File size: 3,446 Bytes
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 | import OAI.Probability.MatroidProphet.Maximum.Score
namespace OAI
namespace MatroidProphet.Maximum
open MeasureTheory Finset
lemma maximumHidden_mask_congr {n bits bits' : ℕ} {K : Type*} [Countable K]
[MeasurableSpace K] [MeasurableSingletonClass K] [LinearOrder K]
(M : Matroid (Fin n)) (key : ℝ → K) (hkey : Measurable key) (active : K → Prop)
(mask : Seed bits → Finset (Fin n)) (mask' : Seed bits' → Finset (Fin n))
(r : Seed bits) (r' : Seed bits') (hm : mask r = mask' r')
(w : Weights n) (π : ArrivalOrder n) (t : ℕ) :
hiddenAcceptedThrough (maximumHidden M key hkey active mask) r w π t =
hiddenAcceptedThrough (maximumHidden M key hkey active mask') r' w π t := by
classical
have hd (k : Fin n) :
decisionAt (maximumRule M key hkey active mask) r
(observed (maximumHidden M key hkey active mask) r w) w π k = true ↔
decisionAt (maximumRule M key hkey active mask') r'
(observed (maximumHidden M key hkey active mask') r' w) w π k = true := by
unfold maximumRule
rw [firstRule_decision, firstRule_decision]
simp only [threshold, observed, maximumHidden, hm]
rfl
ext e
change e ∈ acceptedThrough (maximumRule M key hkey active mask) r
(observed (maximumHidden M key hkey active mask) r w) w π t \ mask r ↔
e ∈ acceptedThrough (maximumRule M key hkey active mask') r'
(observed (maximumHidden M key hkey active mask') r' w) w π t \ mask' r'
rw [Finset.mem_sdiff, Finset.mem_sdiff]
constructor
· rintro ⟨he, hn⟩
obtain ⟨_, ht, ha⟩ := Finset.mem_filter.mp he
exact ⟨Finset.mem_filter.mpr ⟨Finset.mem_univ _, ht, (hd _).mp ha⟩,
by simpa only [hm] using hn⟩
· rintro ⟨he, hn⟩
obtain ⟨_, ht, ha⟩ := Finset.mem_filter.mp he
exact ⟨Finset.mem_filter.mpr ⟨Finset.mem_univ _, ht, (hd _).mpr ha⟩,
by simpa only [hm] using hn⟩
lemma maximumHidden_worst_mask_congr {n bits bits' : ℕ} {K : Type*} [Countable K]
[MeasurableSpace K] [MeasurableSingletonClass K] [LinearOrder K]
(M : Matroid (Fin n)) (key : ℝ → K) (hkey : Measurable key) (active : K → Prop)
(mask : Seed bits → Finset (Fin n)) (mask' : Seed bits' → Finset (Fin n))
(r : Seed bits) (r' : Seed bits') (hm : mask r = mask' r') (w : Weights n) :
hiddenWorstReward (maximumHidden M key hkey active mask) w r =
hiddenWorstReward (maximumHidden M key hkey active mask') w r' := by
classical
unfold hiddenWorstReward
congr 1
funext π
unfold hiddenReward
rw [maximumHidden_mask_congr M key hkey active mask mask' r r' hm]
end MatroidProphet.Maximum
namespace MatroidProphet
noncomputable def maximumForMask {n : ℕ} (M : Matroid (Fin n)) (w : Weights n)
(H : Finset (Fin n)) : ℝ :=
hiddenWorstReward (roundedMaximumHidden M (seedSet (bits := n))) w (setSeed H)
lemma roundedMaximum_worst_mask {n bits : ℕ} (M : Matroid (Fin n))
(mask : Seed bits → Finset (Fin n)) (r : Seed bits) (w : Weights n) :
hiddenWorstReward (roundedMaximumHidden M mask) w r = maximumForMask M w (mask r) := by
apply Maximum.maximumHidden_worst_mask_congr
exact (seedSet_setSeed _).symm
lemma maximumPayoff_eq_masks {n : ℕ} (M : Matroid (Fin n)) (w : Weights n) :
maximumPayoff M w = bitsExpectation (fun _ => (1/2 : ℝ)) Finset.univ (maximumForMask M w) := by
exact integral_bernoulliSeedLaw _ _ _ _
end MatroidProphet
end OAI
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