| import OAI.Probability.MatroidSecretary.Labels.FiniteLabelTransport |
| import OAI.Probability.MatroidSecretary.Labels.RelabelingProphet |
|
|
| namespace OAI |
|
|
| /-! The two preserved arbitrary-label interfaces describe exactly the same |
| offline benchmark and online execution. These are definitional equalities, |
| not extra model-validity premises or comparisons of surrogate rewards. -/ |
|
|
| namespace MatroidProphet.FiniteLabelAgreement |
|
|
| variable {E : Type*} [Fintype E] [MeasurableSpace E] |
| [MeasurableSingletonClass E] {bits : ℕ} |
|
|
| omit [MeasurableSpace E] [MeasurableSingletonClass E] in |
| theorem optimum_eq (M : Matroid E) (w : E → ℝ) : |
| Labeling.finiteOptimum M w = Relabeling.finiteOptimum M w := rfl |
|
|
| theorem decision_eq (A : OnlineRule (Fintype.card E) bits) |
| (k : Fin (Fintype.card E)) (r : Seed bits) (s : E → ℝ) |
| (h : Relabeling.LabeledHistory E k) : |
| (Labeling.relabelRule A).decide k r s h = |
| Relabeling.transportedDecision (Fintype.equivFin E) A k r s h := rfl |
|
|
| theorem acceptedThrough_eq (A : OnlineRule (Fintype.card E) bits) |
| (r : Seed bits) (s v : E → ℝ) (π : Labeling.LabeledOrder E) (t : ℕ) : |
| Labeling.labeledAcceptedThrough (Labeling.relabelRule A) r s v π t = |
| Relabeling.transportedAcceptedThrough (Fintype.equivFin E) A r s v π t := rfl |
|
|
| theorem reward_eq (A : OnlineRule (Fintype.card E) bits) |
| (r : Seed bits) (s v : E → ℝ) (π : Labeling.LabeledOrder E) : |
| (∑ e ∈ Labeling.labeledAcceptedThrough (Labeling.relabelRule A) |
| r s v π (Fintype.card E), v e) = |
| Relabeling.transportedReward (Fintype.equivFin E) A r s v π := rfl |
|
|
| end MatroidProphet.FiniteLabelAgreement |
|
|
| end OAI |
|
|
