| import OAI.Probability.MatroidSecretary.Labels.FiniteLabelTransport |
| import OAI.Probability.MatroidProphet.Main |
|
|
| namespace OAI |
|
|
| /-! |
| # The one-sample endpoint on arbitrary finite labels |
|
|
| This is supporting transport for `thm:main`, not a new manuscript target. |
| The enumeration is fixed from the label type before the input distributions. |
| All labels, including loops, and all probability/online-information hypotheses |
| are retained. The probability-space universe is independent of the label |
| universe. No moment assumption on individual loop values is introduced. |
| -/ |
|
|
| namespace MatroidProphet.Labeling |
|
|
| open MeasureTheory ProbabilityTheory |
|
|
| variable {E : Type*} [Fintype E] |
|
|
| lemma measurable_finWeights : Measurable (finWeights (E := E)) := by |
| apply Measurable.of_eval |
| intro i |
| exact measurable_pi_apply ((Fintype.equivFin E).symm i) |
|
|
| lemma paired_independent_finWeights {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) (S V : Ω → E → ℝ) |
| (hi : iIndepFun (fun i : E × Bool => |
| if i.2 then fun ω => S ω i.1 else fun ω => V ω i.1) μ) : |
| iIndepFun (pairedCoordinates (fun ω => finWeights (S ω)) |
| (fun ω => finWeights (V ω))) μ := by |
| have h := hi.precomp (Equiv.prodCongr |
| (Fintype.equivFin E).symm (Equiv.refl Bool)).injective |
| change iIndepFun (fun i : Fin (Fintype.card E) × Bool => |
| if i.2 then fun ω => S ω ((Fintype.equivFin E).symm i.1) |
| else fun ω => V ω ((Fintype.equivFin E).symm i.1)) μ |
| simpa using h |
|
|
| /-- The complete distribution-independent prophet guarantee after a fixed |
| renaming of arbitrary finite labels. The order may depend on the full seed, |
| all samples, all values, and any other randomness in the probability space. -/ |
| theorem one_sample_finite_labels.{u, v} {E : Type v} [Fintype E] |
| [MeasurableSpace E] [MeasurableSingletonClass E] |
| (M : Matroid E) (hE : M.E = Set.univ) : |
| ∃ (bits : ℕ) (ν : Measure (Seed bits)), IsProbabilityMeasure ν ∧ |
| ∃ A : LabeledOnlineRule E bits, |
| (∀ (r : Seed bits) (s v : E → ℝ) (π : LabeledOrder E) (t : ℕ), |
| (∀ e, 0 ≤ s e) → (∀ e, 0 ≤ v e) → |
| M.Indep (labeledAcceptedThrough A r s v π t : Set E)) ∧ |
| ∀ {Ω : Type u} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] |
| (S V : Ω → E → ℝ) (R : Ω → Seed bits), |
| Measurable S → Measurable V → Measurable R → |
| (∀ᵐ ω ∂μ, ∀ e, 0 ≤ S ω e) → |
| (∀ᵐ ω ∂μ, ∀ e, 0 ≤ V ω e) → |
| iIndepFun (fun i : E × Bool => |
| if i.2 then fun ω => S ω i.1 else fun ω => V ω i.1) μ → |
| (∀ e, μ.map (fun ω => S ω e) = μ.map (fun ω => V ω e)) → |
| IndepFun (fun ω => (S ω, V ω)) R μ → μ.map R = ν → |
| Integrable (fun ω => finiteOptimum M (V ω)) μ → |
| ∀ π : Ω → LabeledOrder E, Measurable π → |
| Integrable (fun ω => ∑ e ∈ labeledAcceptedThrough A |
| (R ω) (S ω) (V ω) (π ω) (Fintype.card E), V ω e) μ ∧ |
| ((2 : ℝ) ^ 310)⁻¹ * (∫ ω, finiteOptimum M (V ω) ∂μ) ≤ |
| ∫ ω, ∑ e ∈ labeledAcceptedThrough A |
| (R ω) (S ω) (V ω) (π ω) (Fintype.card E), V ω e ∂μ := by |
| obtain ⟨bits, ν, hν, A, hA, hbound⟩ := |
| MatroidProphet.one_sample.{u} (Fintype.card E) (finMatroid M) |
| (finMatroid_ground M hE) |
| refine ⟨bits, ν, hν, relabelRule A, relabelRule_feasible M A hA, ?_⟩ |
| intro Ω mΩ μ hμ S V R hS hV hR hSN hVN hi hlaw hseed hRlaw hopt π hπ |
| have hSN' : ∀ᵐ ω ∂μ, ∀ i, 0 ≤ finWeights (S ω) i := |
| hSN.mono fun _ h => (finWeights_nonnegative_iff _).2 h |
| have hVN' : ∀ᵐ ω ∂μ, ∀ i, 0 ≤ finWeights (V ω) i := |
| hVN.mono fun _ h => (finWeights_nonnegative_iff _).2 h |
| have hseed' : IndepFun |
| (fun ω => (finWeights (S ω), finWeights (V ω))) R μ := by |
| exact hseed.comp ((measurable_finWeights.comp measurable_fst).prodMk |
| (measurable_finWeights.comp measurable_snd)) measurable_id |
| have hπ' : Measurable |
| (fun ω => (π ω).trans (Fintype.equivFin E)) := |
| (measurable_of_countable |
| (fun p : LabeledOrder E => p.trans (Fintype.equivFin E))).comp hπ |
| have hopt' : Integrable |
| (fun ω => optimum (finMatroid M) (finWeights (V ω))) μ := by |
| simpa only [optimum_finMatroid] using hopt |
| have hb := hbound μ (fun ω => finWeights (S ω)) |
| (fun ω => finWeights (V ω)) R (measurable_finWeights.comp hS) |
| (measurable_finWeights.comp hV) hR hSN' hVN' |
| (paired_independent_finWeights μ S V hi) |
| (fun i => hlaw ((Fintype.equivFin E).symm i)) hseed' hRlaw hopt' |
| (fun ω => (π ω).trans (Fintype.equivFin E)) hπ' |
| simpa only [relabelRule_reward, optimum_finMatroid] using hb |
|
|
| end MatroidProphet.Labeling |
|
|
| end OAI |
|
|
