| import OAI.Probability.MatroidProphet.Main |
|
|
| namespace OAI |
|
|
| /-! |
| # The stronger constant for the concrete one-sample rule |
|
|
| The introduction explicitly records the stronger `2⁻²⁹³` constant proved in |
| accounting.tex. The root endpoint uses its announced `2⁻³¹⁰` constant. This |
| supporting theorem exposes the stronger claim for the actual constructed rule, |
| without changing the endpoint or imposing moments on individual loop values. |
|
|
| The rule and initial seed law below depend only on the known labeled matroid. |
| The probability space, matching input laws, and full-seed-aware adversary are |
| supplied afterwards. Feasibility is pointwise at every prefix and the reward |
| estimate uses only integrability of the actual offline optimum. |
| -/ |
|
|
| namespace MatroidProphet |
|
|
| open MeasureTheory ProbabilityTheory |
|
|
| / |
| theorem source_one_sample_feasible {n : ℕ} (M : Matroid (Fin n)) |
| (hE : M.E = Set.univ) : |
| Feasible M (sampleSimulation (completeHiddenRule M hE)) := |
| sampleSimulation_feasible M (completeHiddenRule M hE) |
| (completeHiddenRule_feasible M hE) |
|
|
| / |
| `thm:main`, even when the order sees the complete initial seed. This is a |
| consequence for the concrete rule, not a premise asserting its performance. -/ |
| theorem source_one_sample_strong {n : ℕ} (M : Matroid (Fin n)) |
| (hE : M.E = Set.univ) |
| {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] |
| (S V : Ω → Weights n) (R : Ω → Seed (mainSeedBits n)) |
| (hS : Measurable S) (hV : Measurable V) (hR : Measurable R) |
| (hSN : ∀ᵐ ω ∂μ, ∀ e, 0 ≤ S ω e) |
| (hVN : ∀ᵐ ω ∂μ, ∀ e, 0 ≤ V ω e) |
| (hi : iIndepFun (pairedCoordinates S V) μ) |
| (hlaw : ∀ e, μ.map (fun ω => S ω e) = μ.map (fun ω => V ω e)) |
| (hseed : IndepFun (fun ω => (S ω, V ω)) R μ) |
| (hRlaw : μ.map R = sourceSeedLaw n) |
| (hopt : Integrable (fun ω => optimum M (V ω)) μ) |
| (π : Ω → ArrivalOrder n) (hπ : Measurable π) : |
| Integrable (fun ω => |
| reward (sampleSimulation (completeHiddenRule M hE)) |
| (R ω) (S ω) (V ω) (π ω)) μ ∧ |
| ((2 : ℝ) ^ 293)⁻¹ * (∫ ω, optimum M (V ω) ∂μ) ≤ |
| ∫ ω, reward (sampleSimulation (completeHiddenRule M hE)) |
| (R ω) (S ω) (V ω) (π ω) ∂μ := |
| oneSample_of_hidden M (completeHiddenRule M hE) (sourceSeedLaw n) |
| (((2 : ℝ) ^ 293)⁻¹) (completeHiddenRule_feasible M hE) |
| (complete_hidden_guarantee M hE) μ S V R hS hV hR hSN hVN |
| hi hlaw hseed hRlaw hopt π hπ |
|
|
| end MatroidProphet |
|
|
| end OAI |
|
|
