| import Mathlib |
| import OAI.Probability.SKBarriers.Scalar.ScalarJointLoss |
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| namespace OAI |
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| section |
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| noncomputable section |
| open scoped BigOperators NNReal Topology |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
| attribute [local instance 2000] parameterNormedGroup parameterNormedSpace |
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| theorem scalarHierarchy_hessian_square_cramer {f : ℝ → ℝ} |
| (hf : BoundedDerivs f) (hc : ScalarSpinConvex f) (n : ℕ) (m v : Fin n → ℝ) |
| (hm : ∀ i,m i∈Icc (0:ℝ) 1) (x : ℝ) : |
| (∑ i,(v i)^2)*(rootHessian 0 (scalarHierarchy n m v f) x)^2 ≤ |
| (1+2*(∑ i,(v i)^2))*(scalarHierarchyAverage n m v f |
| (fun y => (rootGradient 0 f y)^2) x-(rootGradient 0 (scalarHierarchy n m v f) x)^2) := by |
| let S := ∑ i,(v i)^2 |
| let K := scalarHierarchyAverage n m v f (rootHessian 0 f) x |
| let H := rootHessian 0 (scalarHierarchy n m v f) x |
| let V := scalarHierarchyAverage n m v f (fun y => (rootGradient 0 f y)^2) x- |
| (rootGradient 0 (scalarHierarchy n m v f) x)^2 |
| have hS : 0≤S := Finset.sum_nonneg (fun _ _ => sq_nonneg _) |
| have hd := scalarHierarchy_cramer_data hf hc n m v hm x |
| change 0≤K ∧ K≤H ∧ H≤K+V ∧ S*K^2≤V at hd |
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| have hH1 : H≤1 := (le_abs_self H).trans ((hc.scalarHierarchy hf n m v hm).bounds x).2 |
| have hV : 0≤V := (mul_nonneg hS (sq_nonneg _)).trans hd.2.2.2 |
| have he : H^2≤K^2+2*V := by |
| have hdiff : 0≤H-K := sub_nonneg.mpr hd.2.1 |
| have hsum : H+K≤2 := by linarith [hd.2.1] |
| have hh := mul_le_mul_of_nonneg_left hsum hdiff |
| nlinarith [hd.2.2.1] |
| have heS := mul_le_mul_of_nonneg_left he hS |
| change S*H^2≤(1+2*S)*V |
| nlinarith [hd.2.2.2] |
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| theorem scalarIncrementAverage_hessian_square_cramer (l j : List (ℝ × ℝ)) |
| {f : ℝ → ℝ} (hf : BoundedDerivs f) (hc : ScalarSpinConvex f) |
| (hj : ∀ p∈j,p.1∈Icc (0:ℝ) 1) : |
| let H := scalarIncrementChain j f |
| rawVariance j*scalarIncrementAverage l H (fun x => (rootHessian 0 H x)^2) 0 ≤ |
| (1+2*rawVariance j)*(scalarIncrementAverage (l++j) f (fun x => (rootGradient 0 f x)^2) 0- |
| scalarIncrementAverage l H (fun x => (rootGradient 0 H x)^2) 0) := by |
| dsimp only |
| let H := scalarIncrementChain j f |
| have hH := scalarIncrementChain_regular j hf |
| have hHc : ScalarSpinConvex H := by |
| change ScalarSpinConvex (scalarIncrementChain j f) |
| rw [scalarIncrementChain_get] |
| exact hc.scalarHierarchy hf _ _ _ (fun i => hj _ (List.get_mem _ _)) |
| have hχ := hHc.hessian_bounded hH |
| have hU := hHc.gradient_bounded hH |
| have hV := (scalarIncrementAverage_bounded j hf (hc.gradient_bounded hf).sq).sub hU.sq |
| have hpoint (x : ℝ) : rawVariance j*(rootHessian 0 H x)^2 ≤ |
| (1+2*rawVariance j)*(scalarIncrementAverage j f (fun y => (rootGradient 0 f y)^2) x- |
| (rootGradient 0 H x)^2) := by |
| have G := scalarHierarchy_hessian_square_cramer hf hc j.length |
| (fun i => (j.get i).1) (fun i => (j.get i).2) (fun i => hj _ (List.get_mem _ _)) x |
| rw [sum_get_map j (fun p => p.2^2)] at G |
| simpa only [H,scalarIncrementChain_get,scalarIncrementAverage_get,rawVariance] using G |
| have G := scalarIncrementAverage_mono l hH (hχ.sq.const_mul (rawVariance j)) |
| (hV.const_mul (1+2*rawVariance j)) hpoint 0 |
| change rawVariance j*scalarIncrementAverage l H (fun x => (rootHessian 0 H x)^2) 0 ≤ _ |
| have hleft := scalarHierarchyAverage_const_mul hH hχ.sq |
| (n:=l.length) (m:=fun i => (l.get i).1) (v:=fun i => (l.get i).2) |
| (c:=rawVariance j) (x:=0) |
| have hright := scalarHierarchyAverage_const_mul hH hV |
| (n:=l.length) (m:=fun i => (l.get i).1) (v:=fun i => (l.get i).2) |
| (c:=1+2*rawVariance j) (x:=0) |
| have hsub := scalarHierarchyAverage_sub hH |
| (scalarIncrementAverage_bounded j hf (hc.gradient_bounded hf).sq) hU.sq |
| (n:=l.length) (m:=fun i => (l.get i).1) (v:=fun i => (l.get i).2) (x:=0) |
| simp only [← scalarIncrementAverage_get] at hleft hright hsub |
| rw [hleft,hright] at G |
| change rawVariance j*scalarIncrementAverage l H (fun x => (rootHessian 0 H x)^2) 0 ≤ |
| (1+2*rawVariance j)*scalarIncrementAverage l H (fun x => scalarIncrementAverage j f |
| (fun y => (rootGradient 0 f y)^2) x-(rootGradient 0 H x)^2) 0 at G |
| rw [hsub] at G |
| rw [scalarIncrementAverage_append] |
| exact G |
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| end SK.Analytic |
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| end |
| end |
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| end OAI |
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