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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 | import OAI.Probability.MatroidSecretary.Labels.LabeledTransport
import OAI.Probability.MatroidProphet.Main
namespace OAI
/-! The indexed endpoint transported to any finite labeled ground type. The
encoding is fixed from the labels alone, before distributions and randomness.
No representability, positive-rank, atomlessness, or per-label moment premise
is introduced. This is optional source-correspondence support. -/
namespace MatroidProphet.LabeledTransport
open MeasureTheory ProbabilityTheory
variable {α : Type*} [Fintype α] [MeasurableSpace α] [MeasurableSingletonClass α]
def pairedCoordinates {Ω : Type*} (S V : Ω → α → ℝ) (i : α × Bool) : Ω → ℝ :=
if i.2 then fun ω => S ω i.1 else fun ω => V ω i.1
/-- The main guarantee on arbitrary finite physical labels, with a measurable
history-only rule and the same all-prefix feasibility and arbitrary-law bound. -/
theorem one_sample.{u} (M : Matroid α) (hE : M.E = Set.univ) :
∃ (bits : ℕ) (ν : Measure (Seed bits)), IsProbabilityMeasure ν ∧
∃ (A : OnlineRule α (Fintype.card α) bits),
(∀ (r : Seed bits) (s v : α → ℝ) (π : ArrivalOrder α (Fintype.card α)) (t : ℕ),
(∀ a, 0 ≤ s a) → (∀ a, 0 ≤ v a) →
M.Indep (acceptedThrough A r s v π t : Set α)) ∧
∀ {Ω : Type u} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]
(S V : Ω → α → ℝ) (R : Ω → Seed bits),
Measurable S → Measurable V → Measurable R →
(∀ᵐ ω ∂μ, ∀ a, 0 ≤ S ω a) →
(∀ᵐ ω ∂μ, ∀ a, 0 ≤ V ω a) →
iIndepFun (pairedCoordinates S V) μ →
(∀ a, Measure.map (fun ω => S ω a) μ = Measure.map (fun ω => V ω a) μ) →
IndepFun (fun ω => (S ω, V ω)) R μ →
Measure.map R μ = ν →
Integrable (fun ω => optimum M (V ω)) μ →
∀ (π : Ω → ArrivalOrder α (Fintype.card α)), Measurable π →
Integrable (fun ω => reward A (R ω) (S ω) (V ω) (π ω)) μ ∧
((2 : ℝ) ^ 310)⁻¹ * (∫ ω, optimum M (V ω) ∂μ) ≤
∫ ω, reward A (R ω) (S ω) (V ω) (π ω) ∂μ := by
classical
let e : α ≃ Fin (Fintype.card α) := Fintype.equivFin α
obtain ⟨bits, ν, hν, A, hA, hperf⟩ :=
MatroidProphet.one_sample.{u} (Fintype.card α) (M.mapEquiv e) (mapEquiv_ground M hE e)
refine ⟨bits, ν, hν, decodeRule e A, ?_, ?_⟩
· intro r s v π t hs hv
exact decode_feasible M e A hA r s v π t hs hv
· intro Ω _ μ _ S V R hS hV hR hS0 hV0 hind hmatch hseed hlaw hint π hπ
let S' : Ω → Weights (Fintype.card α) := fun ω => S ω ∘ e.symm
let V' : Ω → Weights (Fintype.card α) := fun ω => V ω ∘ e.symm
have hS' : Measurable S' := by
apply Measurable.of_eval
intro i
exact (measurable_pi_apply (e.symm i)).comp hS
have hV' : Measurable V' := by
apply Measurable.of_eval
intro i
exact (measurable_pi_apply (e.symm i)).comp hV
have hS0' : ∀ᵐ ω ∂μ, ∀ i, 0 ≤ S' ω i :=
hS0.mono (fun _ h i => h (e.symm i))
have hV0' : ∀ᵐ ω ∂μ, ∀ i, 0 ≤ V' ω i :=
hV0.mono (fun _ h i => h (e.symm i))
have hind' : iIndepFun (MatroidProphet.pairedCoordinates S' V') μ := by
have h := hind.precomp (g := fun i : Fin (Fintype.card α) × Bool => (e.symm i.1, i.2))
(by
intro i j hij
apply Prod.ext
· exact e.symm.injective (congrArg Prod.fst hij)
· exact congrArg (fun p : α × Bool => p.2) hij)
exact h
have hmap : Measurable (fun x : (α → ℝ) × (α → ℝ) =>
(x.1 ∘ e.symm, x.2 ∘ e.symm)) := by
apply Measurable.prodMk
· apply Measurable.of_eval
intro i
exact (measurable_pi_apply (e.symm i)).comp measurable_fst
· apply Measurable.of_eval
intro i
exact (measurable_pi_apply (e.symm i)).comp measurable_snd
have hseed' : IndepFun (fun ω => (S' ω, V' ω)) R μ :=
hseed.comp hmap measurable_id
have hint' : Integrable (fun ω => MatroidProphet.optimum (M.mapEquiv e) (V' ω)) μ := by
simpa only [optimum_eq_indexed M e, V'] using hint
have hπ' : Measurable (fun ω => (π ω).trans e) :=
(measurable_of_finite (fun p : ArrivalOrder α (Fintype.card α) => p.trans e)).comp hπ
have h := hperf μ S' V' R hS' hV' hR hS0' hV0' hind'
(fun i => hmatch (e.symm i)) hseed' hlaw hint' (fun ω => (π ω).trans e) hπ'
simpa only [decode_reward, optimum_eq_indexed M e, S', V'] using h
end MatroidProphet.LabeledTransport
end OAI
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