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import OAI.Probability.LogConcave.OraclePrograms.Program
namespace OAI
section
section
noncomputable section
open MeasureTheory Filter
open scoped ENNReal NNReal Topology
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped BigOperators RealInnerProductSpace
namespace LogConcaveSampling.SeedCompiler
open OracleCompiler
open OracleCompiler.Program
def terminalMean (d : ℕ) (r σ : ℝ) :
Program (Point d × (Fin 2 → Point d)) d 1 (Point d) :=
oneQuery (fun p => p.1 + r • p.2 0) (by fun_prop)
(fun p => p.2.2 + σ • p.1.2 1) (by fun_prop)
def terminalSample (d : ℕ) (r η : ℝ) :
Program (Point d × (Fin 3 → Point d)) d 1 (Point d) :=
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 +
(η / 2) • p.1.2 2) (by fun_prop)
lemma terminalMean_run {d : ℕ} (r σ : ℝ) (V : Point d → ℝ)
(x : Point d) (g : Fin 2 → Point d) :
(terminalMean d r σ).run V (x,g) = gradient V (x+r • g 0) + σ • g 1 := by
rw [terminalMean, oneQuery_run]
rfl
lemma terminalSample_run {d : ℕ} (r η : ℝ) (V : Point d → ℝ)
(x : Point d) (g : Fin 3 → Point d) :
(terminalSample d r η).run V (x,g) = g 0 - r • gradient V (x+r • g 0) +
(Real.sqrt 3 * η / 2) • g 1 + (η / 2) • g 2 := by
rw [terminalSample, oneQuery_run]
rfl
def terminalShift (n : ℕ) : Fin (n+1) → ℝ := fun i => if i = 0 then -1 else 0
lemma terminalShift_energy (n : ℕ) : (∑ i, terminalShift n i ^ 2) = 1 := by
simp [terminalShift]
lemma terminalQuery_invariant {d n : ℕ} (x Δ : Point d) (r : ℝ)
(g : Fin (n+1) → Point d) :
(x + r • Δ) + r • (g 0 + terminalShift n 0 • Δ) = x+r • g 0 := by
simp only [terminalShift, ite_true, neg_smul, one_smul]
module
lemma terminalMean_shift {d : ℕ} (r σ : ℝ) (V : Point d → ℝ) :
ShiftInvariant r (terminalShift 1) (fun x g => (terminalMean d r σ).run V (x,g)) := by
intro x Δ g
simp only [terminalMean_run, terminalQuery_invariant]
simp [terminalShift]
lemma terminalSample_shift {d : ℕ} (r η : ℝ) (V : Point d → ℝ)
(x Δ : Point d) (g : Fin 3 → Point d) :
(terminalSample d r η).run V (x+r • Δ, fun i => g i + terminalShift 2 i • Δ) =
(terminalSample d r η).run V (x,g) - Δ := by
simp only [terminalSample_run, terminalQuery_invariant]
simp only [terminalShift, Fin.reduceEq, ite_true, ite_false,
neg_smul, one_smul, zero_smul, add_zero]
module
lemma terminalMean_history_shift {d : ℕ} (r σ : ℝ) (V : Point d → ℝ)
(x Δ : Point d) (g : Fin 2 → Point d) :
(terminalMean d r σ).history V (x+r • Δ,fun i => g i+terminalShift 1 i • Δ) 1 =
(terminalMean d r σ).history V (x,g) 1 := by
simp only [Program.history, terminalMean, oneQuery, Nat.lt_add_one, dite_eq_left,
terminalQuery_invariant]
lemma terminalSample_history_shift {d : ℕ} (r η : ℝ) (V : Point d → ℝ)
(x Δ : Point d) (g : Fin 3 → Point d) :
(terminalSample d r η).history V (x+r • Δ,fun i => g i+terminalShift 2 i • Δ) 1 =
(terminalSample d r η).history V (x,g) 1 := by
simp only [Program.history, terminalSample, oneQuery, Nat.lt_add_one, dite_eq_left,
terminalQuery_invariant]
end LogConcaveSampling.SeedCompiler
end
end
end
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 restored_from_joint_law (κ : Measure Ω) [SFinite κ] (μ : Measure A) [SFinite μ]
(anchor : Ω → A) (noise : Ω → E) (ha : Measurable anchor) (hz : Measurable noise)
(hlaw : κ.map (fun w => (anchor w,noise w))=μ.prod (stdGaussian E))
(a : ι → ℝ) (ha2 : (∑i,a i^2)≤1) :
(κ.prod (Measure.pi fun _ : ι => stdGaussian E)).map
(fun z => (anchor z.1,restored (sqrtCoefficient a) a (z.2,noise z.1)))=
μ.prod (Measure.pi fun _ : ι => stdGaussian E) := by
let γ : Measure (ι → E) := Measure.pi fun _ => stdGaussian E
have hAZ : MeasurePreserving (fun w => (anchor w,noise w)) κ (μ.prod (stdGaussian E)) :=
⟨ha.prodMk hz,hlaw⟩
have hp := hAZ.prod (MeasurePreserving.id γ)
have hs := (MeasurePreserving.id μ).prod (Measure.measurePreserving_swap (μ:=stdGaussian E) (ν:=γ))
have ht := hs.comp ((measurePreserving_prodAssoc μ (stdGaussian E) γ).comp hp)
have hr : MeasurePreserving (Prod.map id (restored (sqrtCoefficient a) a))
(μ.prod (γ.prod (stdGaussian E))) (μ.prod γ) :=
⟨by fun_prop,restored_with_anchors μ a ha2⟩
exact (hr.comp ht).map_eq
theorem restored_error_integral (κ : Measure Ω) [SFinite κ] (μ : Measure A) [SFinite μ]
(anchor : Ω → A) (noise : Ω → E) (ha : Measurable anchor) (hz : Measurable noise)
(hlaw : κ.map (fun w => (anchor w,noise w))=μ.prod (stdGaussian E))
(a : ι → ℝ) (ha2 : (∑i,a i^2)≤1) (f : A × (ι → E) → ℝ)
(hf : Integrable f (μ.prod (Measure.pi fun _ : ι => stdGaussian E))) :
Integrable (fun z : Ω × (ι → E) =>
f (anchor z.1,restored (sqrtCoefficient a) a (z.2,noise z.1)))
(κ.prod (Measure.pi fun _ : ι => stdGaussian E)) ∧
(∫z : Ω × (ι → E),f (anchor z.1,restored (sqrtCoefficient a) a (z.2,noise z.1))
∂κ.prod (Measure.pi fun _ : ι => stdGaussian E))=
∫z,f z ∂μ.prod (Measure.pi fun _ : ι => stdGaussian E) := by
let T : Ω × (ι → E) → A × (ι → E) := fun z =>
(anchor z.1,restored (sqrtCoefficient a) a (z.2,noise z.1))
have hm : Measurable T := (ha.comp measurable_fst).prodMk ((restored (sqrtCoefficient a) a).measurable.comp
(measurable_snd.prodMk (hz.comp measurable_fst)))
have he : (κ.prod (Measure.pi fun _ : ι => stdGaussian E)).map T=
μ.prod (Measure.pi fun _ : ι => stdGaussian E) :=
restored_from_joint_law κ μ anchor noise ha hz hlaw a ha2
have hi : Integrable f ((κ.prod (Measure.pi fun _ : ι => stdGaussian E)).map T) := he ▸ hf
exact ⟨hi.comp_measurable hm,(integral_map hm.aemeasurable hi.aestronglyMeasurable).symm.trans
(by rw [he])⟩
end LogConcaveSampling.SeedCompiler
end
end
end
end OAI
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