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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 | 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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