| import Mathlib |
| import OAI.Probability.SKBarriers.Parisi.CDFOverlapConvergence |
| import OAI.Probability.SKBarriers.Parisi.CDFSemigroup |
| import OAI.Probability.SKBarriers.Scalar.ScalarHierarchyCramer |
| import OAI.Probability.SKBarriers.Parisi.CDFEndpoint |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
| attribute [local instance 2000] parameterNormedGroup parameterNormedSpace |
|
|
| theorem scalarCDFValue_rootGradient (β : ℝ) {α : ℝ → ℝ} |
| (ha : ∀ z, α z∈Icc (0:ℝ) 1) (hm : Monotone α) |
| (s : ℝ) (t : ℝ≥0) (ht : t ≤ 1) : |
| rootGradient 0 (scalarCDFValue β α s t)=scalarCDFGradient β α s t := by |
| funext x |
| exact ((scalarCDFValue_regular β ha hm s t ht).1.differentiable (by norm_num) x |>.hasDerivAt).unique |
| (scalarCDFValue_hasDerivAt β ha hm s t ht x) |
|
|
| theorem scalarCDFValue_rootHessian (β : ℝ) {α : ℝ → ℝ} |
| (ha : ∀ z, α z∈Icc (0:ℝ) 1) (hm : Monotone α) |
| (s : ℝ) (t : ℝ≥0) (ht : t ≤ 1) : |
| rootHessian 0 (scalarCDFValue β α s t)=scalarCDFHessian β α s t := by |
| funext x |
| have H : HasDerivAt (rootGradient 0 (scalarCDFValue β α s t)) |
| (rootHessian 0 (scalarCDFValue β α s t) x) x := by |
| simpa only [parameterAxis,smul_eq_mul,mul_one,zero_add] using |
| rootGradientLine_hasDerivAt 0 (scalarCDFValue_regular β ha hm s t ht) 0 x |
| rw [scalarCDFValue_rootGradient β ha hm s t ht] at H |
| exact H.unique (scalarCDFGradient_hasDerivAt β ha hm s t ht x) |
|
|
| theorem scalarCDFValue_spinConvex (β : ℝ) {α : ℝ → ℝ} |
| (ha : ∀ z, α z∈Icc (0:ℝ) 1) (hm : Monotone α) |
| (s : ℝ) (t : ℝ≥0) (ht : t ≤ 1) : ScalarSpinConvex (scalarCDFValue β α s t) := by |
| intro x |
| rw [scalarCDFValue_rootGradient β ha hm s t ht,scalarCDFValue_rootHessian β ha hm s t ht] |
| exact (scalarCDF_derivative_bounds β ha hm s t ht x).2 |
|
|
| theorem chain_coefficients_variance (β : ℝ) (l : List (ℝ × ℝ≥0)) : |
| (∑ i : Fin l.length, (β*Real.sqrt ((l.get i).2:ℝ))^2)=β^2*(chainDuration l:ℝ) := by |
| simp only [mul_pow,Real.sq_sqrt (NNReal.coe_nonneg _),← Finset.mul_sum] |
| congr 1 |
| induction l with |
| | nil => simp [chainDuration] |
| | cons p l ih => simpa [Fin.sum_univ_succ,chainDuration] using congrArg ((p.2:ℝ)+·) ih |
|
|
| theorem scalarTimeChain_cramer {f : ℝ → ℝ} (hf : BoundedDerivs f) |
| (hc : ScalarSpinConvex f) (β : ℝ) (l : List (ℝ × ℝ≥0)) |
| (hm : ∀ p∈l, p.1∈Icc (0:ℝ) 1) (x : ℝ) : |
| let V := scalarTimeChainAverage β l f (fun y => (rootGradient 0 f y)^2) x- |
| (rootGradient 0 (scalarTimeChain β l f) x)^2 |
| β^2*(chainDuration l:ℝ)*(max (rootHessian 0 (scalarTimeChain β l f) x-V) 0)^2 ≤ V := by |
| have H := scalarHierarchy_cramer hf hc l.length (fun i => (l.get i).1) |
| (fun i => β*Real.sqrt ((l.get i).2:ℝ)) (fun i => hm _ (List.get_mem _ _)) x |
| dsimp only at H ⊢ |
| rw [chain_coefficients_variance] at H |
| simpa only [scalarTimeChainAverage,scalarTimeChain_eq_hierarchy] using H |
|
|
| theorem scalarCDFOverlap_cramer_quantile {k : ℕ} (β : ℝ) (Q : Fin (k+1) → ℝ) |
| (hQ : Q∈admissibleQuantiles k) {q : ℝ} (hq : q∈Icc (0:ℝ) 1) : |
| β^2*q*(max (scalarCDFHessian β (quantileCDF k Q) 0 1 0-scalarCDFOverlap β (quantileCDF k Q) q) 0)^2 ≤ |
| scalarCDFOverlap β (quantileCDF k Q) q := by |
| let α := quantileCDF k Q |
| have ha := quantileCDF_bounds k Q |
| have hm := quantileCDF_monotone k Q |
| let t := Real.toNNReal q |
| let r := Real.toNNReal (1-q) |
| have ht : (t:ℝ)=q := Real.coe_toNNReal q hq.1 |
| have hr : (r:ℝ)=1-q := Real.coe_toNNReal _ (sub_nonneg.mpr hq.2) |
| have ht1 : t ≤ 1 := Real.toNNReal_le_iff_le_coe.mpr hq.2 |
| have hr1 : r ≤ 1 := Real.toNNReal_le_iff_le_coe.mpr (by simp only [NNReal.coe_one]; linarith [hq.1]) |
| obtain ⟨l,hd,hl,hmodel,_⟩ := quantile_full_chain Q hQ.1 (hQ.2 0).1 (hQ.2 (Fin.last k)).2 |
| have htr : chainDuration l=t+r := by |
| rw [hd]; apply NNReal.coe_injective |
| rw [NNReal.coe_add,ht,hr,NNReal.coe_one]; ring |
| obtain ⟨l₁,l₂,h₁,h₂,hm₁,hm₂,hs₁,hs₂,he⟩ := scalarTimeChain_split_at β α l hl 0 hmodel t r htr |
| let f := scalarCDFValue β α q r |
| have hf : BoundedDerivs f := scalarCDFValue_regular β ha hm q r hr1 |
| have hfc : ScalarSpinConvex f := scalarCDFValue_spinConvex β ha hm q r hr1 |
| have hLip := scalarCDFValue_lipschitz β ha hm q r hr1 |
| have hroot : scalarTimeChain β l₁ f=scalarCDFValue β α 0 1 := by |
| funext x |
| rw [← scalarCDFOperator_eq_chain hf hLip β ha hm l₁ hm₁ 0 t ht1 h₁ hs₁ x] |
| exact (congrFun (scalarCDFValue_semigroup_quantile β Q hQ hq) x).symm |
| have htest := scalarCDFGradient_sq_regular β ha hm q r hr1 |
| have hav : scalarTimeChainAverage β l₁ f (fun y => (rootGradient 0 f y)^2) 0= |
| scalarCDFOverlap β α q := by |
| change scalarTimeChainAverage β l₁ (scalarCDFValue β α q r) _ 0=_ |
| rw [scalarCDFValue_rootGradient β ha hm q r hr1] |
| exact (scalarCDFAverage_eq_chainAverage hf htest.1 htest.2.1 hLip |
| (B:=1) htest.2.2.1 (C:=2) htest.2.2.2 β ha hm l₁ hm₁ 0 t ht1 h₁ hs₁ 0).symm |
| have H := scalarTimeChain_cramer hf hfc β l₁ hm₁ 0 |
| dsimp only at H |
| rw [h₁,ht,hav,hroot,scalarCDFValue_rootGradient β ha hm 0 1 le_rfl, |
| scalarCDFValue_rootHessian β ha hm 0 1 le_rfl,scalarCDFGradient_zero β ha hm 0 1 le_rfl, |
| zero_pow (by decide : 2≠0),sub_zero] at H |
| exact H |
|
|
| theorem scalarCDFOverlap_cramer (β : ℝ) (α : StieltjesFunction ℝ) |
| (ha : ∀ z, α z∈Icc (0:ℝ) 1) (h1 : α 1=1) {q : ℝ} (hq : q∈Icc (0:ℝ) 1) : |
| β^2*q*(max (scalarCDFHessian β α 0 1 0-scalarCDFOverlap β α q) 0)^2 ≤ scalarCDFOverlap β α q := by |
| let a (n : ℕ) := quantileCDF n (uniformCDFQuantiles n α) |
| have hn (n : ℕ) := quantileCDF_bounds n (uniformCDFQuantiles n α) |
| have hnm (n : ℕ) := quantileCDF_monotone n (uniformCDFQuantiles n α) |
| have hD := uniformCDFQuantiles_L1_tendsto α ha h1 |
| have Hχ := (scalarCDFHessian_tendstoUniformly β ha α.mono hn hnm hD).tendsto_at 0 |
| have HΓ := (scalarCDFOverlap_tendstoUniformlyOn β ha α.mono hn hnm hD).tendsto_at hq |
| have HL := ((Hχ.sub HΓ).max (tendsto_const_nhds (x := (0:ℝ)))).pow 2 |>.const_mul (β^2*q) |
| exact le_of_tendsto_of_tendsto HL HΓ (Eventually.of_forall (fun n => |
| scalarCDFOverlap_cramer_quantile β (uniformCDFQuantiles n α) (uniformCDFQuantiles_admissible n α h1) hq)) |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
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