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a2ee19a | 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 | import OAI.Probability.MatroidProphet.Main
import Mathlib.MeasureTheory.Integral.Bochner.Basic
namespace OAI
open MeasureTheory
namespace MatroidProphet
/-- For a fixed vector, arbitrary measurable seed-dependent orders give an
integrable reward under every finite measure. No feasibility or sign assumption
is needed, since the seed and order spaces are finite. -/
lemma integrable_hiddenReward_randomOrder {n bits : ℕ} (A : HiddenRule n bits)
(w : Weights n) {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsFiniteMeasure μ]
(R : Ω → Seed bits) (π : Ω → ArrivalOrder n)
(hR : Measurable R) (hπ : Measurable π) :
Integrable (fun ω => hiddenReward A (R ω) w (π ω)) μ := by
let f : Seed bits × ArrivalOrder n → ℝ := fun x => hiddenReward A x.1 w x.2
have hf : Measurable f := measurable_of_finite f
have hpair : Measurable (fun ω => (R ω, π ω)) := hR.prodMk hπ
exact (integrable_map_measure hf.aestronglyMeasurable hpair.aemeasurable).1
(Integrable.of_finite (f := f))
/-- The seedwise minimum is integrable after any measurable finite-seed map. -/
lemma integrable_hiddenWorstReward_comp {n bits : ℕ} (A : HiddenRule n bits)
(w : Weights n) {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsFiniteMeasure μ]
(R : Ω → Seed bits) (hR : Measurable R) :
Integrable (fun ω => hiddenWorstReward A w (R ω)) μ := by
have hf : Measurable (hiddenWorstReward A w) := measurable_of_finite _
exact (integrable_map_measure hf.aestronglyMeasurable hR.aemeasurable).1
(Integrable.of_finite (f := hiddenWorstReward A w))
/-- A reconstructed seed's correct marginal transports the expected minimum to
any measurable full-information order. The order need not be independent of the
seed, and the ambient space may carry arbitrarily much additional information. -/
theorem hiddenWorstReward_integral_le_randomOrder {n bits : ℕ}
(A : HiddenRule n bits) (w : Weights n)
{Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ]
(ν : Measure (Seed bits)) (R : Ω → Seed bits) (π : Ω → ArrivalOrder n)
(hR : Measurable R) (hπ : Measurable π) (hRlaw : μ.map R = ν) :
(∫ r, hiddenWorstReward A w r ∂ν) ≤
∫ ω, hiddenReward A (R ω) w (π ω) ∂μ := by
have hf : Measurable (hiddenWorstReward A w) := measurable_of_finite _
rw [← hRlaw, integral_map_of_stronglyMeasurable hR hf.stronglyMeasurable]
apply integral_mono
(integrable_hiddenWorstReward_comp A w μ R hR)
(integrable_hiddenReward_randomOrder A w μ R π hR hπ)
intro ω
exact hiddenWorstReward_le A w (R ω) (π ω)
/-- A pointwise accepted-reward identity may be replaced by an almost-everywhere
identity. This is useful when a reconstruction kernel has zero-mass seed outcomes. -/
theorem hiddenWorstReward_integral_le_replayed {n bits : ℕ}
(A : HiddenRule n bits) (w : Weights n)
{Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ]
(ν : Measure (Seed bits)) (R : Ω → Seed bits) (π : Ω → ArrivalOrder n)
(hR : Measurable R) (hπ : Measurable π) (hRlaw : μ.map R = ν)
(payoff : Ω → ℝ)
(hreplay : payoff =ᵐ[μ] fun ω => hiddenReward A (R ω) w (π ω)) :
Integrable payoff μ ∧ (∫ r, hiddenWorstReward A w r ∂ν) ≤ ∫ ω, payoff ω ∂μ := by
refine ⟨(integrable_hiddenReward_randomOrder A w μ R π hR hπ).congr hreplay.symm, ?_⟩
rw [integral_congr_ae hreplay]
exact hiddenWorstReward_integral_le_randomOrder A w μ ν R π hR hπ hRlaw
/-- There is a measurable adversary attaining the minimum separately at every
seed. Thus the minimum used above is genuinely inside expectation, not the
minimum of fixed-order expected rewards. -/
lemma exists_measurable_worstOrder {n bits : ℕ} (A : HiddenRule n bits)
(w : Weights n) :
∃ π : Seed bits → ArrivalOrder n, Measurable π ∧
∀ r, hiddenReward A r w (π r) = hiddenWorstReward A w r := by
classical
have hex (r : Seed bits) :
∃ π : ArrivalOrder n, hiddenReward A r w π = hiddenWorstReward A w r := by
obtain ⟨π, _, hπ⟩ := Finset.exists_mem_eq_inf'
(s := (Finset.univ : Finset (ArrivalOrder n))) Finset.univ_nonempty
(hiddenReward A r w)
exact ⟨π, hπ.symm⟩
choose π hπ using hex
exact ⟨π, measurable_of_finite π, hπ⟩
/-- The lower bound is sharp among measurable full-seed order policies. -/
lemma exists_order_integral_eq_hiddenWorstReward {n bits : ℕ}
(A : HiddenRule n bits) (w : Weights n) (ν : Measure (Seed bits)) :
∃ π : Seed bits → ArrivalOrder n, Measurable π ∧
(∫ r, hiddenReward A r w (π r) ∂ν) = ∫ r, hiddenWorstReward A w r ∂ν := by
obtain ⟨π, hπ, heq⟩ := exists_measurable_worstOrder A w
exact ⟨π, hπ, integral_congr_ae (ae_of_all _ heq)⟩
end MatroidProphet
end OAI
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