| import OAI.Probability.MatroidProphet.Main |
|
|
| namespace OAI |
|
|
| /-! The literal conditional cost inequality `eq:expected-cost`. The cost is |
| zero on an unlisted true group; listing is never conditioned upon. -/ |
|
|
| namespace MatroidProphet.AccountingContracts |
| open Set Finset MainAlgorithm |
|
|
| variable {n : ℕ} |
|
|
| /-- The single path-expansion budget of a listed group, with its real-valued |
| density contribution. -/ |
| noncomputable def listedCost (M : Matroid (Fin n)) (d : MainMasks n) |
| (w : Fin n → Option ℤ) (h : ℕ) : ℝ := |
| (groupMask M d w d.C h).ncard + |
| ∑ j ∈ range ((groups M d w).length - (h + 1)), |
| ((groupMask M d w d.C (h + 1 + j)).ncard + |
| (groupMask M d w d.D (h + 1 + j)).ncard / densityThreshold) |
|
|
| /-- The source's `c_i`, extended by zero to unlisted true groups. -/ |
| noncomputable def trueCost (M : Matroid (Fin n)) (d : MainMasks n) |
| (w : Fin n → Option ℤ) (i : ℤ) : ℝ := by |
| classical |
| exact if i ∈ groups M d w then listedCost M d w ((groups M d w).idxOf i) else 0 |
|
|
| theorem listedCost_le_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 + |
| (higherTrueGroups M d w i).card / densityThreshold := by |
| have hc := listed_cost_le_true_counts M d w hi (2^100) |
| rw [← trueGroup_eq_candidate_filter M d w i] at hc |
| have hr : densityThreshold * (groupMask M d w d.C h).ncard + |
| ∑ j ∈ range ((groups M d w).length - (h + 1)), |
| (densityThreshold * (groupMask M d w d.C (h + 1 + j)).ncard + |
| (groupMask M d w d.D (h + 1 + j)).ncard) ≤ |
| densityThreshold * (d.C ∩ trueGroup M d w i).card + |
| densityThreshold * (d.C ∩ higherTrueGroups M d w i).card + |
| (higherTrueGroups M d w i).card := by |
| simpa only [densityThreshold, higherTrueGroups, Nat.cast_add, Nat.cast_mul, |
| Nat.cast_sum, Nat.cast_pow, Nat.cast_ofNat] using |
| (Nat.cast_le.mpr hc : ( _ : ℝ) ≤ _) |
| have hk : 0 < densityThreshold := constants_positive.2.1 |
| apply (mul_le_mul_iff_right₀ hk).mp |
| have hs : densityThreshold * listedCost M d w h = |
| densityThreshold * (groupMask M d w d.C h).ncard + |
| ∑ j ∈ range ((groups M d w).length - (h + 1)), |
| (densityThreshold * (groupMask M d w d.C (h + 1 + j)).ncard + |
| (groupMask M d w d.D (h + 1 + j)).ncard) := by |
| unfold listedCost |
| rw [mul_add, mul_sum] |
| congr 1 |
| apply sum_congr rfl |
| intro j hj |
| rw [mul_add] |
| congr 1 |
| field_simp |
| rw [hs, mul_add, mul_add] |
| have he : densityThreshold * |
| ((higherTrueGroups M d w i).card / densityThreshold) = |
| (higherTrueGroups M d w i).card := by field_simp |
| rw [he] |
| exact hr |
|
|
| theorem trueCost_le_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 + |
| (higherTrueGroups M d w i).card / densityThreshold := by |
| classical |
| unfold trueCost |
| split_ifs with hi |
| · exact listedCost_le_true_counts M d w (groups_get?_idxOf M d w i hi) |
| · have hk : 0 < densityThreshold := constants_positive.2.1 |
| positivity |
|
|
| /-- Conditional on `H`, the actual randomly listed cost is dominated before |
| averaging. The higher-group cardinality is the sum over all higher true groups, |
| including those that the density mask did not list. -/ |
| theorem expected_cost (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 / 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 + |
| (higherTrueGroups M d w i).card / densityThreshold) := by |
| apply mainMean_mono |
| intro d' hH |
| have hc := trueCost_le_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_guard_count, mainMean_const] at hm |
| convert hm using 1 |
| ring |
|
|
| /-- Exact finite true-level sum presentation of `eq:expected-cost`. -/ |
| theorem expected_cost_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 / 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 M d w i |
|
|
| end MatroidProphet.AccountingContracts |
|
|
| end OAI |
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