| import Mathlib |
| import OAI.Probability.LogConcave.Analysis.SymmetricGradientDerivative |
| import OAI.Probability.LogConcave.Sampling.Primitive |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped ENNReal NNReal Topology |
|
|
| section UpperProof |
| namespace LogConcaveSampling |
| namespace Trapezoid |
| open scoped RealInnerProductSpace Topology |
| open Asymptotics |
|
|
| variable {d : ℕ} {H : Point d → ℝ} |
|
|
| def remainder (H : Point d → ℝ) (x y : Point d) : ℝ := |
| H y-H x-inner ℝ (y-x) (gradient H x+gradient H y)/2 |
|
|
| lemma contDiff_gradient (hH : ContDiff ℝ 2 H) : ContDiff ℝ 1 (gradient H) := |
| (InnerProductSpace.toDual ℝ (Point d)).symm.contDiff.comp |
| (hH.fderiv_right (by norm_num)) |
|
|
| lemma hasFDerivAt_remainder (hH : ContDiff ℝ 2 H) (x y : Point d) : |
| HasFDerivAt (remainder H x) |
| (InnerProductSpace.toDual ℝ (Point d) |
| ((1/2:ℝ) • (gradient H y-gradient H x-fderiv ℝ (gradient H) y (y-x)))) y := by |
| have hg := contDiff_gradient hH |
| have hd := (hasFDerivAt_id y).sub_const x |>.inner ℝ |
| ((hasFDerivAt_const (gradient H x) y).add |
| (hg.differentiable (by norm_num) y).hasFDerivAt) |
| have hh := ((hH.differentiable (by norm_num) y).hasFDerivAt.sub_const (H x)).sub |
| (hd.const_mul (1/2:ℝ)) |
| convert! hh using 1 |
| · funext z; dsimp [remainder]; ring |
| ext z |
| have hs : inner ℝ (fderiv ℝ (gradient H) y (y-x)) z = |
| inner ℝ (y-x) (fderiv ℝ (gradient H) y z) := |
| LogConcaveSampling.ContDiff.isSymmetric_gradient_derivative hH y (y-x) z |
| have hgrad : fderiv ℝ H y z = inner ℝ (gradient H y) z := |
| InnerProductSpace.toDual_symm_apply.symm |
| simp only [InnerProductSpace.toDual_apply_apply, sub_apply, smul_apply, |
| ContinuousLinearMap.comp_apply, ContinuousLinearMap.prod_apply, ContinuousLinearMap.id_apply, |
| fderivInnerCLM_apply, Pi.add_apply, id_eq, zero_add, smul_eq_mul] |
| change inner ℝ ((1/2:ℝ) • (gradient H y-gradient H x-fderiv ℝ (gradient H) y (y-x))) z = |
| fderiv ℝ H y z-(1/2:ℝ)*(inner ℝ (y-x) (fderiv ℝ (gradient H) y z)+ |
| inner ℝ z (gradient H x+gradient H y)) |
| rw [real_inner_smul_left,inner_sub_left,inner_sub_left,hgrad,← hs,inner_add_right] |
| nlinarith [real_inner_comm z (gradient H x),real_inner_comm z (gradient H y)] |
|
|
| lemma derivative_remainder_littleO (hH : ContDiff ℝ 2 H) (x : Point d) : |
| (fun y => InnerProductSpace.toDual ℝ (Point d) |
| ((1/2:ℝ) • (gradient H y-gradient H x-fderiv ℝ (gradient H) y (y-x)))) |
| =o[𝓝 x] (fun y => ‖y-x‖) := by |
| have hg := contDiff_gradient hH |
| have hl := (hg.differentiable (by norm_num) x).hasFDerivAt.isLittleO |
| have hc := (hg.continuous_fderiv (by norm_num)).continuousAt (x := x) |
| rw [Asymptotics.isLittleO_iff] |
| intro c hcpos |
| have he₁ := hl.bound hcpos |
| have he₂ : ∀ᶠ y in 𝓝 x, ‖fderiv ℝ (gradient H) y-fderiv ℝ (gradient H) x‖ < c := by |
| simpa only [dist_eq_norm] using (Metric.tendsto_nhds.mp hc c hcpos) |
| filter_upwards [he₁,he₂] with y hy hz |
| rw [LinearIsometryEquiv.norm_map,norm_smul,Real.norm_eq_abs,abs_of_pos (by norm_num : (0:ℝ)<1/2),norm_norm] |
| have hid : gradient H y-gradient H x-fderiv ℝ (gradient H) y (y-x) = |
| (gradient H y-gradient H x-fderiv ℝ (gradient H) x (y-x))- |
| (fderiv ℝ (gradient H) y-fderiv ℝ (gradient H) x) (y-x) := by |
| simp only [sub_apply]; abel |
| rw [hid] |
| have hb := norm_sub_le (gradient H y-gradient H x-fderiv ℝ (gradient H) x (y-x)) |
| ((fderiv ℝ (gradient H) y-fderiv ℝ (gradient H) x) (y-x)) |
| have hh := (fderiv ℝ (gradient H) y-fderiv ℝ (gradient H) x).le_opNorm (y-x) |
| have hc' := mul_le_mul_of_nonneg_right hz.le (norm_nonneg (y-x)) |
| nlinarith only [hy,hb,hh,hc'] |
|
|
| theorem remainder_littleO (hH : ContDiff ℝ 2 H) (x : Point d) : |
| remainder H x =o[𝓝 x] (fun y => ‖y-x‖^2) := by |
| have hl := derivative_remainder_littleO hH x |
| have hh := convex_univ.isLittleO_pow_succ (n := 1) (Set.mem_univ x) |
| (fun y _ => (hasFDerivAt_remainder hH x y).hasFDerivWithinAt) |
| (by simpa only [nhdsWithin_univ,pow_one] using hl) |
| change (fun y => H y-H x-inner ℝ (y-x) (gradient H x+gradient H y)/2) =o[𝓝 x] (fun y => ‖y-x‖^2) |
| simpa [remainder] using hh |
|
|
| lemma remainder_bound (hH : ContDiff ℝ 2 H) {M : ℝ≥0} |
| (hM : LipschitzWith M (gradient H)) (x y : Point d) : |
| |remainder H x y| ≤ (M:ℝ)*‖y-x‖^2 := by |
| have ht := quadratic_remainder_of_gradient_lipschitz hH hM x (y-x) |
| rw [add_sub_cancel] at ht |
| have hnorm : ‖gradient H y-gradient H x‖ ≤ (M:ℝ)*‖y-x‖ := hM.norm_sub_le y x |
| have hi := (abs_real_inner_le_norm (y-x) (gradient H y-gradient H x)).trans |
| (mul_le_mul_of_nonneg_left hnorm (norm_nonneg (y-x))) |
| have he : remainder H x y = |
| (H y-H x-inner ℝ (y-x) (gradient H x))- |
| inner ℝ (y-x) (gradient H y-gradient H x)/2 := by |
| dsimp [remainder] |
| rw [inner_add_right,inner_sub_right] |
| ring |
| rw [he] |
| have hb := abs_sub (H y-H x-inner ℝ (y-x) (gradient H x)) |
| (inner ℝ (y-x) (gradient H y-gradient H x)/2) |
| rw [abs_div,abs_of_pos (by norm_num : (0:ℝ)<2)] at hb |
| nlinarith only [hb,ht,hi] |
|
|
| end Trapezoid |
| end LogConcaveSampling |
|
|
| end UpperProof |
| end |
| end |
| end |
|
|
| end OAI |
|
|
