File size: 4,211 Bytes
92416a2 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 | import Mathlib
import OAI.Computability.VertexCover.Analysis.OwnGradientUnsupported
namespace OAI
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
namespace VertexCover.Average
open MeasureTheory
theorem finite_abs_le {α : Type*} [Fintype α] (g : α → ℝ)
{B : ℝ} (hB : 0 ≤ B) (hg : ∀ a, |g a| ≤ B) :
|(∑ a, g a) / Fintype.card α| ≤ B := by
classical
by_cases he : IsEmpty α
· let := he
simpa using hB
· have : Nonempty α := not_isEmpty_iff.mp he
have hc : (0 : ℝ) < Fintype.card α := Nat.cast_pos.mpr Fintype.card_pos
rw [abs_div, abs_of_nonneg hc.le]
apply (div_le_iff₀ hc).mpr
calc
|∑ a, g a| ≤ ∑ a, |g a| := Finset.abs_sum_le_sum_abs _ _
_ ≤ ∑ _a : α, B := Finset.sum_le_sum (fun a _ => hg a)
_ = B * Fintype.card α := by simp [mul_comm]
theorem integral_lipschitz {α E : Type*} [MeasurableSpace α] [PseudoMetricSpace E]
(μ : Measure α) [IsProbabilityMeasure μ] {C : NNReal} (F : E → α → ℝ)
(hi : ∀ x, Integrable (F x) μ) (hF : ∀ a, LipschitzWith C (fun x => F x a)) :
LipschitzWith C (fun x => ∫ a, F x a ∂μ) := by
apply LipschitzWith.of_dist_le_mul
intro x y
rw [Real.dist_eq, ← integral_sub (hi x) (hi y)]
have h := norm_integral_le_of_norm_le_const (μ := μ)
(f := fun a => F x a - F y a) (C := (C : ℝ) * dist x y)
(Filter.Eventually.of_forall (fun a => by
simpa only [dist_eq_norm] using (hF a).dist_le_mul x y))
simpa only [Real.dist_eq, Real.norm_eq_abs, probReal_univ, mul_one] using h
end VertexCover.Average
namespace VertexCover.LabelCover
open MeasureTheory
theorem ownMean_lipschitz (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
(frozen : Φ.Seeds d) (c0 : Φ.Coordinate d → ℝ)
(A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty)
(j : J) (i : Φ.Query d) :
LipschitzWith 1 (Φ.ownMean J frozen c0 A hA j i) := by
classical
have hI : ∀ hidden : Φ.ownFiber J frozen j i,
LipschitzWith 1 (fun s => ∫ other,
Φ.batchFunction (Φ.spliceSeeds J frozen hidden) J c0 A hA
(Function.update other j s) ∂Φ.batchLaw J) := by
intro hidden
apply VertexCover.Average.integral_lipschitz
· intro s
exact VertexCover.Cube.integrable_continuous_block
((Φ.batchFunction_continuous _ J c0 A hA).comp
(continuous_id.update j continuous_const))
· intro other
exact Φ.batchFunction_own_lipschitz _ J c0 A hA other j
apply LipschitzWith.of_dist_le_mul
intro s t
simp only [NNReal.coe_one, one_mul, Real.dist_eq, ownMean, VertexCover.finiteMean,
← sub_div, ← Finset.sum_sub_distrib]
apply VertexCover.Average.finite_abs_le (hB := dist_nonneg)
intro hidden
simpa only [Real.dist_eq, NNReal.coe_one, one_mul] using (hI hidden).dist_le_mul s t
theorem ownGradient_measurable (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
(frozen : Φ.Seeds d) (c0 : Φ.Coordinate d → ℝ)
(A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty)
(j : J) (i : Φ.Query d) (k : Fin (Φ.WeightDimension d)) :
Measurable (fun s => Φ.ownGradient J frozen c0 A hA j i s k) := by
classical
have hI : ∀ hidden : Φ.ownFiber J frozen j i,
StronglyMeasurable (fun s => ∫ other,
Φ.canonicalGradient (Φ.spliceSeeds J frozen hidden) J c0 A hA
(Function.update other j s) j k ∂Φ.batchLaw J) := by
intro hidden
apply StronglyMeasurable.integral_prod_right
apply Measurable.stronglyMeasurable
exact (measurable_pi_apply k).comp ((measurable_pi_apply j).comp
((Φ.canonicalGradient_measurable _ J c0 A hA).comp
(continuous_snd.update j continuous_fst).measurable))
unfold ownGradient VertexCover.finiteMean
exact (Finset.measurable_sum _ (fun hidden _ => (hI hidden).measurable)).div_const _
end VertexCover.LabelCover
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
|