openAI-math-dataset / lean /OAI /Computability /VertexCover /Analysis /CanonicalGradientInvariant.lean
| import Mathlib | |
| import OAI.Computability.VertexCover.Analysis.CanonicalGradientValid | |
| namespace OAI | |
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| namespace VertexCover.LabelCover | |
| open MeasureTheory | |
| theorem canonicalGradient_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) : | |
| Measurable (Φ.canonicalGradient seed J c0 A hA) := by | |
| classical | |
| apply Measurable.ite (VertexCover.AffineSelector.regular_measurableSet _ _) | |
| · apply Measurable.of_eval; intro batchIndex | |
| apply Measurable.of_eval; intro weightIndex | |
| exact (ContinuousLinearMap.apply ℝ ℝ | |
| (Pi.single batchIndex (Pi.single weightIndex 1))).measurable.comp | |
| (measurable_fderiv ℝ (Φ.batchFunction seed J c0 A hA)) | |
| · exact measurable_const | |
| theorem canonicalGradient_ae_derivative (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) | |
| (μ : Measure (Φ.BatchWeights J)) [μ.IsAddHaarMeasure] : | |
| ∀ᵐ s ∂μ, DifferentiableAt ℝ (Φ.batchFunction seed J c0 A hA) s ∧ | |
| Φ.canonicalGradient seed J c0 A hA s = | |
| Φ.batchGradient J (fderiv ℝ (Φ.batchFunction seed J c0 A hA) s) := by | |
| classical | |
| filter_upwards [VertexCover.AffineSelector.ae_regular μ (Φ.affineSlope seed J) | |
| (Φ.affineOffset c0 (A := A))] with s hs | |
| obtain ⟨im, ip, hder⟩ := Φ.batchFunction_hasFDerivAt seed J c0 A hA s hs | |
| exact ⟨hder.differentiableAt, by simp [canonicalGradient, hs]⟩ | |
| theorem canonicalGradient_block_bound (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) (s : Φ.BatchWeights J) (j : J) : | |
| ∑ k, |Φ.canonicalGradient seed J c0 A hA s j k| ≤ 1 := by | |
| obtain ⟨G, hG, heq⟩ := Φ.canonicalGradient_valid seed J c0 A hA s | |
| simp_rw [heq] | |
| exact Φ.hull_block_bound seed hG j | |
| theorem canonicalGradient_unused (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) (s : Φ.BatchWeights J) | |
| (j : J) (k : Fin (Φ.WeightDimension d)) | |
| (hk : ∀ a : (Φ.query seed j).LocalLabel, (Φ.query seed j).slot a ≠ k) : | |
| Φ.canonicalGradient seed J c0 A hA s j k = 0 := by | |
| obtain ⟨G, hG, heq⟩ := Φ.canonicalGradient_valid seed J c0 A hA s | |
| rw [heq] | |
| exact Φ.hull_unused seed hG j k hk | |
| theorem canonicalGradient_listMass (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) (s : Φ.BatchWeights J) | |
| (L : Φ.PrivateLists d) : | |
| (∑ j : J, ∑ a ∈ (L (Φ.query seed j)).toFinset, | |
| |Φ.canonicalGradient seed J c0 A hA s j ((Φ.query seed j).slot a)|) ≤ | |
| Φ.listHitMaximum L seed J := by | |
| classical | |
| obtain ⟨G, hG, heq⟩ := Φ.canonicalGradient_valid seed J c0 A hA s | |
| simp_rw [heq] | |
| exact (Finset.sum_coe_sort J (fun j => | |
| ∑ a ∈ (L (Φ.query seed j)).toFinset, |G j ((Φ.query seed j).slot a)|)).trans_le | |
| (Φ.listMass_le_hitMaximum L seed J hG) | |
| theorem canonicalGradient_query_invariant (Φ : LabelCover) {d : ℕ} | |
| (seed seed' : Φ.Seeds d) (J : Finset (Fin d)) | |
| (hq : ∀ j : J, Φ.query seed j = Φ.query seed' j) | |
| (c0 : Φ.Coordinate d → ℝ) (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) : | |
| Φ.canonicalGradient seed J c0 A hA = Φ.canonicalGradient seed' J c0 A hA := by | |
| have hS : Φ.batchSum seed J = Φ.batchSum seed' J := by | |
| funext s | |
| unfold batchSum | |
| apply Finset.sum_congr rfl | |
| intro j hj | |
| rw [hq j] | |
| have hC : Φ.batchSumCLM seed J = Φ.batchSumCLM seed' J := by | |
| ext s p | |
| exact congrFun (congrFun hS s) p | |
| have hA' : (Φ.affineSlope seed J (A := A)) = Φ.affineSlope seed' J := by | |
| funext i | |
| simp only [affineSlope, hC] | |
| have hf : Φ.batchFunction seed J c0 A hA = Φ.batchFunction seed' J c0 A hA := by | |
| funext s | |
| simp only [batchFunction, hS] | |
| funext s | |
| unfold canonicalGradient | |
| have hr : VertexCover.AffineSelector.Regular (Φ.affineSlope seed J) | |
| (Φ.affineOffset c0 (A := A)) s ↔ | |
| VertexCover.AffineSelector.Regular (Φ.affineSlope seed' J) | |
| (Φ.affineOffset c0 (A := A)) s := by rw [hA'] | |
| simp only [hf] | |
| split_ifs <;> first | rfl | (exfalso; tauto) | |
| end VertexCover.LabelCover | |
| end | |
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| end | |
| end OAI | |