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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 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 | import OAI.Probability.MatroidProphet.Main
import Mathlib.Probability.Independence.Basic
import Mathlib.MeasureTheory.Integral.Bochner.Basic
namespace OAI
namespace MatroidProphet.General
open MeasureTheory ProbabilityTheory
/-- A decision uses only the entire initial seed, the initially observed vector,
and the label/value history through the current arrival. -/
structure OnlineRule (n : ℕ) (R : Type*) [MeasurableSpace R] where
decide : (k : Fin n) → R → Weights n → History n k → Bool
measurable_decide : ∀ k, Measurable
(fun x : R × (Weights n × History n k) => decide k x.1 x.2.1 x.2.2)
def decisionAt {n : ℕ} {R : Type*} [MeasurableSpace R]
(A : OnlineRule n R) (r : R) (s v : Weights n)
(π : ArrivalOrder n) (k : Fin n) : Bool :=
A.decide k r s (history v π k)
noncomputable def acceptedThrough {n : ℕ} {R : Type*} [MeasurableSpace R]
(A : OnlineRule n R) (r : R) (s v : Weights n)
(π : ArrivalOrder n) (t : ℕ) : Finset (Fin n) := by
classical
exact Finset.univ.filter fun e =>
(π.symm e).val < t ∧ decisionAt A r s v π (π.symm e) = true
noncomputable def reward {n : ℕ} {R : Type*} [MeasurableSpace R]
(A : OnlineRule n R) (r : R) (s v : Weights n) (π : ArrivalOrder n) : ℝ :=
∑ e ∈ acceptedThrough A r s v π n, v e
def Feasible {n : ℕ} {R : Type*} [MeasurableSpace R]
(M : Matroid (Fin n)) (A : OnlineRule n R) : Prop :=
∀ (r : R) (s v : Weights n) (π : ArrivalOrder n) (t : ℕ),
(∀ e, 0 ≤ s e) → (∀ e, 0 ≤ v e) →
M.Indep (acceptedThrough A r s v π t : Set (Fin n))
/-- The observation mask is a measurable function of the seed alone. -/
structure HiddenRule (n : ℕ) (R : Type*) [MeasurableSpace R] where
mask : R → Finset (Fin n)
measurable_mask : Measurable mask
core : OnlineRule n R
noncomputable def observed {n : ℕ} {R : Type*} [MeasurableSpace R]
(A : HiddenRule n R) (r : R) (w : Weights n) : Weights n := by
classical
exact fun e => if e ∈ A.mask r then w e else 0
noncomputable def hiddenAcceptedThrough {n : ℕ} {R : Type*} [MeasurableSpace R]
(A : HiddenRule n R) (r : R) (w : Weights n)
(π : ArrivalOrder n) (t : ℕ) : Finset (Fin n) :=
acceptedThrough A.core r (observed A r w) w π t \ A.mask r
noncomputable def hiddenReward {n : ℕ} {R : Type*} [MeasurableSpace R]
(A : HiddenRule n R) (r : R) (w : Weights n) (π : ArrivalOrder n) : ℝ :=
∑ e ∈ hiddenAcceptedThrough A r w π n, w e
noncomputable def hiddenWorstReward {n : ℕ} {R : Type*} [MeasurableSpace R]
(A : HiddenRule n R) (w : Weights n) (r : R) : ℝ := by
classical
exact Finset.univ.inf' Finset.univ_nonempty (hiddenReward A r w)
def HiddenFeasible {n : ℕ} {R : Type*} [MeasurableSpace R]
(M : Matroid (Fin n)) (A : HiddenRule n R) : Prop :=
∀ (w : Weights n), (∀ e, 0 ≤ w e) →
∀ (r : R) (π : ArrivalOrder n) (t : ℕ),
M.Indep (hiddenAcceptedThrough A r w π t : Set (Fin n))
/-- Sample values on the sacrificed mask, and realized values elsewhere. -/
noncomputable def glue {n : ℕ} (mask : Finset (Fin n)) (s v : Weights n) : Weights n := by
classical
exact fun e => if e ∈ mask then s e else v e
noncomputable def simulationHistory {n : ℕ} {R : Type*} [MeasurableSpace R]
(A : HiddenRule n R) (r : R) (s : Weights n)
{k : Fin n} (h : History n k) : History n k := by
classical
exact fun j => ((h j).1, if (h j).1 ∈ A.mask r then s (h j).1 else (h j).2)
noncomputable def simulationDecision {n : ℕ} {R : Type*} [MeasurableSpace R]
(A : HiddenRule n R) (k : Fin n) (r : R) (s : Weights n)
(h : History n k) : Bool := by
classical
exact if (h ⟨k.val, Nat.lt_succ_self _⟩).1 ∈ A.mask r then false
else A.core.decide k r (observed A r s) (simulationHistory A r s h)
/-- Full source contract for `lem:simulation`, with arbitrary measurable seed
and probability spaces. The simulator is constructed before the distributions
and adversary are supplied. Its decision function is specified concretely;
the conjunction includes exact execution, feasibility, the seed/glued-vector
joint law, integrability, and preservation of the same competitive ratio. -/
def SimulationContract.{u, v} : Prop :=
∀ (n : ℕ) (R : Type v) [MeasurableSpace R]
(M : Matroid (Fin n)) (A : HiddenRule n R)
(ν : Measure R), IsProbabilityMeasure ν →
∀ (c : ℝ), HiddenFeasible M A →
(∀ (w : Weights n), (∀ e, 0 ≤ w e) →
c * optimum M w ≤ ∫ r, hiddenWorstReward A w r ∂ν) →
∃ B : OnlineRule n R,
(∀ (k : Fin n) (r : R) (s : Weights n) (h : History n k),
B.decide k r s h = simulationDecision A k r s h) ∧
Feasible M B ∧
(∀ (r : R) (s v : Weights n) (π : ArrivalOrder n) (t : ℕ),
acceptedThrough B r s v π t =
hiddenAcceptedThrough A r (glue (A.mask r) s v) π t) ∧
(∀ (r : R) (s v : Weights n) (π : ArrivalOrder n),
reward B r s v π = hiddenReward A r (glue (A.mask r) s v) π) ∧
∀ {Ω : Type u} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]
(S V : Ω → Weights n) (seed : Ω → R),
Measurable S → Measurable V → Measurable seed →
(∀ᵐ ω ∂μ, ∀ e, 0 ≤ S ω e) →
(∀ᵐ ω ∂μ, ∀ e, 0 ≤ V ω e) →
iIndepFun (pairedCoordinates S V) μ →
(∀ e, μ.map (fun ω => S ω e) = μ.map (fun ω => V ω e)) →
IndepFun (fun ω => (S ω, V ω)) seed μ → μ.map seed = ν →
Integrable (fun ω => optimum M (V ω)) μ →
∀ (π : Ω → ArrivalOrder n), Measurable π →
MeasurePreserving (fun ω => (seed ω, glue (A.mask (seed ω)) (S ω) (V ω)))
μ (ν.prod (μ.map V)) ∧
Integrable (fun ω => reward B (seed ω) (S ω) (V ω) (π ω)) μ ∧
c * (∫ ω, optimum M (V ω) ∂μ) ≤
∫ ω, reward B (seed ω) (S ω) (V ω) (π ω) ∂μ
end MatroidProphet.General
end OAI
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