| import Mathlib |
| import OAI.Probability.LogConcave.OraclePrograms.CircuitParameters |
| import OAI.Probability.LogConcave.OraclePrograms.MeanSize |
| import OAI.Probability.LogConcave.Numerics.BaseVariationCenters |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace LogConcaveSampling.OracleCompiler.CircuitParameters |
| open MeanTree MeasureTheory Quadrature Expression |
| open scoped Classical |
|
|
| variable (C : CircuitParameters) {d : ℕ} |
|
|
| lemma meanTree_base (r σ : ℝ) : |
| base (C.meanTree (d:=d) r σ)=fun z => (σ/2) • z.2.2 := by |
| exact (literalMean_geometry (d:=d) (r:=r) (σ:=σ) (N:=C.N) (nc:=C.nc) (Nc:=C.Nc) |
| (by linarith [C.T_lower]) C.T_upper C.h_pos C.v_nonneg C.angle_bound C.meanEndpoint).1 |
|
|
| lemma meanCompiled_variation {r σ Lf Li : ℝ} (hr : 0<r) (hσ : 0<σ) |
| (hψ : 0≤C.ψ) (hA : C.A=C.actualA) (hAf : C.Af=C.actualAf) |
| (hLf : 0≤Lf) (hLi : 0≤Li) (hsmall : Li*C.meanCenterBudget r σ≤1) |
| (V : Point d → ℝ) (P : Bool → ℝ → ℝ → Prop) (M : Bool → ℝ → ℝ → SeedProgram d) |
| (hm : ∀r τ,P true r τ → ∀u v g,‖(M true r τ).program.run V (u,g)-(M true r τ).program.run V (v,g)‖≤Lf*‖u-v‖) |
| (hmi : ∀r τ,P false r τ → ∀u v g,‖(M false r τ).program.run V (u,g)-(M false r τ).program.run V (v,g)‖≤Li*‖u-v‖) |
| (n : ℝ) (hready : RootShiftReady C.D (C.meanCenterBudget r σ) C.Af |
| (P true) (P false) (C.meanTree (d:=d) r σ) n) : |
| let E := compileDeclaredRoot C.D (C.meanCenterBudget r σ) C.Af |
| (M true) (M false) (C.meanTree (d:=d) r σ) n |
| let K := C.Af*Lf*(depth (C.meanTree (d:=d) r σ):ℝ) |
| (∀x y Y G g,‖E.eval V (x,(Y,G)) g-E.eval V (y,(Y,G)) g‖≤K*‖x-y‖) ∧ |
| (∀x Y Z G g,‖E.eval V (x,(Y,G)) g-E.eval V (x,(Z,G)) g‖≤(K*r)*‖Y-Z‖) := by |
| have hw := C.meanTree_actual_weights (d:=d) hr hσ hψ |
| rw [←hAf,←hA] at hw |
| have hT0 : 0<C.T := by linarith [C.T_lower] |
| constructor |
| · intro x y Y G g |
| have he := compileDeclaredRoot_variation hLf hLi (norm_nonneg (x-y)) hsmall V P M hm hmi |
| (C.meanTree r σ) n (x,(Y,G)) (y,(Y,G)) |
| (literalMean_variationX hr.le hT0 C.T_upper C.h_pos C.v_nonneg C.angle_bound C.meanEndpoint x y Y G) |
| (Bf:=0) (by rw [C.meanTree_base]; simp) hw.1 hw.2 hready g |
| simpa only [add_zero] using he |
| · intro x Y Z G g |
| have he := compileDeclaredRoot_variation hLf hLi (mul_nonneg hr.le (norm_nonneg (Y-Z))) hsmall V P M hm hmi |
| (C.meanTree r σ) n (x,(Y,G)) (x,(Z,G)) |
| (literalMean_variationY hr.le hT0 C.T_upper C.h_pos C.v_nonneg C.angle_bound C.meanEndpoint x Y Z G) |
| (Bf:=0) (by rw [C.meanTree_base]; simp) hw.1 hw.2 hready g |
| simpa only [add_zero,mul_assoc] using he |
|
|
| lemma sampleCompiled_variation {r η Lf Li : ℝ} (hr : 0<r) (hη : 0<η) (hη1 : η≤1) |
| (hA : C.A=C.actualA) (hAf : C.Af=C.actualAf) |
| (hLf : 0≤Lf) (hLi : 0≤Li) (hsmall : Li*C.sampleCenterBudget r≤1) |
| (V : Point d → ℝ) (P : Bool → ℝ → ℝ → Prop) (M : Bool → ℝ → ℝ → SeedProgram d) |
| (hm : ∀r τ,P true r τ → ∀u v g,‖(M true r τ).program.run V (u,g)-(M true r τ).program.run V (v,g)‖≤Lf*‖u-v‖) |
| (hmi : ∀r τ,P false r τ → ∀u v g,‖(M false r τ).program.run V (u,g)-(M false r τ).program.run V (v,g)‖≤Li*‖u-v‖) |
| (n : ℝ) (hready : RootShiftReady C.D (C.sampleCenterBudget r) (C.sampleFinalBudget r) |
| (P true) (P false) (C.sampleTree (d:=d) r η hη hη1) n) : |
| let E := compileDeclaredRoot C.D (C.sampleCenterBudget r) (C.sampleFinalBudget r) |
| (M true) (M false) (C.sampleTree (d:=d) r η hη hη1) n |
| ∀x y z g,‖E.eval V (x,z) g-E.eval V (y,z) g‖≤ |
| (C.sampleFinalBudget r*Lf*(depth (C.sampleTree (d:=d) r η hη hη1):ℝ))*‖x-y‖ := by |
| have hw := C.sampleTree_actual_weights (d:=d) hr hη hη1 |
| rw [←hAf,←hA] at hw |
| have hp := sampleCorrelation_properties hη hη1 |
| have hT0 : 0<sampleCorrelation η := by linarith [hp.1] |
| dsimp only |
| intro x y z g |
| have hg := literalSample_geometry (d:=d) (r:=r) (N:=C.N) (Nat.succ_pos C.n) |
| hT0 hp.2.1 C.h_pos (C.sampleEndpoint η hη hη1) |
| have he := compileDeclaredRoot_variation (Af:=C.sampleFinalBudget r) hLf hLi (norm_nonneg (x-y)) hsmall V P M hm hmi |
| (C.sampleTree r η hη hη1) n (x,z) (y,z) |
| (literalSample_variationX hr.le (Nat.succ_pos C.n) hT0 hp.2.1 C.h_pos (C.sampleEndpoint η hη hη1) x y z) |
| (Bf:=0) (by unfold sampleTree; rw [hg.1]; simp) hw.1 hw.2 hready g |
| simpa only [add_zero] using he |
|
|
| end LogConcaveSampling.OracleCompiler.CircuitParameters |
|
|
| end |
|
|
| end |
|
|
| end OAI |
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|
