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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 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 | import OAI.Probability.MatroidSecretary.Labels.LabeledHiddenModel
import OAI.Probability.MatroidSecretary.Labels.FiniteLabelTransport
import OAI.Probability.MatroidProphet.Main
namespace OAI
/-!
# The full hidden-vector endpoint on arbitrary finite labels
The enumeration is fixed before weights. These exact transport identities
preserve observation, sacrifice, every prefix, and the order minimum before
integration; they do not merely rename an expected fixed-order reward.
-/
namespace MatroidProphet.Labeling
open Finset MeasureTheory
variable {E : Type*} [Fintype E] [MeasurableSpace E]
[MeasurableSingletonClass E] {bits : ℕ}
noncomputable def relabelHiddenRule (A : HiddenRule (Fintype.card E) bits) :
LabeledHiddenRule E bits where
mask r := (A.mask r).map (Fintype.equivFin E).symm.toEmbedding
core := relabelRule A.core
theorem relabelHiddenRule_observed (A : HiddenRule (Fintype.card E) bits)
(r : Seed bits) (w : E → ℝ) :
finWeights (labeledObserved (relabelHiddenRule A) r w) =
observed A r (finWeights w) := by
classical
funext i
simp [finWeights, labeledObserved, relabelHiddenRule, observed]
theorem relabelHiddenRule_acceptedThrough
(A : HiddenRule (Fintype.card E) bits) (r : Seed bits)
(w : E → ℝ) (π : LabeledOrder E) (t : ℕ) :
(labeledHiddenAcceptedThrough (relabelHiddenRule A) r w π t).map
(Fintype.equivFin E).toEmbedding =
hiddenAcceptedThrough A r (finWeights w)
(π.trans (Fintype.equivFin E)) t := by
classical
unfold labeledHiddenAcceptedThrough hiddenAcceptedThrough
rw [Finset.map_sdiff]
change (labeledAcceptedThrough (relabelRule A.core) r
(labeledObserved (relabelHiddenRule A) r w) w π t).map
(Fintype.equivFin E).toEmbedding \
((A.mask r).map (Fintype.equivFin E).symm.toEmbedding).map
(Fintype.equivFin E).toEmbedding = _
rw [relabelRule_acceptedThrough, relabelHiddenRule_observed]
congr 1
ext i
simp
theorem relabelHiddenRule_reward (A : HiddenRule (Fintype.card E) bits)
(r : Seed bits) (w : E → ℝ) (π : LabeledOrder E) :
labeledHiddenReward (relabelHiddenRule A) r w π =
hiddenReward A r (finWeights w) (π.trans (Fintype.equivFin E)) := by
classical
rw [hiddenReward, ← relabelHiddenRule_acceptedThrough]
simp [labeledHiddenReward, finWeights]
/-- The set of labeled orders is transported bijectively, so the exact
seedwise adversarial minimum, not just any one order, is preserved. -/
theorem relabelHiddenRule_worstReward (A : HiddenRule (Fintype.card E) bits)
(w : E → ℝ) (r : Seed bits) :
labeledHiddenWorstReward (relabelHiddenRule A) w r =
hiddenWorstReward A (finWeights w) r := by
classical
apply le_antisymm
· apply Finset.le_inf'
intro π _
have h := Finset.inf'_le
(labeledHiddenReward (relabelHiddenRule A) r w)
(Finset.mem_univ (π.trans (Fintype.equivFin E).symm))
simpa only [labeledHiddenWorstReward, relabelHiddenRule_reward,
Equiv.trans_assoc, Equiv.symm_trans_self, Equiv.trans_refl] using h
· apply Finset.le_inf'
intro π _
rw [relabelHiddenRule_reward]
exact Finset.inf'_le _ (Finset.mem_univ _)
theorem relabelHiddenRule_feasible (M : Matroid E)
(A : HiddenRule (Fintype.card E) bits)
(hA : ∀ (w : Weights (Fintype.card E)), (∀ e, 0 ≤ w e) →
∀ (r : Seed bits) (π : ArrivalOrder (Fintype.card E)) (t : ℕ),
(finMatroid M).Indep (hiddenAcceptedThrough A r w π t : Set _))
(w : E → ℝ) (hw : ∀ e, 0 ≤ w e) (r : Seed bits)
(π : LabeledOrder E) (t : ℕ) :
M.Indep (labeledHiddenAcceptedThrough (relabelHiddenRule A) r w π t : Set E) := by
have hi := hA (finWeights w) ((finWeights_nonnegative_iff w).2 hw) r
(π.trans (Fintype.equivFin E)) t
rw [← relabelHiddenRule_acceptedThrough A r w π t] at hi
exact (mapEquiv_finset_indep M (Fintype.equivFin E) _).1 hi
omit [MeasurableSingletonClass E] in
/-- A hidden label remains sacrificed for every order and every prefix,
irrespective of what the online core would have decided. -/
theorem labeledHiddenAcceptedThrough_disjoint_mask
(A : LabeledHiddenRule E bits) (r : Seed bits) (w : E → ℝ)
(π : LabeledOrder E) (t : ℕ) :
Disjoint (labeledHiddenAcceptedThrough A r w π t) (A.mask r) := by
classical
exact Finset.sdiff_disjoint
/-- Literal arbitrary-finite-label version of the hidden-vector result.
All masked labels, loops, empty types, ties and zero weights are retained. -/
theorem hidden_vector_finite_labels (M : Matroid E) (hE : M.E = Set.univ) :
∃ (bits : ℕ) (ν : Measure (Seed bits)), IsProbabilityMeasure ν ∧
∃ A : LabeledHiddenRule E bits,
(∀ (w : E → ℝ), (∀ e, 0 ≤ w e) →
∀ (r : Seed bits) (π : LabeledOrder E) (t : ℕ),
M.Indep (labeledHiddenAcceptedThrough A r w π t : Set E)) ∧
∀ (w : E → ℝ), (∀ e, 0 ≤ w e) →
((2 : ℝ) ^ 293)⁻¹ * finiteOptimum M w ≤
∫ r, labeledHiddenWorstReward A w r ∂ν := by
obtain ⟨bits, ν, hν, A, hA, hbound⟩ :=
MatroidProphet.hidden_vector (Fintype.card E) (finMatroid M)
(finMatroid_ground M hE)
refine ⟨bits, ν, hν, relabelHiddenRule A, relabelHiddenRule_feasible M A hA, ?_⟩
intro w hw
have h := hbound (finWeights w) ((finWeights_nonnegative_iff w).2 hw)
simpa only [← relabelHiddenRule_worstReward, optimum_finMatroid] using h
end MatroidProphet.Labeling
end OAI
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