File size: 1,658 Bytes
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 | 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
|