| import Mathlib |
| import OAI.Computability.VertexCover.Analysis.OwnGradientUnsupported |
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| namespace OAI |
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| 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 |
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| 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 |
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|
