| import Mathlib |
| import OAI.Probability.LogConcave.OraclePrograms.CircuitParameters |
| import OAI.Probability.LogConcave.OraclePrograms.MeanTreeBase |
| import OAI.Probability.LogConcave.OraclePrograms.CalibratedMeanCircuit |
| import OAI.Probability.LogConcave.Numerics.BaseVariationCenters |
| import OAI.Probability.LogConcave.Numerics.EvalLipschitz |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace LogConcaveSampling.OracleCompiler.CircuitParameters |
| open MeanTree MeasureTheory Quadrature |
| open scoped Classical NNReal |
|
|
| variable (C : CircuitParameters) {d : ℕ} |
|
|
| lemma pair_dist_bounds (z w : Point d × Point d) : |
| ‖z.1-w.1‖≤dist z w ∧ ‖z.2-w.2‖≤dist z w := by |
| change ‖z.1-w.1‖≤ max (dist z.1 w.1) (dist z.2 w.2) ∧ |
| ‖z.2-w.2‖≤ max (dist z.1 w.1) (dist z.2 w.2) |
| simp only [dist_eq_norm] |
| exact ⟨le_max_left _ _,le_max_right _ _⟩ |
|
|
| lemma meanTree_anchorVariation {r σ : ℝ} (hr : 0≤r) (x : Point d) (z w : Point d × Point d) : |
| BaseVariation ((2*r+|σ/2|)*dist z w) (x,z) (x,w) (C.meanTree r σ) := by |
| have hg := 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 |
| have hd := pair_dist_bounds z w |
| apply baseVariation_of_centers |
| · rw [C.meanTree_base] |
| simp only [←smul_sub,norm_smul,Real.norm_eq_abs] |
| nlinarith [mul_le_mul_of_nonneg_left hd.2 (abs_nonneg (σ/2)),mul_nonneg hr (dist_nonneg (x:=z) (y:=w))] |
| · exact centers_mono _ (fun r' b hb => by |
| have he := hb.variation hr C.T_upper.le (x,z) (x,w) |
| simp only [sub_self,norm_zero,zero_add] at he |
| nlinarith [mul_le_mul_of_nonneg_left (add_le_add hd.1 hd.2) hr, |
| mul_nonneg (abs_nonneg (σ/2)) (dist_nonneg (x:=z) (y:=w))]) hg.2 |
|
|
| lemma sampleTree_anchorVariation {r η : ℝ} (hr : 0≤r) (hη : 0<η) (hη1 : η≤1) |
| (x z w : Point d) : |
| BaseVariation ((r+|(sampleCorrelation η)⁻¹|)*dist z w) (x,z) (x,w) |
| (C.sampleTree r η hη hη1) := by |
| have hp := sampleCorrelation_properties hη hη1 |
| have hg := literalSample_geometry (d:=d) (r:=r) (N:=C.N) (Nat.succ_pos C.n) |
| (by linarith [hp.1]) hp.2.1 C.h_pos (C.sampleEndpoint η hη hη1) |
| apply baseVariation_of_centers |
| · unfold sampleTree |
| rw [hg.1] |
| simp only [←smul_sub,norm_smul,Real.norm_eq_abs,dist_eq_norm] |
| nlinarith [mul_nonneg hr (norm_nonneg (z-w))] |
| · exact centers_mono _ (fun r' b hb => by |
| have he := hb.variation hr hp.2.1.le (x,z) (x,w) |
| simp only [sub_self,norm_zero,zero_add,add_zero] at he |
| rw [dist_eq_norm] |
| nlinarith [mul_nonneg (abs_nonneg ((sampleCorrelation η)⁻¹)) (norm_nonneg (z-w))]) hg.2 |
|
|
| lemma meanTree_anchor_lipschitz {F : Point d → ℝ} {lam : ℝ≥0} (hF : Primitive F lam) |
| {r σ : ℝ} (hr : 0<r) (hl : (lam:ℝ)*r^2≤1/4) (x : Point d) : |
| ∃K : ℝ≥0,LipschitzWith K (fun z : Point d × Point d => (C.meanTree r σ).eval F (x,z)) := by |
| have hg := 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 |
| apply eval_exists_lipschitz hF (C.meanTree r σ) (fun z => (x,z)) |
| · rw [C.meanTree_base] |
| exact ⟨_,(lipschitzWith_smul (σ/2)).comp LipschitzWith.prod_snd⟩ |
| · apply centers_mono _ (fun r' b hb => ?_) hg.2 |
| have hr' := hb.radius_pos hr (by linarith [C.T_lower]) C.T_upper |
| have hb' := hb.radius hr.le (by linarith [C.T_lower]) C.T_upper |
| have hs := mul_le_mul_of_nonneg_left (pow_le_pow_left₀ hr'.le hb'.2 2) lam.coe_nonneg |
| refine ⟨hr',hs.trans hl,?_⟩ |
| obtain ⟨ρ,hρ,θ,hθ,he,rfl⟩ := hb |
| exact ⟨_,(LipschitzWith.const x).add ((lipschitzWith_smul (r*ρ)).comp |
| (((lipschitzWith_smul (Real.cos θ)).comp LipschitzWith.prod_fst).add |
| ((lipschitzWith_smul (Real.sin θ)).comp LipschitzWith.prod_snd)))⟩ |
|
|
| lemma sampleTree_anchor_lipschitz {F : Point d → ℝ} {lam : ℝ≥0} (hF : Primitive F lam) |
| {r η : ℝ} (hr : 0<r) (hη : 0<η) (hη1 : η≤1) (hl : (lam:ℝ)*r^2≤1/4) (x : Point d) : |
| ∃K : ℝ≥0,LipschitzWith K (fun z : Point d => (C.sampleTree r η hη hη1).eval F (x,z)) := by |
| have hp := sampleCorrelation_properties hη hη1 |
| have hg := literalSample_geometry (d:=d) (r:=r) (N:=C.N) (Nat.succ_pos C.n) |
| (by linarith [hp.1]) hp.2.1 C.h_pos (C.sampleEndpoint η hη hη1) |
| apply eval_exists_lipschitz hF (C.sampleTree r η hη hη1) (fun z => (x,z)) |
| · unfold sampleTree |
| rw [hg.1] |
| exact ⟨_,lipschitzWith_smul ((sampleCorrelation η)⁻¹)⟩ |
| · apply centers_mono _ (fun r' b hb => ?_) hg.2 |
| have hr' := hb.radius_pos hr (by linarith [hp.1]) hp.2.1 |
| have hb' := hb.radius hr.le (by linarith [hp.1]) hp.2.1 |
| have hs := mul_le_mul_of_nonneg_left (pow_le_pow_left₀ hr'.le hb'.2 2) lam.coe_nonneg |
| refine ⟨hr',hs.trans hl,?_⟩ |
| obtain ⟨ρ,hρ,θ,hθ,he,rfl⟩ := hb |
| exact ⟨_,(LipschitzWith.const x).add ((lipschitzWith_smul (r*ρ)).comp |
| ((lipschitzWith_smul (Real.cos θ)).add ((lipschitzWith_smul (Real.sin θ)).comp (LipschitzWith.const 0))))⟩ |
|
|
| lemma meanCompiled_anchor_lipschitz {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) (x : Point d) : |
| let E := compileDeclaredRoot C.D (C.meanCenterBudget r σ) C.Af |
| (M true) (M false) (C.meanTree (d:=d) r σ) n |
| ∃K : ℝ,0≤K ∧ ∀z w g,‖E.eval V (x,z) g-E.eval V (x,w) g‖≤K*dist z w := by |
| have hw := C.meanTree_actual_weights (d:=d) hr hσ hψ |
| rw [←hAf,←hA] at hw |
| have hAf0 : 0≤C.Af := by rw [hAf]; exact C.actualAf_pos.le |
| let b := 2*r+|σ/2| |
| refine ⟨C.Af*Lf*(depth (C.meanTree (d:=d) r σ):ℝ)*b+b,by dsimp [b]; positivity,?_⟩ |
| intro z w g |
| have hv := C.meanTree_anchorVariation (σ:=σ) hr.le x z w |
| have hb : ‖base (C.meanTree r σ) (x,z)-base (C.meanTree r σ) (x,w)‖≤b*dist z w := by |
| generalize C.meanTree (d:=d) r σ=E at hv ⊢ |
| cases E |
| exact hv.1 |
| have he := compileDeclaredRoot_variation hLf hLi (show 0≤b*dist z w by dsimp [b]; positivity) |
| hsmall V P M hm hmi (C.meanTree r σ) n (x,z) (x,w) hv hb hw.1 hw.2 hready g |
| convert he using 1 |
| dsimp [b] |
| ring |
|
|
| lemma sampleCompiled_anchor_lipschitz {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) (x : Point d) : |
| let E := compileDeclaredRoot C.D (C.sampleCenterBudget r) (C.sampleFinalBudget r) |
| (M true) (M false) (C.sampleTree (d:=d) r η hη hη1) n |
| ∃K : ℝ,0≤K ∧ ∀z w g,‖E.eval V (x,z) g-E.eval V (x,w) g‖≤K*dist z w := by |
| have hw := C.sampleTree_actual_weights (d:=d) hr hη hη1 |
| rw [←hAf,←hA] at hw |
| have hAf0 : 0≤C.sampleFinalBudget r := by unfold sampleFinalBudget; rw [hAf]; positivity [C.actualAf_pos] |
| let b := r+|(sampleCorrelation η)⁻¹| |
| refine ⟨C.sampleFinalBudget r*Lf*(depth (C.sampleTree (d:=d) r η hη hη1):ℝ)*b+b,by dsimp [b]; positivity,?_⟩ |
| intro z w g |
| have hv := C.sampleTree_anchorVariation hr.le hη hη1 x z w |
| have hb : ‖base (C.sampleTree r η hη hη1) (x,z)-base (C.sampleTree r η hη hη1) (x,w)‖≤b*dist z w := by |
| generalize C.sampleTree (d:=d) r η hη hη1=E at hv ⊢ |
| cases E |
| exact hv.1 |
| have he := compileDeclaredRoot_variation hLf hLi (show 0≤b*dist z w by dsimp [b]; positivity) |
| hsmall V P M hm hmi (C.sampleTree r η hη hη1) n (x,z) (x,w) hv hb hw.1 hw.2 hready g |
| convert he using 1 <;> (try simp only [sampleFinalBudget]) |
| dsimp [b] |
| ring |
|
|
| end LogConcaveSampling.OracleCompiler.CircuitParameters |
|
|
| end |
|
|
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace LogConcaveSampling.OracleCompiler.CircuitParameters |
| open MeasureTheory ProbabilityTheory MeanTree Quadrature Function |
| open scoped Classical NNReal |
|
|
| variable (C : CircuitParameters) {d : ℕ} |
|
|
| lemma meanTree_eval {F : Point d → ℝ} {lam : ℝ≥0} (hF : Primitive F lam) |
| (r σ : ℝ) (x : Point d) (y : Point d × Point d) : |
| (C.meanTree r σ).eval F (x,y)= |
| centeringMeanCircuit F x r C.T C.h C.ψ (σ/2) C.n C.m C.N C.nc C.Nc C.meanEndpoint |
| ((productPointEquiv d d).symm y) := by |
| unfold meanTree literalMean |
| rw [eval_meanCircuit hF (by linarith [C.T_lower]) C.T_upper C.h_pos] |
| simp only [eval_anchor] |
|
|
| lemma sampleTree_eval {F : Point d → ℝ} {lam : ℝ≥0} (hF : Primitive F lam) |
| (r η : ℝ) (hη : 0<η) (hη1 : η≤1) (x Y : Point d) : |
| (C.sampleTree r η hη hη1).eval F (x,Y)= |
| sampleMeanCircuit F x r (sampleCorrelation η) C.h (C.n+1) C.N (C.sampleEndpoint η hη hη1) Y := by |
| unfold sampleTree literalSample |
| rw [eval_scale,eval_forward hF (Nat.succ_pos C.n) |
| (by linarith [(sampleCorrelation_properties hη hη1).1]) |
| (sampleCorrelation_properties hη hη1).2.1 C.h_pos] |
| simp only [eval_anchor,sampleMeanCircuit] |
|
|
| theorem meanTree_numerical {F : Point d → ℝ} {lam : ℝ≥0} (hF : Primitive F lam) |
| (x : Point d) {r σ err : ℝ} (hr : 0<r) (hl : (lam:ℝ)*r^2≤1/4) |
| (hm : C.m=C.n) (hnc : C.nc=C.n) (hNc : C.Nc=C.N) |
| (hcert : CalibratedMeanCircuit (r:=r) (T:=C.T) (h:=C.h) (ψ:=C.ψ) (s:=σ/2) |
| hF x hr (by linarith : (lam:ℝ)*r^2≤1/2) (by linarith [C.T_lower]) C.T_upper |
| C.h_pos C.n C.N err) : |
| ∃w : Point d × Point d → Point d,Measurable w ∧ |
| ((interpolationLaw F x r C.T).prod (stdGaussian (Point d))).map w= |
| (stdGaussian (Point d)).map (fun z => primitiveExpectedField F x r+(σ/2) • z) ∧ |
| Integrable (fun z => ‖(C.meanTree r σ).eval F (x,z)-w z‖^2) |
| ((interpolationLaw F x r C.T).prod (stdGaussian (Point d))) ∧ |
| (∫z,‖(C.meanTree r σ).eval F (x,z)-w z‖^2 |
| ∂(interpolationLaw F x r C.T).prod (stdGaussian (Point d)))≤ |
| ((σ/2)*circuitD F x)^2*err := by |
| obtain ⟨X,hsi,hsb,hw,hlaw,hi,hnum⟩ := hcert |
| let w₀ := stationaryMeanTarget hF x hr (by linarith : (lam:ℝ)*r^2≤1/2) |
| (by linarith [C.T_lower] : 0≤C.T) C.T_upper X |
| let w : Point d × Point d → Point d := fun z => w₀ ((productPointEquiv d d).symm z) |
| have hw' : Measurable w := hw.comp (productPointEquiv d d).symm.continuous.measurable |
| have hjoint := centering_joint_law hF x hr.le (by linarith : (lam:ℝ)*r^2≤1/2) |
| (by linarith [C.T_lower]) C.T_upper |
| obtain ⟨K,hK⟩ := C.meanTree_anchor_lipschitz hF hr hl x |
| have hg : Measurable (fun z => (C.meanTree r σ).eval F (x,z)) := hK.continuous.measurable |
| have hid : ∀y : Point (d+d), |
| (C.meanTree r σ).eval F (x,productPointEquiv d d y)-w (productPointEquiv d d y)= |
| centeringMeanCircuit F x r C.T C.h C.ψ (σ/2) C.n C.n C.N C.n C.N C.meanEndpoint y-w₀ y := by |
| intro y |
| rw [C.meanTree_eval hF] |
| simp only [hm,hnc,hNc,w,ContinuousLinearEquiv.symm_apply_apply] |
| refine ⟨w,hw',?_,?_,?_⟩ |
| · rw [←hjoint,Measure.map_map hw' (productPointEquiv d d).continuous.measurable] |
| simpa only [comp_def,w,ContinuousLinearEquiv.symm_apply_apply] using hlaw |
| · rw [←hjoint] |
| apply (integrable_map_measure (by fun_prop) (productPointEquiv d d).continuous.measurable.aemeasurable).mpr |
| simpa only [hid,comp_def,meanEndpoint,w₀] using hi |
| · rw [←hjoint,integral_map (productPointEquiv d d).continuous.measurable.aemeasurable (by fun_prop)] |
| simpa only [hid,meanEndpoint,w₀] using hnum |
|
|
| lemma meanTree_numerical_noisy {F : Point d → ℝ} {lam : ℝ≥0} (hF : Primitive F lam) |
| (x : Point d) {r σ err : ℝ} (hr : 0<r) (hl : (lam:ℝ)*r^2≤1/4) |
| (hm : C.m=C.n) (hnc : C.nc=C.n) (hNc : C.Nc=C.N) |
| (hcert : CalibratedMeanCircuit (r:=r) (T:=C.T) (h:=C.h) (ψ:=C.ψ) (s:=σ/2) |
| hF x hr (by linarith : (lam:ℝ)*r^2≤1/2) (by linarith [C.T_lower]) C.T_upper |
| C.h_pos C.n C.N err) : |
| ∃w : Point d × Point d → Point d,Measurable w ∧ |
| (((interpolationLaw F x r C.T).prod (stdGaussian (Point d))).prod (stdGaussian (Point d))).map |
| (fun z => w z.1+(Real.sqrt 3*σ/2) • z.2)= |
| (stdGaussian (Point d)).map (fun z => primitiveExpectedField F x r+σ • z) ∧ |
| Integrable (fun z => ‖(C.meanTree r σ).eval F (x,z)-w z‖^2) |
| ((interpolationLaw F x r C.T).prod (stdGaussian (Point d))) ∧ |
| (∫z,‖(C.meanTree r σ).eval F (x,z)-w z‖^2 |
| ∂(interpolationLaw F x r C.T).prod (stdGaussian (Point d)))≤ |
| ((σ/2)*circuitD F x)^2*err := by |
| obtain ⟨w,hw,hlaw,hi,hnum⟩ := C.meanTree_numerical hF x hr hl hm hnc hNc hcert |
| let := interpolationLaw_probability hF x hr.le (by linarith : (lam:ℝ)*r^2<1) C.T |
| refine ⟨w,hw,?_,hi,hnum⟩ |
| apply law_add_gaussian _ w hw (primitiveExpectedField F x r) (σ/2) (Real.sqrt 3*σ/2) σ _ hlaw |
| rw [div_pow,div_pow,mul_pow,Real.sq_sqrt (by norm_num : (0:ℝ)≤3)] |
| ring |
| end LogConcaveSampling.OracleCompiler.CircuitParameters |
|
|
| end |
|
|
| end |
|
|
| end OAI |
|
|
