File size: 2,806 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 | import OAI.Probability.MatroidProphet.Main
import Mathlib.Combinatorics.Matroid.Map
namespace OAI
namespace MatroidProphet.Relabeling
open Finset
/-- The offline benchmark on an arbitrary finite label type. -/
noncomputable def finiteOptimum {α : Type*} [Fintype α]
(M : Matroid α) (w : α → ℝ) : ℝ := by
classical
exact Finset.univ.sup' Finset.univ_nonempty
(fun I : Finset α => if M.Indep (I : Set α) then ∑ e ∈ I, w e else 0)
/-- Reindex all weights, retaining every label including loops and zero weights. -/
def encodeWeights {α β : Type*} (e : α ≃ β) (w : α → ℝ) : β → ℝ :=
fun b => w (e.symm b)
/-- Every sample and online coordinate, indexed by original label and copy. -/
def labeledPairedCoordinates {α Ω : Type*} (S V : Ω → α → ℝ)
(i : α × Bool) : Ω → ℝ :=
if i.2 then fun ω => S ω i.1 else fun ω => V ω i.1
/-- An arrival order on `α` lists every label exactly once. -/
abbrev LabeledOrder (α : Type*) (n : ℕ) := Fin n ≃ α
/-- The available prefix, with arbitrary original labels. -/
abbrev LabeledHistory (α : Type*) {n : ℕ} (k : Fin n) :=
Fin (k.val + 1) → α × ℝ
/-- Reindex the labels of an observed history without changing any observed value. -/
def encodeHistory {α : Type*} {n : ℕ} (e : α ≃ Fin n) (k : Fin n)
(h : LabeledHistory α k) : History n k :=
fun j => (e (h j).1, (h j).2)
/-- The history supplied by a genuine arrival stream on the original labels. -/
def labeledHistory {α : Type*} {n : ℕ} (w : α → ℝ)
(π : LabeledOrder α n) (k : Fin n) : LabeledHistory α k :=
fun j => let a := π (prefixIndex k j); (a, w a)
/-- The transported decision uses only the seed, original samples, and history. -/
def transportedDecision {α : Type*} {n bits : ℕ} (e : α ≃ Fin n)
(A : OnlineRule n bits) (k : Fin n) (r : Seed bits)
(s : α → ℝ) (h : LabeledHistory α k) : Bool :=
A.decide k r (encodeWeights e s) (encodeHistory e k h)
/-- Irrevocable accepted labels in the original labeled stream. -/
noncomputable def transportedAcceptedThrough {α : Type*} [Fintype α]
{n bits : ℕ} (e : α ≃ Fin n) (A : OnlineRule n bits)
(r : Seed bits) (s v : α → ℝ) (π : LabeledOrder α n) (t : ℕ) : Finset α := by
classical
exact Finset.univ.filter fun a => (π.symm a).val < t ∧
transportedDecision e A (π.symm a) r s (labeledHistory v π (π.symm a)) = true
/-- Real reward, with no rounding or loss in the relabeling. -/
noncomputable def transportedReward {α : Type*} [Fintype α]
{n bits : ℕ} (e : α ≃ Fin n) (A : OnlineRule n bits)
(r : Seed bits) (s v : α → ℝ) (π : LabeledOrder α n) : ℝ :=
∑ a ∈ transportedAcceptedThrough e A r s v π n, v a
end MatroidProphet.Relabeling
end OAI
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