| import Mathlib |
| import OAI.Computability.VertexCover.Analysis.BatchFunctionOwnLipschitz |
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| namespace OAI |
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| namespace VertexCover.Average |
| open MeasureTheory |
|
|
| theorem integral_l1_le {α ι : Type*} [MeasurableSpace α] [Fintype ι] |
| (μ : Measure α) [IsProbabilityMeasure μ] (g : α → ι → ℝ) |
| (hg : ∀ i, AEStronglyMeasurable (fun a => g a i) μ) |
| (hb : ∀ a, ∑ i, |g a i| ≤ 1) : |
| ∑ i, |∫ a, g a i ∂μ| ≤ 1 := by |
| have hbi : ∀ a i, |g a i| ≤ 1 := by |
| intro a i |
| exact (Finset.single_le_sum (fun j _ => abs_nonneg (g a j)) (Finset.mem_univ i)).trans (hb a) |
| have hi : ∀ i, Integrable (fun a => |g a i|) μ := by |
| intro i |
| exact (Integrable.of_bound (hg i) 1 |
| (Filter.Eventually.of_forall (fun a => by simpa only [Real.norm_eq_abs] using hbi a i))).abs |
| calc |
| (∑ i, |∫ a, g a i ∂μ|) ≤ ∑ i, ∫ a, |g a i| ∂μ := by |
| apply Finset.sum_le_sum |
| intro i _ |
| simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm (fun a => g a i) |
| _ = ∫ a, ∑ i, |g a i| ∂μ := (integral_finsetSum _ (fun i _ => hi i)).symm |
| _ ≤ ∫ _a, (1 : ℝ) ∂μ := integral_mono |
| (integrable_finsetSum _ (fun i _ => hi i)) (integrable_const 1) hb |
| _ = 1 := by simp |
|
|
| theorem finite_l1_le {α ι : Type*} [Fintype α] [Fintype ι] |
| (g : α → ι → ℝ) (hb : ∀ a, ∑ i, |g a i| ≤ 1) : |
| ∑ i, |(∑ a, g a i) / Fintype.card α| ≤ 1 := by |
| classical |
| by_cases he : IsEmpty α |
| · let := he |
| simp |
| · have : Nonempty α := not_isEmpty_iff.mp he |
| have hc : (0 : ℝ) < Fintype.card α := Nat.cast_pos.mpr Fintype.card_pos |
| simp only [abs_div, abs_of_nonneg hc.le, ← Finset.sum_div] |
| apply (div_le_iff₀ hc).mpr |
| calc |
| (∑ i, |∑ a, g a i|) ≤ ∑ i, ∑ a, |g a i| := by |
| exact Finset.sum_le_sum (fun i _ => Finset.abs_sum_le_sum_abs _ _) |
| _ = ∑ a, ∑ i, |g a i| := Finset.sum_comm |
| _ ≤ ∑ _a : α, (1 : ℝ) := Finset.sum_le_sum (fun a _ => hb a) |
| _ = 1 * Fintype.card α := by simp |
|
|
| end VertexCover.Average |
|
|
| namespace VertexCover.LabelCover |
| open MeasureTheory ProbabilityTheory |
|
|
| noncomputable def ownFiber (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d)) |
| (frozen : Φ.Seeds d) (j : J) (i : Φ.Query d) : Finset (Φ.HiddenSeeds J) := by |
| classical |
| exact Finset.univ.filter (fun hidden => Φ.query (Φ.spliceSeeds J frozen hidden) j = i) |
|
|
| @[simp] theorem mem_ownFiber (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d)) |
| (frozen : Φ.Seeds d) (j : J) (i : Φ.Query d) (hidden : Φ.HiddenSeeds J) : |
| hidden ∈ Φ.ownFiber J frozen j i ↔ Φ.query (Φ.spliceSeeds J frozen hidden) j = i := by |
| classical |
| simp [ownFiber] |
|
|
| noncomputable def ownMean (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) (s : Fin (Φ.WeightDimension d) → ℝ) : ℝ := by |
| classical |
| exact VertexCover.finiteMean (fun hidden : Φ.ownFiber J frozen j i => |
| ∫ other, Φ.batchFunction (Φ.spliceSeeds J frozen hidden) J c0 A hA |
| (Function.update other j s) ∂Φ.batchLaw J) |
|
|
| noncomputable def ownGradient (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) (s : Fin (Φ.WeightDimension d) → ℝ) |
| (k : Fin (Φ.WeightDimension d)) : ℝ := by |
| classical |
| exact VertexCover.finiteMean (fun hidden : Φ.ownFiber J frozen j i => |
| ∫ other, Φ.canonicalGradient (Φ.spliceSeeds J frozen hidden) J c0 A hA |
| (Function.update other j s) j k ∂Φ.batchLaw J) |
|
|
| theorem canonicalGradient_update_measurable (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d) |
| (J : Finset (Fin d)) (c0 : Φ.Coordinate d → ℝ) |
| (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) |
| (j : J) (s : Fin (Φ.WeightDimension d) → ℝ) (k : Fin (Φ.WeightDimension d)) : |
| Measurable (fun other : Φ.BatchWeights J => |
| Φ.canonicalGradient seed J c0 A hA (Function.update other j s) j k) := by |
| classical |
| exact (measurable_pi_apply k).comp ((measurable_pi_apply j).comp |
| ((Φ.canonicalGradient_measurable seed J c0 A hA).comp |
| (continuous_id.update j continuous_const).measurable)) |
|
|
| theorem ownGradient_block_bound (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) (s : Fin (Φ.WeightDimension d) → ℝ) : |
| ∑ k, |Φ.ownGradient J frozen c0 A hA j i s k| ≤ 1 := by |
| classical |
| apply VertexCover.Average.finite_l1_le |
| intro hidden |
| apply VertexCover.Average.integral_l1_le |
| · intro k |
| exact (Φ.canonicalGradient_update_measurable _ J c0 A hA j s k).aestronglyMeasurable |
| · intro other |
| exact Φ.canonicalGradient_block_bound _ J c0 A hA (Function.update other j s) j |
|
|
| theorem ownGradient_unused (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) (s : Fin (Φ.WeightDimension d) → ℝ) |
| (k : Fin (Φ.WeightDimension d)) (hk : ∀ a : i.LocalLabel, i.slot a ≠ k) : |
| Φ.ownGradient J frozen c0 A hA j i s k = 0 := by |
| classical |
| unfold ownGradient VertexCover.finiteMean |
| have hz : ∀ hidden : Φ.ownFiber J frozen j i, ∀ other : Φ.BatchWeights J, |
| Φ.canonicalGradient (Φ.spliceSeeds J frozen hidden) J c0 A hA |
| (Function.update other j s) j k = 0 := by |
| intro hidden other |
| apply Φ.canonicalGradient_unused |
| have hq := (Φ.mem_ownFiber J frozen j i hidden).mp hidden.property |
| rw [hq] |
| exact hk |
| simp only [hz, integral_zero, Finset.sum_const_zero, zero_div] |
|
|
| theorem ownGradient_unsupported (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) (hi : Φ.ownFiber J frozen j i = ∅) |
| (s : Fin (Φ.WeightDimension d) → ℝ) : Φ.ownGradient J frozen c0 A hA j i s = 0 := by |
| classical |
| funext k |
| unfold ownGradient VertexCover.finiteMean |
| simp only [Fintype.card_coe, hi, Finset.card_empty, Nat.cast_zero, div_zero, Pi.zero_apply] |
|
|
| end VertexCover.LabelCover |
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| end |
| end |
| end |
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| end |
| end |
|
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| end OAI |
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|
