝ) := by |
| exact (sum_weightGroups_mass a _ _ |
| (weightLevels_mono a (positiveGreedy_survivors_subset M a H)) weightBase).symm |
|
|
| lemma small_comparison_mass (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (w : Weights n) (H : Finset (Fin n)) : |
| comparisonMass M (fun e => roundedLevel weightBase (w e)) H ≤ |
| analyzedComparisonMass M (fun e => roundedLevel weightBase (w e)) H + |
| 2 * densityThreshold * maximumRounded M w := by |
| classical |
| let a : Fin n → Option ℤ := fun e => roundedLevel weightBase (w e) |
| let Y := positiveGreedy M a \ H |
| let U := candidateLabels M a H |
| let L := trueLevels M a H |
| let A := analyzedLevels M a H |
| have hAU : A ⊆ L := Finset.filter_subset _ _ |
| have hsmall : ∀ i ∈ L \ A, ((Y.filter (fun e => a e = some i)).card : ℝ) ≤ densityThreshold := by |
| intro i hi |
| obtain ⟨hiL, hiA⟩ := Finset.mem_sdiff.mp hi |
| have hn : ¬densityThreshold ≤ ((U.filter (fun e => a e = some i)).card : ℝ) := by |
| intro hh |
| exact hiA (Finset.mem_filter.mpr ⟨hiL, hh⟩) |
| have hc : (Y.filter (fun e => a e = some i)).card ≤ (U.filter (fun e => a e = some i)).card := |
| Finset.card_le_card (Finset.filter_subset_filter _ (positiveGreedy_survivors_subset M a H)) |
| have hcR : ((Y.filter (fun e => a e = some i)).card : ℝ) ≤ |
| ((U.filter (fun e => a e = some i)).card : ℝ) := by exact_mod_cast hc |
| exact hcR.trans (le_of_lt (lt_of_not_ge hn)) |
| have hbound := small_group_mass_bound weightBase densityThreshold (maximumRounded M w) |
| (by norm_num [weightBase]) constants_positive.2.1.le (maximumRounded_nonneg M w) |
| (L \ A) (fun i => ((Y.filter (fun e => a e = some i)).card : ℝ)) hsmall |
| (fun i hi => candidate_level_le_maximumRounded M hE w H i (Finset.mem_sdiff.mp hi).1) |
| have hpartition := Finset.sum_sdiff hAU |
| (f := fun i => weightBase ^ i * ((Y.filter (fun e => a e = some i)).card : ℝ)) |
| rw [comparisonMass_partition] |
| change (∑ i ∈ L, weightBase ^ i * ((Y.filter (fun e => a e = some i)).card : ℝ)) ≤ |
| (∑ i ∈ A, weightBase ^ i * ((Y.filter (fun e => a e = some i)).card : ℝ)) + _ |
| linarith |
|
|
| end MatroidProphet.MainAlgorithm |
|
|
| end OAI |
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|
