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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 | import OAI.Probability.MatroidProphet.Algorithm.Statistics
import OAI.Probability.MatroidProphet.Algorithm.Mean
import OAI.Probability.MatroidProphet.WeightedAccounting
namespace OAI
namespace MatroidProphet
open Finset
lemma cumulative_option_weight_bound {α : Type*} [DecidableEq α]
(B : ℝ) (hB : 2 ≤ B) (A : Finset α) (level : α → Option ℤ) (analyzed : Finset ℤ) :
(∑ i ∈ analyzed, B ^ i * ((A.filter (fun e => ∃ j, level e = some j ∧ i ≤ j)).card : ℝ)) ≤
2 * ∑ e ∈ A, levelWeight B (level e) := by
classical
let P := A.filter (fun e => level e ≠ none)
have hfilter (i : ℤ) : P.filter (fun e => i ≤ (level e).getD 0) =
A.filter (fun e => ∃ j, level e = some j ∧ i ≤ j) := by
ext e
cases he : level e <;> simp [P, he]
have hsum : (∑ e ∈ P, B ^ (level e).getD 0) = ∑ e ∈ A, levelWeight B (level e) := by
unfold P
rw [sum_filter]
apply sum_congr rfl
intro e he
cases level e <;> simp [levelWeight]
have hc := cumulative_weight_bound B hB P (fun e => (level e).getD 0) analyzed
simpa only [hfilter, hsum] using hc
namespace MainAlgorithm
variable {n : ℕ}
noncomputable def mainWorstRounded (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (w : Fin n → Option ℤ) : ℝ := by
classical
exact univ.inf' univ_nonempty (fun π : ArrivalOrder n =>
∑ e ∈ selection M hE d w π, levelWeight weightBase (w e))
lemma mainWorstRounded_le (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (w : Fin n → Option ℤ) (π : ArrivalOrder n) :
mainWorstRounded M hE d w ≤ ∑ e ∈ selection M hE d w π, levelWeight weightBase (w e) := by
classical
exact inf'_le _ (mem_univ π)
lemma mainWorstRounded_nonneg (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (w : Fin n → Option ℤ) : 0 ≤ mainWorstRounded M hE d w := by
classical
apply le_inf'
intro π hπ
apply sum_nonneg
intro e he
cases w e with
| none => exact le_rfl
| some i => exact (zpow_pos (by norm_num [weightBase]) i).le
lemma weighted_trueLambda_le_selection (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (w : Fin n → Option ℤ) (analyzed : Finset ℤ) (π : ArrivalOrder n) :
(1/2 : ℝ) * ∑ i ∈ analyzed, weightBase ^ i * (trueLambda M hE d w i : ℝ) ≤
∑ e ∈ selection M hE d w π, levelWeight weightBase (w e) := by
have hc := cumulative_option_weight_bound weightBase (by norm_num [weightBase])
(selection M hE d w π) w analyzed
have hr : (∑ i ∈ analyzed, weightBase ^ i * (trueLambda M hE d w i : ℝ)) ≤
∑ i ∈ analyzed, weightBase ^ i *
(((selection M hE d w π).filter (fun e => ∃ j, w e = some j ∧ i ≤ j)).card : ℝ) := by
apply sum_le_sum
intro i hi
apply mul_le_mul_of_nonneg_left _ (zpow_pos (by norm_num [weightBase]) i).le
exact_mod_cast trueLambda_le_count M hE d w π i
linarith
lemma weighted_trueLambda_le_mainWorstRounded (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (w : Fin n → Option ℤ) (analyzed : Finset ℤ) :
(1/2 : ℝ) * ∑ i ∈ analyzed, weightBase ^ i * (trueLambda M hE d w i : ℝ) ≤
mainWorstRounded M hE d w := by
classical
apply le_inf'
intro π hπ
exact weighted_trueLambda_le_selection M hE d w analyzed π
lemma mainMean_sum {β : Type*} (d : MainMasks n) (I : Finset β) (f : β → MainMasks n → ℝ) :
mainMean d (fun d' => ∑ i ∈ I, f i d') = ∑ i ∈ I, mainMean d (f i) := by
classical
induction I using Finset.induction_on with
| empty => simp only [sum_empty, mainMean_const]
| @insert a I ha ih => simp only [sum_insert ha, mainMean_add, ih]
lemma weighted_mean_trueLambda_le_mainWorstRounded (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (w : Fin n → Option ℤ) (analyzed : Finset ℤ) :
(1/2 : ℝ) * ∑ i ∈ analyzed, weightBase ^ i * mainMean d (fun d' => (trueLambda M hE d' w i : ℝ)) ≤
mainMean d (fun d' => mainWorstRounded M hE d' w) := by
have hh := mainMean_mono d (fun d' _ => weighted_trueLambda_le_mainWorstRounded M hE d' w analyzed)
simpa only [mainMean_mul, mainMean_sum] using hh
lemma mainWorstRounded_le_hiddenWorstReward (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(r : Seed (mainSeedBits n)) (w : Weights n) (hw : ∀ e, 0 ≤ w e) :
mainWorstRounded M hE (mainMasks r) (fun e => roundedLevel weightBase (w e)) ≤
hiddenWorstReward (hidden M hE) w r := by
classical
apply le_inf'
intro π hπ
exact (mainWorstRounded_le M hE (mainMasks r) _ π).trans
(rounded_selection_le_hiddenReward M hE r w hw π)
end MainAlgorithm
end MatroidProphet
end OAI
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