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a2ee19a | 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 | import OAI.Probability.MatroidSecretary.Selection.AssignedSafe
namespace OAI
/-!
# The exact density-averaged safety loss
This retains the first, sharper inequality in `eq:averaged-safe-count`
(`sections/transfer.tex`, line 330). The density mask is averaged only after
the exact conditional loss has been bounded by `|D ∩ U_i| / κ`. The previous
endpoint route weakened this to `|U_i| / κ` before averaging, which suffices
there but does not state the displayed `|U_i| / (4κ)` estimate.
The statistic remains zero on unlisted outcomes. No conditioning on listing,
focal density bits, or an arrival order is introduced.
-/
namespace MatroidProphet.MainAlgorithm
open Finset
variable {n : ℕ}
/-- Literal first inequality of `eq:averaged-safe-count`, for a fixed true
group and any independent comparison subset of that group. -/
theorem listedSafeStatistic_averaged_exact
(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)))
(hI : M.Indep Y) (q : Fin n → ℝ)
(hq0 : ∀ e, 0 ≤ q e) (hq1 : ∀ e, q e ≤ 1) :
retainedFraction * Y.ncard - (trueGroup M d s i).card / (4 * (κ : ℝ)) ≤
tripleMaskExpectation (fun _ => (1 : ℝ) / 4) q q univ (fun D C T =>
fairParityExpectation (fun p =>
(listedSafeStatistic M hE κ (withMasks d D C T) s i Y
(boolParity p) : ℝ))) := by
classical
have hfixed (D : Finset (Fin n)) :
exitThreshold / 4 * ((D : Set (Fin n)) ∩ Y).ncard -
(D ∩ trueGroup M d s i).card / (κ : ℝ) ≤
bitsExpectation q univ (fun C => bitsExpectation q univ (fun T =>
fairParityExpectation (fun p =>
(listedSafeStatistic M hE κ (withMasks d D C T) s i Y
(boolParity p) : ℝ)))) :=
listedSafeStatistic_exact_density M hE κ hκ (withMasks d D ∅ ∅)
s i Y hY hI q hq0 hq1
have hmean := bitsExpectation_mono (fun _ : Fin n => (1 : ℝ) / 4)
(by intro e; norm_num) (by intro e; norm_num) univ (fun D _ => hfixed D)
have hcardY : bitsExpectation (fun _ : Fin n => (1 : ℝ) / 4) univ
(fun D => (((D : Set (Fin n)) ∩ Y).ncard : ℝ)) = Y.ncard / 4 := by
have hset (D : Finset (Fin n)) : (D : Set (Fin n)) ∩ Y =
((Y.toFinset ∩ D : Finset (Fin n)) : Set (Fin n)) := by
ext e
simp only [Set.mem_inter_iff, Finset.mem_coe, mem_inter, Set.mem_toFinset]
exact and_comm
simp only [hset, Set.ncard_coe_finset]
rw [bitsExpectation_card_inter (fun _ : Fin n => (1 : ℝ) / 4)
univ Y.toFinset (subset_univ _)]
simp only [sum_const, nsmul_eq_mul, ← Set.ncard_eq_toFinset_card']
ring
have hcardU : bitsExpectation (fun _ : Fin n => (1 : ℝ) / 4) univ
(fun D => ((D ∩ trueGroup M d s i).card : ℝ)) =
(trueGroup M d s i).card / 4 := by
simpa [Finset.inter_comm, div_eq_mul_inv] using
bitsExpectation_card_inter (fun _ : Fin n => (1 : ℝ) / 4)
univ (trueGroup M d s i) (subset_univ _)
have heq : bitsExpectation (fun _ : Fin n => (1 : ℝ) / 4) univ
(fun D => exitThreshold / 4 * ((D : Set (Fin n)) ∩ Y).ncard -
(D ∩ trueGroup M d s i).card / (κ : ℝ)) =
retainedFraction * Y.ncard - (trueGroup M d s i).card / (4 * (κ : ℝ)) := by
calc
_ = bitsExpectation (fun _ : Fin n => (1 : ℝ) / 4) univ
(fun D => exitThreshold / 4 * ((D : Set (Fin n)) ∩ Y).ncard +
(-(κ : ℝ)⁻¹) * (D ∩ trueGroup M d s i).card) := by
apply bitsExpectation_congr
intro D _
ring
_ = exitThreshold / 4 * (Y.ncard / 4) +
(-(κ : ℝ)⁻¹) * ((trueGroup M d s i).card / 4) := by
rw [bitsExpectation_add, bitsExpectation_mul_const,
bitsExpectation_mul_const, hcardY, hcardU]
_ = _ := by
norm_num [exitThreshold, retainedFraction]
ring
rw [heq] at hmean
exact hmean
/-- Source specialization for its global independent comparison set `Y`.
The intersection appearing here is exactly the source's `Y_i`; all statistics
are those of the actual density-dependent listed groups. -/
theorem averaged_safe_sampled_labels
(M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (s : Fin n → Option ℤ) (i : ℤ)
(Y : Set (Fin n)) (hI : M.Indep Y) :
retainedFraction * (Y ∩ (trueGroup M d s i : Set (Fin n))).ncard -
(trueGroup M d s i).card / (4 * densityThreshold) ≤
tripleMaskExpectation (fun _ => (1 : ℝ) / 4)
(fun _ => thinningRate) (fun _ => thinningRate) univ (fun D C T =>
fairParityExpectation (fun p =>
(listedSafeStatistic M hE (2 ^ 100) (withMasks d D C T) s i Y
(boolParity p) : ℝ))) := by
have hs := listedSafeStatistic_averaged_exact M hE (2 ^ 100) (by positivity)
d s i (Y ∩ (trueGroup M d s i : Set (Fin n))) Set.inter_subset_right
(hI.subset Set.inter_subset_left) (fun _ => thinningRate)
(fun _ => constants_positive.2.2.1.le)
(fun _ => by norm_num [thinningRate])
have heq (D C T : Finset (Fin n)) (p : Bool) :
listedSafeStatistic M hE (2 ^ 100) (withMasks d D C T) s i
(Y ∩ (trueGroup M d s i : Set (Fin n))) (boolParity p) =
listedSafeStatistic M hE (2 ^ 100) (withMasks d D C T) s i Y
(boolParity p) := by
simpa only [trueGroup_withMasks] using
listedSafeStatistic_inter_trueGroup M hE (2 ^ 100)
(withMasks d D C T) s i Y (boolParity p)
simpa only [Nat.cast_pow, Nat.cast_ofNat, densityThreshold, heq] using hs
end MatroidProphet.MainAlgorithm
end OAI
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