openAI-math-dataset / lean /OAI /Probability /LogConcave /OraclePrograms /CalibratedMeanCircuit.lean
| import Mathlib | |
| import OAI.Probability.LogConcave.OraclePrograms.CenteringMeanCircuitFull | |
| import OAI.Probability.LogConcave.Sampling.NumericalScalars | |
| namespace OAI | |
| section | |
| noncomputable section | |
| namespace LogConcaveSampling | |
| open Filter Set MeasureTheory Quadrature ProbabilityTheory | |
| open scoped Classical BigOperators NNReal | |
| def CalibratedMeanCircuit {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} | |
| (hF : Primitive F lam) (x : Point d) {r T h ψ s : ℝ} (hr : 0<r) | |
| (hl : (lam:ℝ)*r^2≤1/2) (hT0 : 0<T) (hT1 : T<1) (hh : 0<h) | |
| (n N : ℕ) (err : ℝ) : Prop := | |
| let e := probabilityEndpoint hT0 hT1 hh (n+1) | |
| let μ := gibbs (centeringPotential F x r T) | |
| ∃X : StationaryPath μ | |
| (skewLieField (centeringPotential F x r T) | |
| (jointSkew (centeringKernel hF x hr hl hT0.le hT1) s)), | |
| (∀i,Integrable (fun y => ‖stationaryPicard n N | |
| (centeringVelocityJoint F x r T h ψ s n n N e) y i-X.path y (probabilityNodes n i)‖^2) μ) ∧ | |
| (∀i,(∫y,‖stationaryPicard n N | |
| (centeringVelocityJoint F x r T h ψ s n n N e) y i-X.path y (probabilityNodes n i)‖^2 ∂μ)≤ | |
| (circuitD F x)^2*err) ∧ | |
| Measurable (stationaryMeanTarget hF x hr hl hT0.le hT1 X) ∧ | |
| μ.map (stationaryMeanTarget hF x hr hl hT0.le hT1 X)= | |
| (stdGaussian (Point d)).map (fun g => primitiveExpectedField F x r+s • g) ∧ | |
| Integrable (fun y => ‖centeringMeanCircuit F x r T h ψ s n n N n N e y- | |
| stationaryMeanTarget hF x hr hl hT0.le hT1 X y‖^2) μ ∧ | |
| (∫y,‖centeringMeanCircuit F x r T h ψ s n n N n N e y- | |
| stationaryMeanTarget hF x hr hl hT0.le hT1 X y‖^2 ∂μ)≤(s*circuitD F x)^2*err | |
| theorem exists_calibrated_mean_circuit {κ J b t : ℝ} | |
| (hκ : 0<κ) (hJ : 1≤J) (hb : 1/2≤b) (ht : 0<t) (htsmall : t<1/100) : | |
| ∀{w : ℝ},0<w → w<t → ∃n : ℕ, | |
| J+10<w*((n:ℝ)-3) ∧ J+10<t*((n:ℝ)-3) ∧ 4*(J+10)<(n:ℝ) ∧ | |
| ∀{S : ℝ},0≤S → ∀ksize : ℕ,∀ᶠ d : ℕ in atTop, | |
| let T := sourceCorrelation (κ*t) d | |
| let h := (d:ℝ)^(-(κ*w)) | |
| let ψ := (d:ℝ)^(-(κ*(4*t))) | |
| ∃hT0 : 0<T,∃hT1 : T<1,∃hh : 0<h, | |
| ∀{F : Point d → ℝ} {lam : ℝ≥0},∀hF : Primitive F lam,∀x : Point d, | |
| ∀{r s : ℝ},∀hr : 0<r,r≤1 → 0<lam → (lam:ℝ)*r≤1 → 0<s → | |
| (lam:ℝ)*r^2≤S*dimensionLog d^ksize*(d:ℝ)^(-(κ*b)) → | |
| (lam:ℝ)*r/s≤S*dimensionLog d^ksize*(d:ℝ)^(-(κ*b)) → | |
| ∃hl : (lam:ℝ)*r^2≤1/2, | |
| CalibratedMeanCircuit (ψ:=ψ) (s:=s) hF x hr hl hT0 hT1 hh n (numericalLayerCount J) | |
| ((d:ℝ)^(-(2*κ*J))) := by | |
| intro w hw hwt | |
| obtain ⟨n,hnw,hnt,hnn⟩ := exists_numerical_order (J:=J) ht hw | |
| have hn : 0<n := by | |
| have he : (0:ℝ)<n := by linarith | |
| exact_mod_cast he | |
| have hn1 : 1≤n := hn | |
| obtain ⟨A,B,Ap,Ah,Lp,Lh,C,J₀,D,hC,hJ₀,hD,k,hproducer⟩ := centeringMeanCircuit_full_rms n | |
| let U := A+B+Ap+Ah+Lp+Lh | |
| have hU : A≤U ∧ B≤U ∧ Ap≤U ∧ Ah≤U ∧ Lp≤U ∧ Lh≤U := by | |
| dsimp only [U] | |
| have ha : (0:ℝ≥0)≤A := bot_le | |
| have hb : (0:ℝ≥0)≤B := bot_le | |
| have hc : (0:ℝ≥0)≤Ap := bot_le | |
| have hd : (0:ℝ≥0)≤Ah := bot_le | |
| have he : (0:ℝ≥0)≤Lp := bot_le | |
| have hf : (0:ℝ≥0)≤Lh := bot_le | |
| constructor <;> first | nlinarith | skip | |
| constructor <;> first | nlinarith | skip | |
| constructor <;> first | nlinarith | skip | |
| constructor <;> first | nlinarith | skip | |
| constructor <;> nlinarith | |
| refine ⟨n,hnw,hnt,hnn,?_⟩ | |
| intro S hS ksize | |
| have hc := centeringScalarCalibration_uniform k n (numericalLayerCount J) ksize A Ap Ah Lp U C J₀ D | |
| hS hκ hJ hb ht htsmall hw hwt hnw hnt hnn (numericalLayerCount_bound J) | |
| have hT := sourceCorrelation_eventually (mul_pos hκ ht) | |
| have hmesh := sourceMesh_eventually (mul_pos hκ ht) (mul_pos hκ hw) | |
| have hangle := sourceAngle_eventually (show 0<κ*(4*t) by positivity) n | |
| filter_upwards [hc,hT,hmesh,hangle,eventually_ge_atTop (1:ℕ)] with d hdc hdT hdmesh hdangle hd | |
| intro T h ψ | |
| let R := (d:ℝ)^(-(κ*t)) | |
| have hd0 : (0:ℝ)<d := by exact_mod_cast hd | |
| have hh : 0<h := Real.rpow_pos_of_pos hd0 _ | |
| have hR : 0<R := Real.rpow_pos_of_pos hd0 _ | |
| have hψ : 0<ψ := Real.rpow_pos_of_pos hd0 _ | |
| refine ⟨hdT.1,hdT.2.1,hh,?_⟩ | |
| intro F lam hF x r s hr hr1 hlam hL hs hl₀ hα | |
| have hgood := hdc ⟨lam,r,s⟩ ⟨hr,hs,hl₀,hα⟩ | |
| rcases hgood with ⟨hl,hcontr,hq,hz,hm⟩ | |
| have bound {K : ℝ≥0} (hK : K≤U) : K*probabilityMeanLipschitz lam r≤1/2 := | |
| (mul_le_mul_of_nonneg_right hK bot_le).trans hcontr | |
| refine ⟨hl,?_⟩ | |
| obtain ⟨X,hsi,hsb,hmeas,hlaw,hmi,hmb⟩ := hproducer hF x hr hr1 hlam hl hL hd | |
| hdT.1 hdT.2.1 hh hdmesh.1 hdmesh.2 | |
| (bound hU.2.2.1) (bound hU.2.2.2.1) (bound hU.2.2.2.2.1) (bound hU.2.2.2.2.2) | |
| (bound hU.1) (bound hU.2.1) (numericalLayerCount J) R hR hdT.2.2.le n hn1 ψ hψ hdangle | |
| s hs n (numericalLayerCount J) hn hq | |
| refine ⟨X,hsi,?_,hmeas,hlaw,hmi,?_⟩ | |
| · intro i | |
| exact (hsb i).trans (mul_le_mul_of_nonneg_left hz (sq_nonneg _)) | |
| · exact hmb.trans (mul_le_mul_of_nonneg_left hm (sq_nonneg _)) | |
| end LogConcaveSampling | |
| end | |
| end | |
| end OAI | |