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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 | import OAI.Probability.MatroidProphet.BranchMix
namespace OAI
namespace MatroidProphet
open MeasureTheory Finset
noncomputable def maximumRounded {n : ℕ} (M : Matroid (Fin n)) (w : Weights n) : ℝ := by
classical
exact Finset.univ.sup' Finset.univ_nonempty (fun o : Option (Fin n) =>
match o with
| none => 0
| some e => if M.Indep ({e} : Set (Fin n)) then roundedWeight weightBase (w e) else 0)
lemma maximumRounded_nonneg {n : ℕ} (M : Matroid (Fin n)) (w : Weights n) :
0 ≤ maximumRounded M w := by
classical
exact Finset.le_sup' (s := Finset.univ)
(f := fun o : Option (Fin n) => match o with
| none => 0
| some e => if M.Indep ({e} : Set (Fin n)) then roundedWeight weightBase (w e) else 0)
(Finset.mem_univ none)
lemma roundedWeight_le_maximumRounded {n : ℕ} (M : Matroid (Fin n)) (w : Weights n)
(e : Fin n) (he : M.Indep ({e} : Set (Fin n))) :
roundedWeight weightBase (w e) ≤ maximumRounded M w := by
classical
have h := Finset.le_sup' (s := Finset.univ)
(f := fun o : Option (Fin n) => match o with
| none => 0
| some e => if M.Indep ({e} : Set (Fin n)) then roundedWeight weightBase (w e) else 0)
(Finset.mem_univ (some e))
change (if M.Indep ({e} : Set (Fin n)) then roundedWeight weightBase (w e) else 0) ≤
maximumRounded M w at h
rwa [ite_eq_left he] at h
noncomputable def maximumPayoff {n : ℕ} (M : Matroid (Fin n)) (w : Weights n) : ℝ :=
∫ r, hiddenWorstReward (roundedMaximumHidden M (seedSet (bits := n))) w r
∂bernoulliSeedLaw (fun _ => (1 / 2 : ℝ)) (fun _ => by norm_num) (fun _ => by norm_num)
lemma maximumPayoff_nonneg {n : ℕ} (M : Matroid (Fin n)) (w : Weights n) (hw : ∀ e, 0 ≤ w e) :
0 ≤ maximumPayoff M w :=
integral_nonneg (fun r => hiddenWorstReward_nonneg _ w hw r)
lemma roundedWeight_le_four_maximumPayoff {n : ℕ} (M : Matroid (Fin n)) (w : Weights n)
(hw : ∀ e, 0 ≤ w e) (e : Fin n) (he : M.Indep ({e} : Set (Fin n))) :
roundedWeight weightBase (w e) ≤ 4 * maximumPayoff M w := by
classical
let a : Fin n → WithBot ℤ := fun f => Priority.asLevel (roundedLevel weightBase (w f))
by_cases hea : a e = ⊥
· have hu : roundedWeight weightBase (w e) = 0 := by
change levelWeight weightBase (a e) = 0
rw [hea]
rfl
rw [hu]
exact mul_nonneg (by norm_num) (maximumPayoff_nonneg M w hw)
· let P : Finset (Fin n) := Finset.univ.filter fun f => M.Indep ({f} : Set (Fin n)) ∧ a f ≠ ⊥
have heP : e ∈ P := Finset.mem_filter.mpr ⟨Finset.mem_univ e, he, hea⟩
obtain ⟨g, hgP, htop⟩ := Maximum.exists_top a P ⟨e, heP⟩
have hg := (Finset.mem_filter.mp hgP).2
have hwin : w g / 4 ≤ maximumPayoff M w :=
Maximum.maximum_integral_top M (fun x => Priority.asLevel (roundedLevel weightBase x))
((measurable_of_countable Priority.asLevel).comp (measurable_roundedLevel weightBase))
(fun x => x ≠ ⊥) w hw g hg.1 hg.2
(fun f hf haf hfg => htop f (Finset.mem_filter.mpr ⟨Finset.mem_univ f, hf, haf⟩) hfg)
have hkey : a e ≤ a g := by
by_cases heg : e = g
· exact heg ▸ le_rfl
· rcases htop e heP heg with hlt | ⟨heq, _⟩
· exact hlt.le
· exact heq.symm.le
have hu : roundedWeight weightBase (w e) ≤ roundedWeight weightBase (w g) :=
levelWeight_mono (by norm_num [weightBase]) hkey
have hraw := roundedWeight_le (by norm_num [weightBase] : 1 < weightBase) (hw g)
nlinarith
theorem maximum_branch_guarantee {n : ℕ} (M : Matroid (Fin n)) (w : Weights n)
(hw : ∀ e, 0 ≤ w e) : maximumRounded M w / 4 ≤ maximumPayoff M w := by
classical
have hnonneg := maximumPayoff_nonneg M w hw
have hfour : maximumRounded M w ≤ 4 * maximumPayoff M w := by
apply Finset.sup'_le
intro o _
cases o with
| none => exact mul_nonneg (by norm_num) hnonneg
| some e =>
change (if M.Indep ({e} : Set (Fin n)) then roundedWeight weightBase (w e) else 0) ≤ _
split_ifs with he
· exact roundedWeight_le_four_maximumPayoff M w hw e he
· exact mul_nonneg (by norm_num) hnonneg
nlinarith
end MatroidProphet
end OAI
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