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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 | import OAI.Probability.MatroidProphet.Algorithm.Observed
namespace OAI
namespace MatroidProphet.MainAlgorithm
open Set Finset
variable {n : ℕ}
noncomputable def selection (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (w : Fin n → Option ℤ) (π : ArrivalOrder n) : Finset (Fin n) :=
layerGreedy M (assignedEligible (assign M hE d w) w)
(fun b => finalPath M hE d w (b-2)) (assignedLayer (assign M hE d w) w)
(fun e => (π.symm e).val)
lemma selection_not_observed (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(d : MainMasks n) (w : Fin n → Option ℤ) (π : ArrivalOrder n)
(e : Fin n) (he : e ∈ selection M hE d w π) : e ∉ d.H ∪ d.D ∪ d.C := by
have hel := layerGreedy_subset M _ _ _ _ he
have ha := assigned_some_of_eligible (assign M hE d w) w e hel
obtain ⟨i, _, hei, _⟩ := (assign_spec M hE d w e _ _).mp ha
exact hei.1
theorem hiddenAccepted_eq_selection (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(r : Seed (mainSeedBits n)) (w : Weights n) (π : ArrivalOrder n) :
hiddenAcceptedThrough (hidden M hE) r w π n =
selection M hE (mainMasks r) (fun e => roundedLevel weightBase (w e)) π := by
classical
have hA : accepted (core M hE) r (observed (hidden M hE) r w) w π =
selection M hE (mainMasks r) (fun e => roundedLevel weightBase (w e)) π := by
unfold core
rw [layerOnlineRule_accepted]
obtain ⟨hf, _, ha⟩ := observed_preprocessing M hE r w
rw [hf, ha]
rfl
change accepted (core M hE) r (observed (hidden M hE) r w) w π \ (hidden M hE).mask r = _
rw [hA]
ext e
simp only [Finset.mem_sdiff]
constructor
· exact fun he => he.1
· intro he
exact ⟨he, selection_not_observed M hE (mainMasks r) _ π e he⟩
theorem hiddenReward_eq_selection (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(r : Seed (mainSeedBits n)) (w : Weights n) (π : ArrivalOrder n) :
hiddenReward (hidden M hE) r w π =
∑ e ∈ selection M hE (mainMasks r) (fun e => roundedLevel weightBase (w e)) π, w e := by
unfold hiddenReward
rw [hiddenAccepted_eq_selection]
theorem rounded_selection_le_hiddenReward (M : Matroid (Fin n)) (hE : M.E = Set.univ)
(r : Seed (mainSeedBits n)) (w : Weights n) (hw : ∀ e, 0 ≤ w e) (π : ArrivalOrder n) :
(∑ e ∈ selection M hE (mainMasks r) (fun e => roundedLevel weightBase (w e)) π,
roundedWeight weightBase (w e)) ≤ hiddenReward (hidden M hE) r w π := by
rw [hiddenReward_eq_selection]
apply Finset.sum_le_sum
intro e he
exact roundedWeight_le (by norm_num [weightBase]) (hw e)
end MatroidProphet.MainAlgorithm
end OAI
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