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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 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 | import OAI.Probability.MatroidSecretary.Accounting.ExpectedCost
namespace OAI
/-!
# Retaining the density-mask rate in the accounting budget
In accounting.tex immediately after `eq:expected-cost`, the manuscript notes
that the exact density-mask contribution is `1 / (4 * κ)`, although the proof
uses the more generous `1 / κ`. This supplement proves that sharper inequality
for the actual zero-extended cost, without conditioning on random listing.
-/
namespace MatroidProphet.AccountingContracts
open Set Finset MainAlgorithm
variable {n : ℕ}
/-- Keep the sampled higher-group density count until after expectation. -/
theorem listedCost_le_sampled_true_counts (M : Matroid (Fin n)) (d : MainMasks n)
(w : Fin n → Option ℤ) {h : ℕ} {i : ℤ}
(hi : (groups M d w)[h]? = some i) :
listedCost M d w h ≤
(d.C ∩ trueGroup M d w i).card +
(d.C ∩ higherTrueGroups M d w i).card +
(d.D ∩ higherTrueGroups M d w i).card / densityThreshold := by
classical
let J := range ((groups M d w).length - (h + 1))
have count (mask : Finset (Fin n)) :
(∑ j ∈ J, (groupMask M d w mask (h + 1 + j)).ncard) ≤
(mask ∩ higherTrueGroups M d w i).card := by
apply sum_groupMask_card_le M d w J (fun j => h + 1 + j)
(by intro a _ b _ hab; change h + 1 + a = h + 1 + b at hab; omega)
intro j hj
exact groupMask_later_subset M d w mask hi
(by have := mem_range.mp hj; have := (List.getElem?_eq_some_iff.mp hi).1; omega)
(by omega)
have hc : (∑ j ∈ J, ((groupMask M d w d.C (h + 1 + j)).ncard : ℝ)) ≤
(d.C ∩ higherTrueGroups M d w i).card := by exact_mod_cast count d.C
have hd : (∑ j ∈ J, ((groupMask M d w d.D (h + 1 + j)).ncard : ℝ)) ≤
(d.D ∩ higherTrueGroups M d w i).card := by exact_mod_cast count d.D
have hf : (groupMask M d w d.C h).ncard = (d.C ∩ trueGroup M d w i).card := by
rw [Set.ncard_eq_toFinset_card', groupMask_eq_focal M d w d.C hi,
trueGroup_eq_candidate_filter]
have hdiv := div_le_div_of_nonneg_right hd constants_positive.2.1.le
unfold listedCost
rw [hf, sum_add_distrib, ← sum_div]
dsimp only [J] at hc hdiv
linarith
/-- The density mask is averaged at its original rate, independently of `H`. -/
lemma mainMean_density_count (d : MainMasks n) (K : Finset (Fin n)) :
mainMean d (fun d' => ((d'.D ∩ K).card : ℝ)) = (1 / 4 : ℝ) * K.card := by
unfold mainMean tripleMaskExpectation MainAlgorithm.withMasks
simp only [fairParityExpectation_const, bitsExpectation_const]
simp_rw [Finset.inter_comm _ K]
rw [bitsExpectation_card_inter _ univ K (subset_univ _)]
simp [mul_comm]
lemma mainMean_div (d : MainMasks n) (f : MainMasks n → ℝ) (c : ℝ) :
mainMean d (fun d' => f d' / c) = mainMean d f / c := by
simpa only [div_eq_mul_inv, mul_comm] using mainMean_mul d c⁻¹ f
/-- Listing is not a conditioning event: unlisted true groups have cost zero. -/
theorem trueCost_le_sampled_true_counts (M : Matroid (Fin n)) (d : MainMasks n)
(w : Fin n → Option ℤ) (i : ℤ) :
trueCost M d w i ≤
(d.C ∩ trueGroup M d w i).card +
(d.C ∩ higherTrueGroups M d w i).card +
(d.D ∩ higherTrueGroups M d w i).card / densityThreshold := by
classical
unfold trueCost
split_ifs with hi
· exact listedCost_le_sampled_true_counts M d w (groups_get?_idxOf M d w i hi)
· have hk : 0 < densityThreshold := constants_positive.2.1
positivity
/-- Source's sharper density contribution before relaxing `1/(4κ)` to `1/κ`. -/
theorem expected_cost_sharp (M : Matroid (Fin n)) (d : MainMasks n)
(w : Fin n → Option ℤ) (i : ℤ) :
mainMean d (fun d' => trueCost M d' w i) ≤
thinningRate * (trueGroup M d w i).card +
(thinningRate + 1 / (4 * densityThreshold)) * (higherTrueGroups M d w i).card := by
have hm : mainMean d (fun d' => trueCost M d' w i) ≤
mainMean d (fun d' => (d'.C ∩ trueGroup M d w i).card +
(d'.C ∩ higherTrueGroups M d w i).card +
(d'.D ∩ higherTrueGroups M d w i).card / densityThreshold) := by
apply mainMean_mono
intro d' hH
have hc := trueCost_le_sampled_true_counts M d' w i
rwa [trueGroup_eq_of_H_eq M d d' w i hH,
higherTrueGroups_eq_of_H_eq M d d' w i hH] at hc
simp only [mainMean_add, mainMean_div, mainMean_guard_count, mainMean_density_count] at hm
convert hm using 1
ring
/-- The equivalent finite sum over *all* higher true levels, listed or not. -/
theorem expected_cost_sharp_sum (M : Matroid (Fin n)) (d : MainMasks n)
(w : Fin n → Option ℤ) (i : ℤ) :
mainMean d (fun d' => trueCost M d' w i) ≤
thinningRate * (trueGroup M d w i).card +
(thinningRate + 1 / (4 * densityThreshold)) *
∑ j ∈ trueLevels M w d.H with i < j, ((trueGroup M d w j).card : ℝ) := by
rw [← higherTrueGroups_card_eq_sum M d w i]
exact expected_cost_sharp M d w i
end MatroidProphet.AccountingContracts
end OAI
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