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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 98 | 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
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