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dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 | 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
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