openAI-math-dataset / lean /OAI /Probability /LogConcave /OraclePrograms /NoisyCenterSquaredError.lean
| import Mathlib | |
| import OAI.Probability.LogConcave.OraclePrograms.TerminalMean | |
| import OAI.Probability.LogConcave.Numerics.NormSubSqTwo | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| namespace LogConcaveSampling.SeedCompiler | |
| open MeasureTheory ProbabilityTheory Function | |
| open scoped BigOperators | |
| variable {Ω A ι E : Type*} [MeasurableSpace Ω] [MeasurableSpace A] | |
| [Fintype ι] [DecidableEq ι] | |
| [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] | |
| [MeasurableSpace E] [BorelSpace E] | |
| theorem noisy_center_squared_error (κ : Measure Ω) [SFinite κ] | |
| (μ : Measure A) [SFinite μ] (anchor : Ω → A) (noise center : Ω → E) | |
| (ha : Measurable anchor) (hz : Measurable noise) (hc : Measurable center) | |
| (hlaw : κ.map (fun w => (anchor w,noise w))=μ.prod (stdGaussian E)) | |
| (x : A → E) (hx : Measurable x) {r D K S B : ℝ} | |
| (hD : 0<D) (hK : 0≤K) (v : ι → ℝ) (hv : (∑i,v i^2)≤D^2) | |
| (M : E → (ι → E) → E) (hm : Measurable (uncurry M)) | |
| (hshift : ShiftInvariant r v M) | |
| (hLip : ∀x y g,‖M x g-M y g‖≤K*‖x-y‖) | |
| (Q : A × (ι → E) → E) (hQ : Measurable Q) | |
| (hci : Integrable (fun w => ‖center w-(x (anchor w)+(r/(2*D)) • noise w)‖^2) κ) | |
| (hce : (∫w,‖center w-(x (anchor w)+(r/(2*D)) • noise w)‖^2 ∂κ)≤S) | |
| (hbi : Integrable (fun z : A × (ι → E) => ‖M (x z.1) z.2-Q z‖^2) | |
| (μ.prod (Measure.pi fun _ : ι => stdGaussian E))) | |
| (hbe : (∫z : A × (ι → E),‖M (x z.1) z.2-Q z‖^2 | |
| ∂μ.prod (Measure.pi fun _ : ι => stdGaussian E))≤B) : | |
| let a := fun i => v i/(2*D) | |
| let T := fun z : Ω × (ι → E) => (anchor z.1,restored (sqrtCoefficient a) a (z.2,noise z.1)) | |
| Integrable (fun z : Ω × (ι → E) => | |
| ‖M (center z.1) (slotTransform (sqrtCoefficient a) a z.2)-Q (T z)‖^2) | |
| (κ.prod (Measure.pi fun _ : ι => stdGaussian E)) ∧ | |
| (∫z : Ω × (ι → E),‖M (center z.1) (slotTransform (sqrtCoefficient a) a z.2)-Q (T z)‖^2 | |
| ∂κ.prod (Measure.pi fun _ : ι => stdGaussian E))≤2*K^2*S+2*B := by | |
| dsimp only | |
| let a : ι → ℝ := fun i => v i/(2*D) | |
| let γ : Measure (ι → E) := Measure.pi fun _ => stdGaussian E | |
| let T : Ω × (ι → E) → A × (ι → E) := fun z => | |
| (anchor z.1,restored (sqrtCoefficient a) a (z.2,noise z.1)) | |
| let C : Ω → ℝ := fun w => ‖center w-(x (anchor w)+(r/(2*D)) • noise w)‖^2 | |
| let J : A × (ι → E) → ℝ := fun z => ‖M (x z.1) z.2-Q z‖^2 | |
| let F : Ω × (ι → E) → ℝ := fun z => | |
| ‖M (center z.1) (slotTransform (sqrtCoefficient a) a z.2)-Q (T z)‖^2 | |
| have ha2 : (∑i,a i^2)≤1 := (normalized_shift_bound hD v hv).trans (by norm_num) | |
| have measurableError : Measurable J := | |
| ((hm.comp ((hx.comp measurable_fst).prodMk measurable_snd)).sub hQ).norm.pow_const 2 | |
| have hJ := restored_error_integral κ μ anchor noise ha hz hlaw a ha2 J | |
| ⟨measurableError.aestronglyMeasurable, hbi.hasFiniteIntegral⟩ | |
| have hC : Integrable (fun z : Ω × (ι → E) => C z.1) (κ.prod γ) := hci.comp_fst γ | |
| have hJ' : Integrable (fun z => J (T z)) (κ.prod γ) := hJ.1 | |
| have hdom : Integrable (fun z : Ω × (ι → E) => 2*K^2*C z.1+2*J (T z)) (κ.prod γ) := | |
| (hC.const_mul _).add (hJ'.const_mul _) | |
| have hTm : Measurable T := (ha.comp measurable_fst).prodMk | |
| ((restored (sqrtCoefficient a) a).measurable.comp | |
| (measurable_snd.prodMk (hz.comp measurable_fst))) | |
| have hPm : Measurable (slotTransform (sqrtCoefficient a) a : (ι → E) → (ι → E)) := by | |
| unfold slotTransform slotSum | |
| fun_prop | |
| have hFm : Measurable F := by | |
| exact ((hm.comp ((hc.comp measurable_fst).prodMk (hPm.comp measurable_snd))).sub | |
| (hQ.comp hTm)).norm.pow_const 2 | |
| have hb (z : Ω × (ι → E)) : F z≤2*K^2*C z.1+2*J (T z) := by | |
| let u := M (center z.1) (slotTransform (sqrtCoefficient a) a z.2) | |
| let w := M (x (anchor z.1)) (T z).2 | |
| have hw : M (x (anchor z.1)+(r/(2*D)) • noise z.1) | |
| (slotTransform (sqrtCoefficient a) a z.2)=w := | |
| absorption_pathwise v M hshift (x (anchor z.1)) _ z.2 (noise z.1) | |
| have hd : ‖u-w‖≤K*‖center z.1-(x (anchor z.1)+(r/(2*D)) • noise z.1)‖ := by | |
| rw [←hw] | |
| exact hLip _ _ _ | |
| have hsq := (sq_le_sq₀ (norm_nonneg (u-w)) | |
| (mul_nonneg hK (norm_nonneg _))).2 hd | |
| rw [mul_pow] at hsq | |
| have ht := RMSIntegral.norm_sub_sq_le_two (u-w) (Q (T z)-w) | |
| rw [show u-w-(Q (T z)-w)=u-Q (T z) from by abel,norm_sub_rev (Q (T z)) w] at ht | |
| change ‖u-Q (T z)‖^2≤2*K^2*C z.1+2*‖w-Q (T z)‖^2 | |
| dsimp only [C] at * | |
| nlinarith | |
| have hi : Integrable F (κ.prod γ) := hdom.mono' hFm.aestronglyMeasurable | |
| (Filter.Eventually.of_forall (fun z => by simpa only [F,Real.norm_eq_abs,abs_sq] using hb z)) | |
| refine ⟨hi,(integral_mono hi hdom hb).trans ?_⟩ | |
| rw [integral_add (hC.const_mul _) (hJ'.const_mul _),integral_const_mul,integral_const_mul] | |
| have hCeq : (∫z : Ω × (ι → E),C z.1 ∂κ.prod γ)=∫w,C w ∂κ := by | |
| rw [integral_prod _ hC] | |
| simp only [integral_const,probReal_univ,one_smul] | |
| rw [hCeq,hJ.2] | |
| exact add_le_add (mul_le_mul_of_nonneg_left hce (by positivity)) | |
| (mul_le_mul_of_nonneg_left hbe (by norm_num)) | |
| end LogConcaveSampling.SeedCompiler | |
| end | |
| end | |
| end | |
| end OAI | |