| import Mathlib |
| import OAI.Probability.SKGap.Entropy.ExponentiallyRare |
|
|
| namespace OAI |
|
|
| section |
| open scoped BigOperators |
| open scoped BigOperators |
| open scoped BigOperators |
| open scoped BigOperators |
| open scoped BigOperators |
| open scoped BigOperators NNReal |
| open MeasureTheory ProbabilityTheory |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators NNReal |
| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators NNReal ENNReal |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators NNReal ENNReal |
| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators Matrix Matrix.Norms.Elementwise |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators Matrix Matrix.Norms.Elementwise |
| open scoped BigOperators |
| open scoped BigOperators NNReal ENNReal |
| open MeasureTheory Metric Set |
| open scoped BigOperators NNReal ENNReal |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators NNReal ENNReal Matrix.Norms.L2Operator |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Matrix.Norms.L2Operator |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Matrix Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter Set |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators ENNReal NNReal |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators NNReal ENNReal Matrix |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators ENNReal NNReal |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators NNReal ENNReal |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators Matrix Matrix.Norms.Elementwise NNReal ENNReal |
| open scoped BigOperators |
| open Filter Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal ENNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators NNReal ENNReal Topology |
| namespace SKGapCutoff.Regression |
|
|
| theorem ExponentialEmpiricalConcentration.parameter_map_weight |
| {E F A : Type*} [PseudoMetricSpace E] [MeasurableSpace E] [BorelSpace E] |
| [PseudoMetricSpace F] [MeasurableSpace F] [BorelSpace F] [PseudoMetricSpace A] |
| {H : ℕ → Type*} [∀ n, MeasurableSpace (H n)] |
| {ρ : ∀ n, Measure (H n)} {X : ∀ n, H n → Fin n → E} {ν : Measure E} |
| (hX : ExponentialEmpiricalConcentration ρ X ν) |
| (C : ∀ n, H n → A) (c : A) (hC : ExponentialConvergence ρ C c) |
| (g : A → E → F) {L : ℝ≥0} (hg : LipschitzWith L (g c)) |
| (W : E → ℝ) (D : ℝ) (hD : 0 < D) |
| (hgc : ∀ a x, dist (g a x) (g c x) ≤ dist a c * W x) |
| (hrare : ExponentiallyRare ρ (fun n => {h | D ≤ (∑ i, W (X n h i))/(n:ℝ)})) : |
| ExponentialEmpiricalConcentration ρ (fun n h i => g (C n h) (X n h i)) |
| (ν.map (g c)) := by |
| have hxmap := hX.map (g c) hg |
| intro f K hf B hB ε hε |
| have hs : ExponentiallyRare ρ (fun n => {h | ε/2 ≤ |
| |(∑ i, f (g c (X n h i)))/(n:ℝ) - ∫ x, f x ∂ν.map (g c)|}) := |
| hxmap f K hf B hB (ε/2) (by positivity) |
| let δ := ε/(2*((K:ℝ)*D+1)) |
| have hδ : 0 < δ := by dsimp [δ] |
| apply ((hs.union (hC δ hδ)).union hrare).mono |
| filter_upwards [eventually_ge_atTop 1] with n hn |
| intro h hh |
| by_cases hb : ε/2 ≤ |(∑ i, f (g c (X n h i)))/(n:ℝ) - ∫ x, f x ∂ν.map (g c)| |
| · exact Or.inl (Or.inl hb) |
| by_cases hcnot : δ ≤ dist (C n h) c |
| · exact Or.inl (Or.inr hcnot) |
| apply Or.inr |
| by_contra hlarge |
| have hcclose : dist (C n h) c < δ := lt_of_not_ge hcnot |
| have hsmall : (∑ i, W (X n h i))/(n:ℝ) < D := lt_of_not_ge hlarge |
| have hnR : (0:ℝ) < n := Nat.cast_pos.mpr hn |
| have hdiff : |(∑ i, f (g (C n h) (X n h i)))/(n:ℝ) - |
| (∑ i, f (g c (X n h i)))/(n:ℝ)| ≤ (K:ℝ)*dist (C n h) c*D := by |
| rw [← sub_div, ← Finset.sum_sub_distrib, abs_div, abs_of_pos hnR] |
| calc |
| |∑ i, (f (g (C n h) (X n h i))-f (g c (X n h i)))|/(n:ℝ) ≤ |
| (∑ i, |f (g (C n h) (X n h i))-f (g c (X n h i))|)/(n:ℝ) := by |
| gcongr |
| exact Finset.abs_sum_le_sum_abs _ _ |
| _ ≤ (∑ i : Fin n, (K:ℝ)*dist (C n h) c*W (X n h i))/(n:ℝ) := by |
| gcongr with i |
| exact (hf.dist_le_mul _ _).trans |
| (by calc |
| (K:ℝ)*dist (g (C n h) (X n h i)) (g c (X n h i)) ≤ |
| (K:ℝ)*(dist (C n h) c*W (X n h i)) := |
| mul_le_mul_of_nonneg_left (hgc _ _) K.coe_nonneg |
| _ = (K:ℝ)*dist (C n h) c*W (X n h i) := by ring) |
| _ = (K:ℝ)*dist (C n h) c*((∑ i, W (X n h i))/(n:ℝ)) := by |
| rw [← Finset.mul_sum]; ring |
| _ ≤ (K:ℝ)*dist (C n h) c*D := |
| mul_le_mul_of_nonneg_left hsmall.le (mul_nonneg K.coe_nonneg dist_nonneg) |
| have hδbound : (K:ℝ)*δ*D ≤ ε/2 := by |
| dsimp [δ] |
| have hp : 0 < 2*((K:ℝ)*D+1) := by positivity |
| apply (le_div_iff₀ (by norm_num : (0:ℝ)<2)).mpr |
| field_simp |
| nlinarith [mul_nonneg K.coe_nonneg hD.le] |
| have hstep : (K:ℝ)*dist (C n h) c*D ≤ ε/2 := |
| (mul_le_mul_of_nonneg_right |
| (mul_le_mul_of_nonneg_left hcclose.le K.coe_nonneg) hD.le).trans hδbound |
| have ht := abs_add_le |
| ((∑ i, f (g (C n h) (X n h i)))/(n:ℝ) - (∑ i, f (g c (X n h i)))/(n:ℝ)) |
| ((∑ i, f (g c (X n h i)))/(n:ℝ) - ∫ x, f x ∂ν.map (g c)) |
| rw [sub_add_sub_cancel] at ht |
| have hb' := lt_of_not_ge hb |
| exact (not_lt_of_ge hh) (ht.trans_lt (by linarith)) |
| |
| lemma average_lipschitz_metric_sq {E : Type*} [PseudoMetricSpace E] |
| {n : ℕ} (hn : 0 < n) (f : E → ℝ) {K : ℝ≥0} (hf : LipschitzWith K f) |
| (v w : Fin n → E) : |
| |((∑ i, f (v i))-(∑ i, f (w i)))/(n:ℝ)|^2 ≤ |
| (K:ℝ)^2 * (∑ i, (dist (v i) (w i))^2)/(n:ℝ) := by |
| have hnR : (0:ℝ) < n := Nat.cast_pos.mpr hn |
| rw [sq_abs, div_pow, ← Finset.sum_sub_distrib] |
| have hcs := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ |
| (fun i : Fin n => (1:ℝ)) (fun i => f (v i)-f (w i)) |
| simp only [one_mul, one_pow, Finset.sum_const, Finset.card_univ, |
| Fintype.card_fin, nsmul_eq_mul, mul_one] at hcs |
| have hsite (i : Fin n) : (f (v i)-f (w i))^2 ≤ (K:ℝ)^2*(dist (v i) (w i))^2 := by |
| have h := hf.dist_le_mul (v i) (w i) |
| rw [Real.dist_eq] at h |
| have hs := mul_self_le_mul_self (abs_nonneg _) h |
| simpa [← pow_two, mul_pow] using hs |
| apply (div_le_iff₀ (sq_pos_of_pos hnR)).mpr |
| calc |
| _ ≤ (n:ℝ) * ∑ i, (f (v i)-f (w i))^2 := hcs |
| _ ≤ (n:ℝ) * ∑ i, (K:ℝ)^2*(dist (v i) (w i))^2 := |
| mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun i _ => hsite i)) hnR.le |
| _ = _ := by rw [← Finset.mul_sum]; field_simp |
| |
| theorem ExponentialEmpiricalConcentration.l2_stability |
| {E : Type*} [PseudoMetricSpace E] [MeasurableSpace E] |
| {H : ℕ → Type*} [∀ n, MeasurableSpace (H n)] |
| {ρ : ∀ n, Measure (H n)} {X Y : ∀ n, H n → Fin n → E} {ν : Measure E} |
| (hX : ExponentialEmpiricalConcentration ρ X ν) |
| (hXY : ExponentialConvergence ρ |
| (fun n h => (∑ i, (dist (X n h i) (Y n h i))^2)/(n:ℝ)) 0) : |
| ExponentialEmpiricalConcentration ρ Y ν := by |
| intro f K hf B hB ε hε |
| have hs : ExponentiallyRare ρ (fun n => {h | ε/2 ≤ |
| |(∑ i, f (X n h i))/(n:ℝ) - ∫ x, f x ∂ν|}) := |
| hX f K hf B hB (ε/2) (by positivity) |
| let δ := ε^2/(4*((K:ℝ)^2+1)) |
| have hδ : 0 < δ := by dsimp [δ]; positivity |
| apply (hs.union (hXY δ hδ)).mono |
| filter_upwards [eventually_ge_atTop 1] with n hn |
| intro h hh |
| by_cases hb : ε/2 ≤ |(∑ i, f (X n h i))/(n:ℝ) - ∫ x, f x ∂ν| |
| · exact Or.inl hb |
| apply Or.inr |
| by_contra hbad |
| have hclose : (∑ i, (dist (X n h i) (Y n h i))^2)/(n:ℝ) < δ := by |
| change ¬δ ≤ dist ((∑ i, (dist (X n h i) (Y n h i))^2)/(n:ℝ)) 0 at hbad |
| have hk := lt_of_not_ge hbad |
| simpa only [Real.dist_eq, sub_zero, abs_of_nonneg (by positivity : |
| 0 ≤ (∑ i, (dist (X n h i) (Y n h i))^2)/(n:ℝ))] using hk |
| have hbound := average_lipschitz_metric_sq hn f hf (X n h) (Y n h) |
| rw [mul_div_assoc] at hbound |
| have hδbound : (K:ℝ)^2*δ < ε^2/4 := by |
| dsimp [δ] |
| have hp : 0 < 4*((K:ℝ)^2+1) := by positivity |
| apply (lt_div_iff₀ (by norm_num : (0:ℝ)<4)).mpr |
| field_simp |
| nlinarith [sq_pos_of_pos hε] |
| have hsmall : |(∑ i, f (X n h i))/(n:ℝ) - (∑ i, f (Y n h i))/(n:ℝ)| < ε/2 := by |
| rw [← sub_div] |
| have ht : |((∑ i, f (X n h i))-(∑ i, f (Y n h i)))/(n:ℝ)|^2 < ε^2/4 := |
| hbound.trans_lt ((mul_le_mul_of_nonneg_left hclose.le (sq_nonneg _)).trans_lt hδbound) |
| nlinarith [abs_nonneg (((∑ i, f (X n h i))-(∑ i, f (Y n h i)))/(n:ℝ))] |
| have ht := abs_add_le |
| ((∑ i, f (Y n h i))/(n:ℝ) - (∑ i, f (X n h i))/(n:ℝ)) |
| ((∑ i, f (X n h i))/(n:ℝ) - ∫ x, f x ∂ν) |
| rw [sub_add_sub_cancel, abs_sub_comm ((∑ i, f (Y n h i))/(n:ℝ)) |
| ((∑ i, f (X n h i))/(n:ℝ))] at ht |
| have hb' := lt_of_not_ge hb |
| exact (not_lt_of_ge hh) (ht.trans_lt (by linarith)) |
|
|
| lemma ExponentialConvergence.dominated_zero |
| {H : ℕ → Type*} [∀ n, MeasurableSpace (H n)] |
| {ρ : ∀ n, Measure (H n)} {X Y : ∀ n, H n → ℝ} |
| (hX : ExponentialConvergence ρ X 0) {A : ∀ n, Set (H n)} |
| (hA : ExponentiallyRare ρ A) (B : ℝ) (hB : 0 ≤ B) |
| (hXY : ∀ᶠ n in atTop, ∀ h ∉ A n, |Y n h| ≤ B*|X n h|) : |
| ExponentialConvergence ρ Y 0 := by |
| intro ε hε |
| let δ := ε/(B+1) |
| have hδ : 0 < δ := by dsimp [δ]; positivity |
| apply (hA.union (hX δ hδ)).mono |
| filter_upwards [hXY] with n hn |
| intro h hh |
| by_cases ha : h ∈ A n |
| · exact Or.inl ha |
| apply Or.inr |
| by_contra hx |
| change ¬δ ≤ dist (X n h) 0 at hx |
| have hx' : |X n h| < δ := by simpa only [Real.dist_eq, sub_zero] using lt_of_not_ge hx |
| have hy : ε ≤ |Y n h| := by simpa only [Set.mem_ofPred_eq, Real.dist_eq, sub_zero] using hh |
| have hδB : B*δ < ε := by |
| dsimp [δ] |
| rw [← mul_div_assoc] |
| apply (div_lt_iff₀ (by positivity : 0 < B+1)).mpr |
| nlinarith |
| exact (not_lt_of_ge hy) ((hn h ha).trans_lt |
| ((mul_le_mul_of_nonneg_left hx'.le hB).trans_lt hδB)) |
| |
| theorem ExponentialConvergence.operator_omit |
| {H E F : ℕ → Type*} [∀ n, MeasurableSpace (H n)] |
| [∀ n, NormedAddCommGroup (E n)] [∀ n, NormedSpace ℝ (E n)] |
| [∀ n, NormedAddCommGroup (F n)] [∀ n, NormedSpace ℝ (F n)] |
| {ρ : ∀ n, Measure (H n)} (A : ∀ n, H n → E n →L[ℝ] F n) |
| (X : ∀ n, H n → E n) |
| (hX : ExponentialConvergence ρ (fun n h => ‖X n h‖^2/(n:ℝ)) 0) |
| (M : ℝ) |
| (hA : ExponentiallyRare ρ (fun n => {h | M ≤ ‖A n h‖})) : |
| ExponentialConvergence ρ (fun n h => ‖A n h (X n h)‖^2/(n:ℝ)) 0 := by |
| apply hX.dominated_zero hA (M^2) (sq_nonneg _) |
| refine Filter.Eventually.of_forall ?_ |
| intro n h hn |
| have ha : ‖A n h‖ < M := lt_of_not_ge hn |
| rw [abs_of_nonneg (by positivity : 0 ≤ ‖A n h (X n h)‖^2/(n:ℝ)), |
| abs_of_nonneg (by positivity : 0 ≤ ‖X n h‖^2/(n:ℝ)), ← mul_div_assoc] |
| apply div_le_div_of_nonneg_right _ (Nat.cast_nonneg _) |
| calc |
| ‖A n h (X n h)‖^2 ≤ (M*‖X n h‖)^2 := |
| pow_le_pow_left₀ (norm_nonneg _) ((A n h).le_opNorm (X n h) |>.trans |
| (mul_le_mul_of_nonneg_right ha.le (norm_nonneg _))) 2 |
| _ = _ := by ring |
| |
| lemma empirical_quadratic_expansion {n : ℕ} {ι : Type*} [Fintype ι] |
| (f : ι → Fin n → ℝ) (c : ι → ℝ) : |
| (∑ a, (∑ i, c i*f i a)^2)/(n:ℝ) = |
| ∑ i, ∑ j, c i*((∑ a, f i a*f j a)/(n:ℝ))*c j := by |
| calc |
| _ = (∑ i, ∑ j, ∑ a, c i*f i a*(c j*f j a))/(n:ℝ) := by |
| congr 1 |
| simp only [pow_two, Finset.sum_mul, Finset.mul_sum] |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro i _ |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro j _ |
| apply Finset.sum_congr rfl |
| intro a _ |
| ring |
| _ = _ := by |
| simp_rw [Finset.sum_div, Finset.mul_sum, Finset.sum_mul] |
| apply Finset.sum_congr rfl |
| intro i _ |
| apply Finset.sum_congr rfl |
| intro j _ |
| apply Finset.sum_congr rfl |
| intro a _ |
| ring |
| |
| lemma ExponentialEmpiricalConcentration.linear_combination_second_moment |
| {E : Type*} [PseudoMetricSpace E] [MeasurableSpace E] |
| {H : ℕ → Type*} [∀ n, MeasurableSpace (H n)] |
| {ρ : ∀ n, Measure (H n)} {X : ∀ n, H n → Fin n → E} {ν : Measure E} |
| (hX : ExponentialEmpiricalConcentration ρ X ν) |
| {ι : Type*} [Fintype ι] (f : ι → E → ℝ) {K B : ℝ≥0} |
| (hf : ∀ i, LipschitzWith K (f i)) (hB : ∀ i x, |f i x| ≤ B) |
| (C : ∀ n, H n → ι → ℝ) (c : ι → ℝ) (hC : ExponentialConvergence ρ C c) : |
| ExponentialConvergence ρ (fun n h => (∑ a, (∑ i, C n h i*f i (X n h a))^2)/(n:ℝ)) |
| (∑ i, ∑ j, c i*(∫ x, f i x*f j x ∂ν)*c j) := by |
| have hc : Continuous (fun z : (ι → ι → ℝ) × (ι → ℝ) => |
| ∑ i, ∑ j, z.2 i*z.1 i j*z.2 j) := by fun_prop |
| have hh := ((hX.gram f hf hB).prod hC).continuous_map hc.continuousAt |
| simpa only [empirical_quadratic_expansion] using hh |
| |
| end SKGapCutoff.Regression |
| |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology |
| |
| end |
| |
| end OAI |
| |
