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a2ee19a | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 | import OAI.Probability.MatroidProphet.Main
import Mathlib.MeasureTheory.Integral.Bochner.Basic
namespace OAI
namespace MatroidProphet.ZeroBenchmark
open MeasureTheory Finset
open scoped Classical
/-- Only singleton-independent labels can affect the offline optimum.
No sign or integrability assumption is necessary for this identity. -/
theorem optimum_congr_on_singletons {n : ℕ} (M : Matroid (Fin n))
(w v : Weights n)
(h : ∀ e, M.Indep ({e} : Set (Fin n)) → w e = v e) :
optimum M w = optimum M v := by
classical
unfold optimum
apply Finset.sup'_congr Finset.univ_nonempty rfl
intro I _
by_cases hI : M.Indep (I : Set (Fin n))
· simp only [ite_eq_left hI]
apply Finset.sum_congr rfl
intro e he
exact h e (hI.subset (Set.singleton_subset_iff.mpr he))
· simp only [ite_eq_right hI]
/-- The zero-weight benchmark is zero, also for empty or rank-zero matroids. -/
@[simp] theorem optimum_zero {n : ℕ} (M : Matroid (Fin n)) :
optimum M (fun _ => 0) = 0 := by
classical
simp [optimum]
/-- On nonnegative inputs, a zero benchmark is exactly the condition that
every label that could be accepted on its own has zero weight. -/
theorem optimum_eq_zero_iff {n : ℕ} (M : Matroid (Fin n))
(w : Weights n) (hw : ∀ e, 0 ≤ w e) :
optimum M w = 0 ↔ ∀ e, M.Indep ({e} : Set (Fin n)) → w e = 0 := by
constructor
· intro h e he
have hb := sum_le_optimum M w {e} (by simpa using he)
simp only [Finset.sum_singleton, h] at hb
exact le_antisymm hb (hw e)
· intro h
rw [optimum_congr_on_singletons M w (fun _ => 0) h]
exact optimum_zero M
/-- Replacing all loop coordinates by zero never changes the objective.
Here ground-set outsiders, if any, are also harmless. -/
theorem optimum_ignore_infeasible_coordinates {n : ℕ} (M : Matroid (Fin n))
(w : Weights n) :
optimum M (fun e => if M.Indep ({e} : Set (Fin n)) then w e else 0) =
optimum M w := by
classical
apply optimum_congr_on_singletons
intro e he
simp only [ite_eq_left he]
/-- Rank-zero behavior needs no bound at all on the values of its labels. -/
theorem optimum_eq_zero_of_no_independent_singleton {n : ℕ}
(M : Matroid (Fin n))
(hM : ∀ e, ¬ M.Indep ({e} : Set (Fin n))) (w : Weights n) :
optimum M w = 0 := by
rw [optimum_congr_on_singletons M w (fun _ => 0) (fun e he => (hM e he).elim)]
exact optimum_zero M
/-- The empty instance has the zero benchmark without a positive-size premise. -/
@[simp] theorem optimum_empty (M : Matroid (Fin 0)) (w : Weights 0) :
optimum M w = 0 := by
apply optimum_eq_zero_of_no_independent_singleton
intro e
exact Fin.elim0 e
/-- Every feasible realized hidden reward vanishes when the benchmark does. -/
theorem hiddenReward_eq_zero {n bits : ℕ} (M : Matroid (Fin n))
(A : HiddenRule n bits) (r : Seed bits) (w : Weights n)
(π : ArrivalOrder n) (hw : ∀ e, 0 ≤ w e)
(hA : M.Indep (hiddenAcceptedThrough A r w π n : Set (Fin n)))
(hopt : optimum M w = 0) : hiddenReward A r w π = 0 := by
apply le_antisymm
· exact (sum_le_optimum M w _ hA).trans_eq hopt
· exact Finset.sum_nonneg fun e _ => hw e
/-- The zero-benchmark conclusion also holds for the minimum taken separately
at each complete seed, before any expectation. -/
theorem hiddenWorstReward_eq_zero {n bits : ℕ} (M : Matroid (Fin n))
(A : HiddenRule n bits) (w : Weights n) (hw : ∀ e, 0 ≤ w e)
(hA : ∀ r π, M.Indep (hiddenAcceptedThrough A r w π n : Set (Fin n)))
(hopt : optimum M w = 0) (r : Seed bits) :
hiddenWorstReward A w r = 0 := by
apply le_antisymm
· exact (hiddenWorstReward_le A w r (Equiv.refl _)).trans_eq
(hiddenReward_eq_zero M A r w (Equiv.refl _) hw (hA r _) hopt)
· exact hiddenWorstReward_nonneg A w hw r
/-- No probability-law or moment premise is needed in the zero case. -/
theorem integral_hiddenWorstReward_eq_zero {n bits : ℕ} (M : Matroid (Fin n))
(A : HiddenRule n bits) (w : Weights n) (hw : ∀ e, 0 ≤ w e)
(hA : ∀ r π, M.Indep (hiddenAcceptedThrough A r w π n : Set (Fin n)))
(hopt : optimum M w = 0) (ν : Measure (Seed bits)) :
(∫ r, hiddenWorstReward A w r ∂ν) = 0 := by
simp only [hiddenWorstReward_eq_zero M A w hw hA hopt]
exact integral_zero _ _
/-- Even completely unrestricted loop coordinates cannot create a moment
requirement when the usable coordinates vanish almost everywhere. -/
theorem integrable_optimum_of_usable_coordinates_zero {n : ℕ}
(M : Matroid (Fin n)) {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) (V : Ω → Weights n)
(hV : ∀ᵐ ω ∂μ, ∀ e, M.Indep ({e} : Set (Fin n)) → V ω e = 0) :
Integrable (fun ω => optimum M (V ω)) μ := by
apply (integrable_zero Ω ℝ μ).congr
filter_upwards [hV] with ω hω
symm
rw [optimum_congr_on_singletons M (V ω) (fun _ => 0) hω]
exact optimum_zero M
theorem integral_optimum_of_usable_coordinates_zero {n : ℕ}
(M : Matroid (Fin n)) {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) (V : Ω → Weights n)
(hV : ∀ᵐ ω ∂μ, ∀ e, M.Indep ({e} : Set (Fin n)) → V ω e = 0) :
(∫ ω, optimum M (V ω) ∂μ) = 0 := by
calc
(∫ ω, optimum M (V ω) ∂μ) = ∫ _ : Ω, (0 : ℝ) ∂μ := by
apply integral_congr_ae
filter_upwards [hV] with ω hω
rw [optimum_congr_on_singletons M (V ω) (fun _ => 0) hω]
exact optimum_zero M
_ = 0 := integral_zero _ _
end MatroidProphet.ZeroBenchmark
end OAI
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