| import OAI.Probability.MatroidSecretary.Labels.RelabelingProbability |
| import OAI.Probability.MatroidProphet.Main |
|
|
| namespace OAI |
|
|
| /-! An unconditional prophet witness on arbitrary finite labels, reusing the |
| canonical relabeling implementation. This supplies the existential endpoint |
| rather than requiring a hidden-rule performance hypothesis from the caller. -/ |
|
|
| namespace MatroidProphet.Relabeling |
|
|
| open MeasureTheory ProbabilityTheory |
|
|
| variable {α : Type*} [Fintype α] [MeasurableSpace α] [MeasurableSingletonClass α] |
| {n : ℕ} [MeasurableSpace (LabeledOrder α n)] [MeasurableSingletonClass (LabeledOrder α n)] |
|
|
| /-- A distribution-independent measurable one-sample rule on the original |
| labels, with exact prefix feasibility and the full almighty-adversary bound. |
| The enumeration is fixed before all probability laws and realizations. -/ |
| theorem exists_labeled_one_sample.{u} (M : Matroid α) (hE : M.E = Set.univ) |
| (e : α ≃ Fin n) : |
| ∃ (bits : ℕ) (ν : Measure (Seed bits)), IsProbabilityMeasure ν ∧ |
| ∃ (A : OnlineRule n bits), |
| (∀ k : Fin n, Measurable (fun x : Seed bits × ((α → ℝ) × LabeledHistory α k) => |
| transportedDecision e A k x.1 x.2.1 x.2.2)) ∧ |
| (∀ (r : Seed bits) (s v : α → ℝ) (π : LabeledOrder α n) (t : ℕ), |
| (∀ a, 0 ≤ s a) → (∀ a, 0 ≤ v a) → |
| M.Indep (transportedAcceptedThrough e 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 (labeledPairedCoordinates S V) μ → |
| (∀ a, μ.map (fun ω => S ω a) = μ.map (fun ω => V ω a)) → |
| IndepFun (fun ω => (S ω, V ω)) R μ → μ.map R = ν → |
| Integrable (fun ω => finiteOptimum M (V ω)) μ → |
| ∀ (π : Ω → LabeledOrder α n), Measurable π → |
| Integrable (fun ω => transportedReward e A (R ω) (S ω) (V ω) (π ω)) μ ∧ |
| ((2 : ℝ) ^ 310)⁻¹ * (∫ ω, finiteOptimum M (V ω) ∂μ) ≤ |
| ∫ ω, transportedReward e A (R ω) (S ω) (V ω) (π ω) ∂μ := by |
| obtain ⟨bits, ν, hν, A, hA, hbound⟩ := |
| MatroidProphet.one_sample.{u} n (M.mapEquiv e) (mapEquiv_ground_univ M e hE) |
| refine ⟨bits, ν, hν, A, measurable_transportedDecision e A, ?_, ?_⟩ |
| · exact transported_feasible M e A hA |
| · intro Ω _ μ _ S V R hS hV hR hS0 hV0 hind hmatch hseed hlaw hint π hπ |
| have ho (w : α → ℝ) : |
| optimum (M.mapEquiv e) (encodeWeights e w) = finiteOptimum M w := |
| finiteOptimum_mapEquiv M e w |
| have hS0' : ∀ᵐ ω ∂μ, ∀ i, 0 ≤ encodeWeights e (S ω) i := |
| hS0.mono (fun _ h => (encodeWeights_nonnegative_iff e _).2 h) |
| have hV0' : ∀ᵐ ω ∂μ, ∀ i, 0 ≤ encodeWeights e (V ω) i := |
| hV0.mono (fun _ h => (encodeWeights_nonnegative_iff e _).2 h) |
| have hint' : Integrable (fun ω => optimum (M.mapEquiv e) (encodeWeights e (V ω))) μ := by |
| simpa only [ho] using hint |
| have hb := hbound μ (fun ω => encodeWeights e (S ω)) |
| (fun ω => encodeWeights e (V ω)) R |
| ((measurable_encodeWeights e).comp hS) ((measurable_encodeWeights e).comp hV) hR |
| hS0' hV0' (paired_independent_encode e μ S V hind) |
| (matching_laws_encode e μ S V hmatch) (seed_independent_encode e μ S V R hseed) |
| hlaw hint' (fun ω => (π ω).trans e) (measurable_encoded_order e π hπ) |
| simpa only [ho, transportedReward_eq_reward] using hb |
| |
| end MatroidProphet.Relabeling |
| |
| end OAI |
| |
