File size: 3,034 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 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 | import OAI.Probability.MatroidProphet.Main
import Mathlib.Combinatorics.Matroid.Map
namespace OAI
namespace MatroidProphet.Labeling
open Finset
/-- The same offline maximum as `MatroidProphet.optimum`, on an arbitrary
finite ground type. The empty independent set is among the competitors. -/
noncomputable def finiteOptimum {E : Type*} [Fintype E]
(M : Matroid E) (w : E → ℝ) : ℝ := by
classical
exact Finset.univ.sup' Finset.univ_nonempty
(fun I : Finset E => if M.Indep (I : Set E) then ∑ e ∈ I, w e else 0)
/-- Relabel all elements, without restricting the matroid or dropping loops. -/
noncomputable def finMatroid {E : Type*} [Fintype E] (M : Matroid E) :
Matroid (Fin (Fintype.card E)) := M.mapEquiv (Fintype.equivFin E)
/-- Transport each weight along the inverse of the chosen enumeration. -/
noncomputable def finWeights {E : Type*} [Fintype E] (w : E → ℝ) :
Weights (Fintype.card E) := fun i => w ((Fintype.equivFin E).symm i)
/-- A labeled arrival order uses arrival times `Fin |E|` and actual labels `E`.
It is a bijection, so every label arrives exactly once. -/
abbrev LabeledOrder (E : Type*) [Fintype E] := Fin (Fintype.card E) ≃ E
/-- Arrival orders form a finite discrete measurable space, just as in the
`Fin n` information model. -/
instance labeledOrderMeasurableSpace (E : Type*) [Fintype E] :
MeasurableSpace (LabeledOrder E) := ⊤
instance labeledOrderMeasurableSingletonClass (E : Type*) [Fintype E] :
MeasurableSingletonClass (LabeledOrder E) := ⟨fun _ => trivial⟩
/-- Only the label coordinates of a history are relabeled. -/
def encodeHistory {E : Type*} {n : ℕ} (e : E ≃ Fin n) (k : Fin n)
(h : Fin (k.val + 1) → E × ℝ) : History n k :=
fun j => (e (h j).1, (h j).2)
/-- A rule on arbitrary finite labels has exactly the same information
restriction as `OnlineRule`: seed, initial samples, and the observed prefix. -/
structure LabeledOnlineRule (E : Type*) [Fintype E] [MeasurableSpace E]
(bits : ℕ) where
decide : (k : Fin (Fintype.card E)) → Seed bits → (E → ℝ) →
(Fin (k.val + 1) → E × ℝ) → Bool
measurable_decide : ∀ k, Measurable
(fun x : Seed bits × ((E → ℝ) × (Fin (k.val + 1) → E × ℝ)) =>
decide k x.1 x.2.1 x.2.2)
/-- The actual observed history for an arbitrary labeled arrival order. -/
def labeledHistory {E : Type*} [Fintype E] (v : E → ℝ)
(π : LabeledOrder E) (k : Fin (Fintype.card E)) :
Fin (k.val + 1) → E × ℝ :=
fun j => let e := π (prefixIndex k j); (e, v e)
/-- Accepted labels after `t` arrivals in the arbitrary-label model. -/
noncomputable def labeledAcceptedThrough {E : Type*} [Fintype E] [MeasurableSpace E]
{bits : ℕ} (A : LabeledOnlineRule E bits) (r : Seed bits)
(s v : E → ℝ) (π : LabeledOrder E) (t : ℕ) : Finset E := by
classical
exact Finset.univ.filter fun e => (π.symm e).val < t ∧
A.decide (π.symm e) r s (labeledHistory v π (π.symm e)) = true
end MatroidProphet.Labeling
end OAI
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