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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 | import Mathlib
import OAI.Probability.LogConcave.Analysis.GaussianAffineNormMoment
import OAI.Probability.LogConcave.Dynamics.Squared
import OAI.Probability.LogConcave.OraclePrograms.ReservedProgram
namespace OAI
section
noncomputable section
namespace LogConcaveSampling
open MeasureTheory ProbabilityTheory OracleCompiler Program Coupling
open scoped Classical BigOperators NNReal
variable {d : ℕ}
def terminalMeanProgram (d : ℕ) (r σ : ℝ) : SeedProgram d where
slots := 2
calls := 1
program := SeedCompiler.terminalMean d r σ
shift := SeedCompiler.terminalShift 1
def terminalReservedProgram (d : ℕ) (r η : ℝ) : ReservedProgram d where
slots := 2
calls := 1
pre := oneQuery (fun p => p.1+r • p.2 0) (by fun_prop)
(fun p => p.1.2 0-r • p.2.2+(Real.sqrt 3*η/2) • p.1.2 1) (by fun_prop)
shift := SeedCompiler.terminalShift 1
reserve := η/2
lemma terminalReservedProgram_pre_run (r η : ℝ) (V : Point d → ℝ)
(x : Point d) (g : Fin 2 → Point d) :
(terminalReservedProgram d r η).pre.run V (x,g)=
g 0-r • gradient V (x+r • g 0)+(Real.sqrt 3*η/2) • g 1 := by
exact oneQuery_run (fun p : Point d × (Fin 2 → Point d) => p.1+r • p.2 0) (by fun_prop)
(fun p => p.1.2 0-r • p.2.2+(Real.sqrt 3*η/2) • p.1.2 1) (by fun_prop) V (x,g)
abbrev gaussianTape (d n : ℕ) : Measure (Fin n → Point d) :=
Measure.pi fun _ => stdGaussian (Point d)
variable {slotCount : ℕ}
lemma gaussianTape_eval (i : Fin slotCount) :
(gaussianTape d slotCount).map (fun g => g i)=stdGaussian (Point d) :=
(measurePreserving_eval (fun _ : Fin slotCount => stdGaussian (Point d)) i).map_eq
lemma gaussianTape_integrable_eval (f : Point d → ℝ) (_hf : Measurable f)
(hi : Integrable f (stdGaussian (Point d))) (i : Fin slotCount) :
Integrable (fun g : Fin slotCount → Point d => f (g i)) (gaussianTape d slotCount) := by
have hh : Integrable f ((gaussianTape d slotCount).map (fun g => g i)) := by rw [gaussianTape_eval]; exact hi
exact hh.comp_measurable (measurable_pi_apply i)
lemma gaussianTape_integral_eval (f : Point d → ℝ) (hf : Measurable f) (i : Fin slotCount) :
(∫g : Fin slotCount → Point d,f (g i) ∂gaussianTape d slotCount)=∫z,f z ∂stdGaussian (Point d) :=
(measurePreserving_eval (fun _ : Fin slotCount => stdGaussian (Point d)) i).hasLaw.integral_comp hf.aestronglyMeasurable
theorem terminalMeanProgram_squared {F : Point d → ℝ} {lam : ℝ≥0} (hF : Primitive F lam)
(x : Point d) {r : ℝ} (hr : 0≤r) (hl : (lam:ℝ)*r^2≤1/2) (σ : ℝ) :
SquaredAt (gaussianTape d 2) (stdGaussian (Point d))
(fun g => (terminalMeanProgram d r σ).program.run F (x,g))
(fun g => primitiveExpectedField F x r+σ • g)
((2*((lam:ℝ)*r))^2*(2*(Real.pi*Real.sqrt d+r*‖primitiveField F x r 0‖)^2+2*d)) := by
have hm : Measurable (fun z => ‖primitiveField F x r z-primitiveExpectedField F x r‖^2) :=
((primitiveField_contDiff hF x r).continuous.measurable.sub measurable_const).norm.pow_const 2
have he (g : Fin 2 → Point d) :
‖(terminalMeanProgram d r σ).program.run F (x,g)-
(primitiveExpectedField F x r+σ • g 1)‖^2=
‖primitiveField F x r (g 0)-primitiveExpectedField F x r‖^2 := by
change ‖(SeedCompiler.terminalMean d r σ).run F (x,g)-_‖^2=_
rw [SeedCompiler.terminalMean_run,add_sub_add_right_eq_sub]
rfl
have ht := terminal_mean_squared_error hF x hr hl
apply SquaredAt.of_joint (gaussianTape d 2) (gaussianTape d 2) (stdGaussian (Point d))
id (fun g => g 1) measurable_id (measurable_pi_apply 1) Measure.map_id (gaussianTape_eval 1)
_ _ ?_ (by fun_prop)
· simpa only [id_eq,he] using gaussianTape_integrable_eval _ hm ht.1 (0:Fin 2)
· simpa only [id_eq,he,gaussianTape_integral_eval _ hm (0:Fin 2)] using ht.2
· change Measurable (fun g => (SeedCompiler.terminalMean d r σ).run F (x,g))
simp only [SeedCompiler.terminalMean_run]
have hg := hF.gradient_lipschitz.continuous.measurable
fun_prop
end LogConcaveSampling
end
end
end OAI
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