| 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 |
|
|
