| import OAI.Probability.GaussianPropeller.Attainment |
|
|
| namespace OAI |
|
|
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal |
| open scoped RealInnerProductSpace |
| open scoped RealInnerProductSpace |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal RealInnerProductSpace |
| open Filter |
| open scoped Topology |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped Topology |
| open scoped RealInnerProductSpace |
| open Set Filter |
| open scoped Topology RealInnerProductSpace |
| open scoped NNReal |
| open Set Filter |
| open scoped Topology RealInnerProductSpace NNReal |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped Topology RealInnerProductSpace |
| open MeasureTheory Set Filter |
| open scoped Topology BigOperators |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped RealInnerProductSpace Topology |
|
|
| namespace GaussianPropeller.Reduction |
| open Set GaussianPropeller.Quantile GaussianPropeller.OneCell GaussianPropeller.ProbabilityBounds |
| open scoped RealInnerProductSpace |
| variable {d k : ℕ} [NeZero k] |
|
|
| lemma winning_inner_nonneg (z : Fin k → Space d) (hz : ∑ j,z j=0) |
| (i : Fin k) {x : Space d} (hx : x ∈ closedCell z i) : 0 ≤ ⟪z i,x⟫ := by |
| have hh : (0:ℝ) ≤ ∑ j : Fin k, (⟪z i,x⟫-⟪z j,x⟫) := |
| Finset.sum_nonneg (fun j _ => sub_nonneg.mpr (hx j)) |
| have hz' : (∑ j,⟪z j,x⟫)=0 := by rw [← sum_inner,hz,inner_zero_left] |
| rw [Finset.sum_sub_distrib, hz', sub_zero, |
| Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul] at hh |
| exact nonneg_of_mul_nonneg_right hh (by exact_mod_cast (NeZero.pos k)) |
|
|
| lemma active_probability_mem (_hk : 2 ≤ k) {A : Fin k → Set (Space d)} |
| (hA : MinimalOptimal A) (hpos : 0 < value A) {i : Fin k} |
| (hi : gaussian d (A i) ≠ 0) : 0 < (gaussian d).real (A i) ∧ (gaussian d).real (A i) ≤ 1/2 := by |
| let : IsProbabilityMeasure (gaussian d) := inferInstanceAs (IsProbabilityMeasure (stdGaussian (Space d))) |
| have hP : 0 < (gaussian d).real (A i) := ENNReal.toReal_pos hi (measure_ne_top _ _) |
| have hzn : centroid (A i) ≠ 0 := active_centroid_ne_zero hA hpos hi |
| let e : Space d := - (‖centroid (A i)‖⁻¹ • centroid (A i)) |
| have he : ‖e‖=1 := by simp [e, norm_smul, norm_ne_zero_iff.mpr hzn] |
| have hmap := OneCell.map_unit_inner_gaussian e he |
| have hs : closedCell (fun j => centroid (A j)) i ⊆ {x | ⟪e,x⟫ ≤ 0} := by |
| intro x hx |
| have hn := winning_inner_nonneg (fun j => centroid (A j)) (sum_centroid_eq_zero hA.1.1) i hx |
| simp only [Set.mem_ofPred_eq,e,inner_neg_left,real_inner_smul_left] |
| exact neg_nonpos.mpr (mul_nonneg (inv_nonneg.mpr (norm_nonneg _)) hn) |
| have heq : (gaussian d) {x | ⟪e,x⟫ ≤ 0} = (gaussianReal 0 1) (Iic 0) := by |
| rw [← hmap, Measure.map_apply (by fun_prop) measurableSet_Iic] |
| rfl |
| refine ⟨hP, ?_⟩ |
| have hh := measure_mono (μ:=gaussian d) hs |
| rw [← measure_congr (active_eq_closedCell_ae hA hpos hi), heq] at hh |
| have hh' := ENNReal.toReal_mono (measure_ne_top _ _) hh |
| change (gaussian d).real (A i) ≤ (gaussianReal 0 1).real (Iic 0) at hh' |
| rw [← Φ_eq_gaussianReal, Φ_zero] at hh' |
| exact hh' |
|
|
| lemma p_eq_Φ_neg (x : ℝ) : p x=Φ (-x) := by |
| rw [Φ_neg, Φ_eq_one_sub_tail] |
| simp only [φ, OneCell.gaussianConst, p, OneCell.tail, OneCell.density] |
| rw [integral_const_mul] |
| ring |
|
|
| lemma loss_eq_quantile {P : ℝ} (_hP : P ∈ Ioo (0:ℝ) 1) {x : ℝ} (hx : p x=P) : |
| loss x = φ (q P)*(q P*P+φ (q P)) := by |
| have hq : q P = -x := by rw [← hx, p_eq_Φ_neg, q_Φ] |
| rw [hq, φ_neg, loss_eq] |
| rw [← hx] |
| simp only [p, φ, OneCell.gaussianConst, OneCell.density] |
| ring |
|
|
| lemma loss_rationalize {C α r L : ℝ} (_hC : 0 < C) (hα : 0 < α) |
| (_hr : 0 ≤ r) (hs : 0 ≤ 1-α*r^2) |
| (hl : C*(1-Real.sqrt (1-α*r^2)) ≤ α*L) : |
| C*r^2/(1+Real.sqrt (1-α*r^2)) ≤ L := by |
| have hq := Real.sqrt_nonneg (1-α*r^2) |
| have hq2 := Real.sq_sqrt hs |
| rw [div_le_iff₀ (by positivity)] |
| have hm := mul_le_mul_of_nonneg_right hl (show 0 ≤ 1+Real.sqrt (1-α*r^2) by positivity) |
| have heq : C*(1-Real.sqrt (1-α*r^2))*(1+Real.sqrt (1-α*r^2)) = α*(C*r^2) := by |
| calc |
| _ = C*(1-Real.sqrt (1-α*r^2)^2) := by ring |
| _ = _ := by rw [hq2]; ring |
| rw [heq] at hm |
| exact (mul_le_mul_iff_right₀ hα).mp (by simpa only [mul_assoc] using hm) |
|
|
| lemma minimal_loss_constraint (hk : 2 ≤ k) {A : Fin k → Set (Space d)} |
| (hA : MinimalOptimal A) (hC : 9/(8*Real.pi) < value A) |
| (hall : ∀ j, gaussian d (A j) ≠ 0) (i : Fin k) : |
| ∃ x ≥ (0:ℝ), p x=(gaussian d).real (A i) ∧ |
| (value A)*(‖centroid (A i)‖/Real.sqrt (value A))^2 / |
| (1+Real.sqrt (1-((k:ℝ)/((k:ℝ)-1))*(‖centroid (A i)‖/Real.sqrt (value A))^2)) ≤ loss x := by |
| let C := value A |
| let r := ‖centroid (A i)‖/Real.sqrt C |
| let α := (k:ℝ)/((k:ℝ)-1) |
| have hCp : 0 < C := (by positivity : (0:ℝ)<9/(8*Real.pi)).trans hC |
| have hk1 : (1:ℝ)<k := by exact_mod_cast (show 1<k by omega) |
| have hα : 0<α := div_pos (by linarith) (by linarith) |
| have hP := active_probability_mem hk hA hCp (hall i) |
| obtain ⟨x,hx,hpx⟩ := exists_p hP.1 hP.2 |
| refine ⟨x,hx,hpx,?_⟩ |
| have hl := (optimal_loss_lower hk hA.1 i).trans (minimal_loss_upper hk hA hCp hall i) |
| rw [mul_assoc, ← loss_eq_quantile ⟨hP.1,by linarith [hP.2]⟩ hpx] at hl |
| have hsq : ‖centroid (A i)‖^2 = C*r^2 := by |
| dsimp [r] |
| rw [div_pow, Real.sq_sqrt hCp.le] |
| field_simp |
| have hnon : 0 ≤ C-α*‖centroid (A i)‖^2 := by |
| change 0 ≤ (∑ j, ‖centroid (A j)‖^2) - ((k:ℝ)/((k:ℝ)-1))*‖centroid (A i)‖^2 |
| rw [← recentered_norm_sq hk (fun j => centroid (A j)) (sum_centroid_eq_zero hA.1.1) i] |
| positivity |
| have hs : 0 ≤ 1-α*r^2 := by |
| rw [hsq] at hnon |
| nlinarith only [hnon,hCp] |
| have hsqrt : Real.sqrt (C-α*‖centroid (A i)‖^2) = Real.sqrt C*Real.sqrt (1-α*r^2) := by |
| rw [hsq, show C-α*(C*r^2)=C*(1-α*r^2) by ring, Real.sqrt_mul hCp.le] |
| change C-Real.sqrt C*Real.sqrt (C-α*‖centroid (A i)‖^2) ≤ α*loss x at hl |
| rw [hsqrt] at hl |
| have ht := Real.sq_sqrt hCp.le |
| apply loss_rationalize hCp hα (by positivity : 0≤r) hs |
| have he : C-Real.sqrt C*(Real.sqrt C*Real.sqrt (1-α*r^2)) = C*(1-Real.sqrt (1-α*r^2)) := by |
| rw [←mul_assoc, ←pow_two, ht]; ring |
| rwa [he] at hl |
|
|
| lemma minimal_universal_loss (hk : 2 ≤ k) {A : Fin k → Set (Space d)} |
| (hA : MinimalOptimal A) (hC : 9/(8*Real.pi) < value A) |
| (hall : ∀ j, gaussian d (A j) ≠ 0) (i : Fin k) : |
| ∃ x ≥ (0:ℝ), p x=(gaussian d).real (A i) ∧ |
| (9/(8*Real.pi))*(‖centroid (A i)‖/Real.sqrt (value A))^2 / |
| (1+Real.sqrt (1-(‖centroid (A i)‖/Real.sqrt (value A))^2)) ≤ loss x := by |
| obtain ⟨x,hx,hp,hl⟩ := minimal_loss_constraint hk hA hC hall i |
| refine ⟨x,hx,hp, le_trans ?_ hl⟩ |
| have hCp : 0<value A := (by positivity : (0:ℝ)<9/(8*Real.pi)).trans hC |
| apply div_le_div₀ (by positivity) (mul_le_mul_of_nonneg_right hC.le (sq_nonneg _)) (by positivity) |
| apply add_le_add_right |
| apply Real.sqrt_le_sqrt |
| have hk1 : (1:ℝ)<k := by exact_mod_cast (show 1<k by omega) |
| have hα : 1≤(k:ℝ)/((k:ℝ)-1) := (le_div_iff₀ (by linarith)).mpr (by linarith) |
| nlinarith only [mul_nonneg (sub_nonneg.mpr hα) (sq_nonneg (‖centroid (A i)‖/Real.sqrt (value A)))] |
|
|
| lemma minimal_linear_quadratic (hk : 2 ≤ k) {A : Fin k → Set (Space d)} |
| (hA : MinimalOptimal A) (hC : 9/(8*Real.pi) < value A) |
| (hall : ∀ j, gaussian d (A j) ≠ 0) (i : Fin k) : |
| 0.415*(‖centroid (A i)‖/Real.sqrt (value A))+ |
| 0.15*(‖centroid (A i)‖/Real.sqrt (value A))^2 < (gaussian d).real (A i) := by |
| have hCp : 0<value A := (by positivity : (0:ℝ)<9/(8*Real.pi)).trans hC |
| obtain ⟨x,hx,hp,hl⟩ := minimal_universal_loss hk hA hC hall i |
| rw [← hp] |
| exact linear_quadratic_of_loss (div_pos (norm_pos_iff.mpr (active_centroid_ne_zero hA hCp (hall i))) |
| (Real.sqrt_pos.mpr hCp)) (normalized_centroid_lt_two_thirds (hA.1.1.1 i) hC).le hx hl |
|
|
| end GaussianPropeller.Reduction |
|
|
| end OAI |
|
|
