| 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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|
