| import OAI.Probability.MatroidSecretary.Accounting.ExpectedCost |
|
|
| namespace OAI |
|
|
| /-! |
| |
|
|
| 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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