| 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 |
|
|
