| import Mathlib |
| import OAI.Probability.LogConcave.Numerics.OrdinaryDerivativeWeight |
| import OAI.Probability.LogConcave.Sampling.TerminalActionChainQuotient |
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| namespace OAI |
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| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open Set MeasureTheory ProbabilityTheory Quadrature RMSIntegral |
| open scoped Classical BigOperators NNReal |
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| lemma terminalDerivativeBudget_polylog (n d : ℕ) (lam : ℝ≥0) (r : ℝ) : |
| terminalDerivativeBudget n d lam r≤(lam:ℝ)*r^2* |
| (Real.exp (Real.pi^2/4)*(TensorSum.terminalQ n).val.logCoefficient 0 2)* |
| (1+Real.log ((d:ℝ)+1))^((TensorSum.terminalQ n).val.logDegree 0) := by |
| have he := mul_le_mul_of_nonneg_left |
| ((TensorSum.terminalQ n).val.momentBudget_polylog d 0 2) |
| (show 0≤r*((lam:ℝ)*r)*Real.exp (Real.pi^2/4) by |
| have hh : 0≤r*((lam:ℝ)*r) := by nlinarith [sq_nonneg r,lam.coe_nonneg] |
| positivity) |
| convert! he using 1 |
| ring |
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|
| lemma logMeshCount_mul_step {T h : ℝ} (hT0 : 0<T) (hT1 : T<1) |
| (hh : 0<h) (hL : h≤logMeshLength T) : |
| (logMeshCount T h:ℝ)*h≤2*logMeshLength T := by |
| have he := mul_lt_mul_of_pos_right (logMeshCount_bound hT0 hT1 hh) hh |
| have hh0 := ne_of_gt hh |
| field_simp at he |
| nlinarith |
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| def kernelAnalyticBudget (n d : ℕ) (lam : ℝ≥0) (r T h D : ℝ) : ℝ := |
| 32*(1+D^2+(∑i,|zeroHermiteWeight (probabilityNodes (n+1)) 0 i|)^2)* |
| (2*logMeshLength T*terminalDerivativeBudget (n+1) d lam r*2^(n+1)*h^n)^2*d |
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| variable {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r T h : ℝ} (hr : 0<r) (hlam : 0<lam) |
| (hl : (lam:ℝ)*r^2≤1/2) (hT0 : 0<T) (hT1 : T<1) |
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| include hlam in |
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| theorem terminalQuadrature_kernel_rms (hd : 1≤d) (n : ℕ) (D : ℝ) (hD : 1≤D) |
| (hDu : ∀t∈Icc (0:ℝ) 1,∑i,|basisDerivative (probabilityNodes (n+1)) i t|≤D) |
| (hh : 0<h) (hs : h≤Real.log 2) (hL : h≤logMeshLength T) : |
| let J := terminalActionChain hF x hr hl hT0 hT1 |
| let μ := (interpolationLaw F x r T).prod (stdGaussian (Point d)) |
| let err := fun z => (∑s,terminalQuadratureWeight T h n s • |
| J 0 (probabilityNodeTime T h (n+1) s,z))- |
| r • (centeringKernel hF x hr hl hT0.le hT1 z.1).adjoint z.2 |
| Integrable (fun z => ‖err z‖^2) μ ∧ |
| (∫z,‖err z‖^2 ∂μ)≤kernelAnalyticBudget n d lam r T h D := by |
| dsimp only |
| have he := terminalQuadrature_analytic_rms hF x hr hlam hl hT0 hT1 hd n D hD hDu hh hs hL |
| dsimp only at he |
| simp_rw [terminalActionChain_integral hF x hr hlam hl hT0 hT1,norm_sub_rev] at he |
| refine ⟨he.1,he.2.trans ?_⟩ |
| have hN := logMeshCount_mul_step hT0 hT1 hh hL |
| have hmul := mul_le_mul_of_nonneg_right hN |
| (show 0≤terminalDerivativeBudget (n+1) d lam r*2^(n+1)*h^n from |
| mul_nonneg (mul_nonneg (terminalDerivativeBudget_nonneg _ _ _ _) (by positivity)) (by positivity)) |
| have hsq := pow_le_pow_left₀ |
| (show 0≤(logMeshCount T h:ℝ)*h*(terminalDerivativeBudget (n+1) d lam r*2^(n+1)*h^n) by |
| exact mul_nonneg (mul_nonneg (Nat.cast_nonneg _) hh.le) |
| (by positivity [terminalDerivativeBudget_nonneg (n+1) d lam r])) hmul 2 |
| have hscaled := mul_le_mul_of_nonneg_right |
| (mul_le_mul_of_nonneg_left hsq |
| (show 0≤32*(1+D^2+(∑i,|zeroHermiteWeight (probabilityNodes (n+1)) 0 i|)^2) by positivity)) |
| (Nat.cast_nonneg (α:=ℝ) d) |
| convert! hscaled using 1 <;> (try dsimp only [kernelAnalyticBudget]) <;> (try simp only [mul_pow,pow_succ]) <;> first | rfl | ring |
| end LogConcaveSampling |
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| end |
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| end |
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| end OAI |
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