File size: 3,122 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 | import OAI.Probability.MatroidProphet.Maximum.Probability
import OAI.Probability.MatroidProphet.SeedLaw
import OAI.Probability.MatroidProphet.Simulation
namespace OAI
namespace MatroidProphet.Maximum
open MeasureTheory Finset Set
variable {n : ℕ} {K : Type*} [Countable K] [MeasurableSpace K]
[MeasurableSingletonClass K] [LinearOrder K]
lemma le_hiddenWorstReward {bits : ℕ} (A : HiddenRule n bits) (w : Weights n)
(r : Seed bits) (c : ℝ) (h : ∀ π, c ≤ hiddenReward A r w π) :
c ≤ hiddenWorstReward A w r := by
classical
apply Finset.le_inf'
intro π _
exact h π
theorem maximum_integral_top (M : Matroid (Fin n))
(key : ℝ → K) (hkey : Measurable key) (active : K → Prop)
(w : Weights n) (hw : ∀ e, 0 ≤ w e) (g : Fin n)
(hg : M.Indep ({g} : Set (Fin n))) (hag : active (key (w g)))
(htop : ∀ f, M.Indep ({f} : Set (Fin n)) → active (key (w f)) → f ≠ g →
higher (g, key (w g)) (f, key (w f))) :
w g / 4 ≤ ∫ r, hiddenWorstReward
(maximumHidden M key hkey active (seedSet (bits := n))) w r
∂bernoulliSeedLaw (fun _ => (1 / 2 : ℝ))
(fun _ => by norm_num) (fun _ => by norm_num) := by
classical
let A := maximumHidden M key hkey active (seedSet (bits := n))
let F : Finset (Fin n) → ℝ := fun H => hiddenWorstReward A w (setSeed H)
rw [integral_bernoulliSeedLaw]
change w g / 4 ≤ bitsExpectation (fun _ => (1 / 2 : ℝ)) Finset.univ F
have hF : ∀ H ⊆ Finset.univ, 0 ≤ F H :=
fun H _ => hiddenWorstReward_nonneg A w hw (setSeed H)
let P : Finset (Fin n) := Finset.univ.filter fun f =>
M.Indep ({f} : Set (Fin n)) ∧ active (key (w f)) ∧ f ≠ g
by_cases hP : P.Nonempty
· obtain ⟨h, hhP, hsecond⟩ := exists_top (fun f => key (w f)) P hP
have hh := (Finset.mem_filter.mp hhP).2
apply maximum_expectation_two Finset.univ g h (Finset.mem_univ _) (Finset.mem_univ _)
(Ne.symm hh.2.2) F (w g) hF
intro H _ hgH hhH
apply le_hiddenWorstReward
intro π
have hqual := threshold_top_two M active (seedSet (bits := n)) (setSeed H)
(fun f => key (w f)) g h hg hag hh.1 hh.2.1 htop
(fun f hf haf hfg hfh => hsecond f (Finset.mem_filter.mpr ⟨Finset.mem_univ _, hf, haf, hfg⟩) hfh)
(by simpa using hgH) (by simpa using hhH)
exact (maximumHidden_single_qualifier M key hkey active seedSet w (setSeed H) g
(by simpa using hgH) hqual π).symm.le
· apply maximum_expectation_one Finset.univ g (Finset.mem_univ _) F (w g) (hw g) hF
intro H _ hgH
apply le_hiddenWorstReward
intro π
have honly : ∀ f, M.Indep ({f} : Set (Fin n)) → active (key (w f)) → f = g := by
intro f hf haf
by_contra hfg
exact hP ⟨f, Finset.mem_filter.mpr ⟨Finset.mem_univ _, hf, haf, hfg⟩⟩
have hqual := threshold_only_positive M active (seedSet (bits := n)) (setSeed H)
(fun f => key (w f)) g hg hag honly (by simpa using hgH)
exact (maximumHidden_single_qualifier M key hkey active seedSet w (setSeed H) g
(by simpa using hgH) hqual π).symm.le
end MatroidProphet.Maximum
end OAI
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