| import Mathlib |
| import OAI.Probability.SKGap.Entropy.EntropyTransport |
| import OAI.Probability.SKGap.Entropy.EntropyEll |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory InformationTheory Real Set Filter |
| open scoped NNReal ENNReal Topology |
| noncomputable section |
| open Real Set |
| namespace SKGap |
| theorem gaussian_tanh_integration_by_parts {d : ℝ} {s : ℝ≥0} (hs : s ≠ 0) : |
| (∫ y, (y - d) * tanh y ∂gaussianReal d s) = |
| (s : ℝ) * (1 - ∫ y, tanh y ^ 2 ∂gaussianReal d s) := by |
| have ht : Integrable tanh (gaussianReal d s) := Integrable.of_bound |
| continuous_tanh.aestronglyMeasurable 1 (ae_of_all _ (fun y => (abs_tanh_lt_one y).le)) |
| have h := gaussian_integration_by_parts hs hasDerivAt_tanh ht |
| ((integrable_const 1).sub integrable_tanh_sq) |
| (integrable_sub_mul_tanh IsGaussian.integrable_id d) |
| rw [integral_sub (integrable_const 1) integrable_tanh_sq] at h |
| simpa using h |
|
|
| theorem scalar_entropy_finite {P : Measure ℝ} [IsProbabilityMeasure P] |
| (hP : Integrable (fun y : ℝ => y ^ 2) P) |
| {j d : ℝ} {s : ℝ≥0} (hj0 : 0 < j) (hj1 : j < 1) (hs : s ≠ 0) |
| (hcons : j * (∫ y, tanh y ^ 2 ∂P) ≤ (s : ℝ)) |
| (hD : klDiv P (gaussianReal d s) ≠ ⊤) : |
| let q := ∫ y, tanh y ^ 2 ∂P |
| let a := ∫ y, (y - d) * tanh y ∂P |
| let T := j * (a - (s : ℝ) * (1 - q)) ^ 2 / (2 * (s : ℝ) * ((s : ℝ) + j * q)) |
| T ≤ (klDiv P (gaussianReal d s)).toReal ∧ |
| (a ≠ (s : ℝ) * (1 - q) → T < (klDiv P (gaussianReal d s)).toReal) := by |
| dsimp only |
| let q := ∫ y, tanh y ^ 2 ∂P |
| let a := ∫ y, (y - d) * tanh y ∂P |
| let k := j * q / (s : ℝ) |
| let B := j * (1 - q) |
| let C := j * a / (s : ℝ) |
| let D := (klDiv P (gaussianReal d s)).toReal |
| have hsp : 0 < (s : ℝ) := NNReal.coe_pos.mpr (pos_iff_ne_zero.mpr hs) |
| have hq0 : 0 < q := entropy_q_pos hs hD |
| have hq1 : q < 1 := entropy_q_lt_one |
| have hk0 : 0 < k := div_pos (mul_pos hj0 hq0) hsp |
| have hk1 : k ≤ 1 := (div_le_one hsp).mpr hcons |
| have hB0 : 0 < B := mul_pos hj0 (sub_pos.mpr hq1) |
| have hB1 : B < 1 := by dsimp [B]; nlinarith |
| have htransport (u : ℝ) (hu : u < 1) : |
| u * (C - B) - k * u ^ 2 / 2 - B * entropyEll u ≤ j * D := by |
| have h := mul_le_mul_of_nonneg_left (entropy_transport_lower (secondMoment_integrable hP) hu hs hD) hj0.le |
| change j * (u / (s : ℝ) * a - u ^ 2 / (2 * (s : ℝ)) * q + (1 - q) * log (1 - u)) ≤ j * D at h |
| convert h using 1 |
| dsimp [k, B, C, entropyEll] |
| field_simp |
| ring |
| have hvariance (hC : 1 < C) : C ^ 2 / k - 1 - log (C ^ 2 / k) ≤ 2 * (j * D) := by |
| let v := (∫ y, (y - d) ^ 2 ∂P) / (s : ℝ) |
| have hcauchy := entropy_moment_cauchy hP d hq0 |
| have hcv : C ^ 2 / (j * k) ≤ v := by |
| have heq : C ^ 2 / (j * k) * (s : ℝ) = a ^ 2 / q := by |
| dsimp [C, k] |
| field_simp |
| dsimp [v] |
| rw [le_div_iff₀ hsp, heq, div_le_iff₀ hq0] |
| simpa only [q, a, mul_comm] using hcauchy |
| have hv : 0 < v := (div_pos (pow_pos (by linarith : 0 < C) 2) (mul_pos hj0 hk0)).trans_le hcv |
| have hvar : (∫ y, (y - d) ^ 2 ∂P) = (s : ℝ) * v := by dsimp [v]; field_simp |
| have hbound := entropy_variance_lower hs (centered_secondMoment_integrable hP d) hD hv hvar |
| exact entropy_variance_scaled_lower hj0 hj1.le hk0 hk1 hC hcv hbound |
| have hopt := entropy_scalar_optimization hk0 hk1 hB0 hB1 |
| (mul_nonneg hj0.le ENNReal.toReal_nonneg) htransport hvariance |
| have heq : (C - B) ^ 2 / (2 * (1 + k)) = |
| j * (j * (a - (s : ℝ) * (1 - q)) ^ 2 / (2 * (s : ℝ) * ((s : ℝ) + j * q))) := by |
| dsimp [C, B, k] |
| field_simp |
| rw [heq] at hopt |
| refine ⟨le_of_mul_le_mul_left hopt.1 hj0, ?_⟩ |
| intro ha |
| have hCB : C ≠ B := by |
| intro h |
| apply ha |
| dsimp [C, B] at h |
| have he := (div_eq_iff hsp.ne').mp h |
| nlinarith |
| exact lt_of_mul_lt_mul_left (hopt.2 hCB) hj0.le |
| |
| def scalarEntropyTarget (P : Measure ℝ) (j d : ℝ) (s : ℝ≥0) : ℝ := |
| j * ((∫ y, (y - d) * tanh y ∂P) - (s : ℝ) * (1 - ∫ y, tanh y ^ 2 ∂P)) ^ 2 / |
| (2 * (s : ℝ) * ((s : ℝ) + j * (∫ y, tanh y ^ 2 ∂P))) |
| |
| theorem scalarEntropyTarget_nonneg {P : Measure ℝ} {j d : ℝ} {s : ℝ≥0} (hj : 0 ≤ j) : |
| 0 ≤ scalarEntropyTarget P j d s := by |
| unfold scalarEntropyTarget |
| apply div_nonneg (mul_nonneg hj (sq_nonneg _)) |
| positivity |
| |
| theorem scalar_entropy_inequality {P : Measure ℝ} [IsProbabilityMeasure P] |
| (hP : Integrable (fun y : ℝ => y ^ 2) P) |
| {j d : ℝ} {s : ℝ≥0} (hj0 : 0 < j) (hj1 : j < 1) (hs : s ≠ 0) |
| (hcons : j * (∫ y, tanh y ^ 2 ∂P) ≤ (s : ℝ)) : |
| ENNReal.ofReal (scalarEntropyTarget P j d s) ≤ klDiv P (gaussianReal d s) := by |
| by_cases hD : klDiv P (gaussianReal d s) = ⊤ |
| · simp [hD] |
| · rw [ENNReal.ofReal_le_iff_le_toReal hD] |
| exact (scalar_entropy_finite hP hj0 hj1 hs hcons hD).1 |
| |
| theorem scalar_entropy_equality_iff {P : Measure ℝ} [IsProbabilityMeasure P] |
| (hP : Integrable (fun y : ℝ => y ^ 2) P) |
| {j d : ℝ} {s : ℝ≥0} (hj0 : 0 < j) (hj1 : j < 1) (hs : s ≠ 0) |
| (hcons : j * (∫ y, tanh y ^ 2 ∂P) ≤ (s : ℝ)) : |
| klDiv P (gaussianReal d s) = ENNReal.ofReal (scalarEntropyTarget P j d s) ↔ |
| P = gaussianReal d s := by |
| constructor |
| · intro he |
| have hD : klDiv P (gaussianReal d s) ≠ ⊤ := by rw [he]; exact ENNReal.ofReal_ne_top |
| have hreal : (klDiv P (gaussianReal d s)).toReal = scalarEntropyTarget P j d s := by |
| rw [he, ENNReal.toReal_ofReal (scalarEntropyTarget_nonneg hj0.le)] |
| have hm : (∫ y, (y - d) * tanh y ∂P) = (s : ℝ) * (1 - ∫ y, tanh y ^ 2 ∂P) := by |
| by_contra h |
| have hstrict := (scalar_entropy_finite hP hj0 hj1 hs hcons hD).2 h |
| change scalarEntropyTarget P j d s < _ at hstrict |
| rw [hreal] at hstrict |
| exact lt_irrefl _ hstrict |
| have hzero : klDiv P (gaussianReal d s) = 0 := by simpa [scalarEntropyTarget, hm] using he |
| exact klDiv_eq_zero_iff.mp hzero |
| · rintro rfl |
| simp [scalarEntropyTarget, gaussian_tanh_integration_by_parts hs, klDiv_self] |
| |
| end SKGap |
| |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Real Set Filter |
| open scoped Topology NNReal ENNReal BoundedContinuousFunction |
| namespace SKGap |
| |
| def EntropyTestBound (P Q : Measure ℝ) (D : ℝ) : Prop := |
| ∀ f : ℝ →ᵇ ℝ, (∫ x, f x ∂P) - log (∫ x, exp (f x) ∂Q) ≤ D |
| |
| def realClip (n : ℕ) (x : ℝ) : ℝ := max (-(n:ℝ)) (min (n:ℝ) x) |
| |
| lemma realClip_bounds (n : ℕ) (x : ℝ) : -(n:ℝ) ≤ realClip n x ∧ realClip n x ≤ n := by |
| constructor |
| · exact le_max_left _ _ |
| · exact max_le (by linarith [Nat.cast_nonneg (α := ℝ) n]) (min_le_left _ _) |
| |
| lemma realClip_abs (n : ℕ) (x : ℝ) : |realClip n x| ≤ |x| := by |
| rw [abs_le] |
| constructor |
| · exact le_sup_of_le_right (le_min (by linarith [abs_nonneg x]) (neg_abs_le x)) |
| · exact sup_le (by linarith [abs_nonneg x]) ((min_le_right _ _).trans (le_abs_self x)) |
| |
| lemma realClip_le (n : ℕ) (x : ℝ) : realClip n x ≤ max 0 x := |
| max_le (by linarith [le_max_left (0:ℝ) x]) ((min_le_right _ _).trans (le_max_right _ _)) |
| |
| lemma tendsto_realClip (x : ℝ) : Tendsto (fun n : ℕ => realClip n x) atTop (𝓝 x) := by |
| have he : ∀ᶠ n : ℕ in atTop, realClip n x = x := by |
| obtain ⟨N, hN⟩ := exists_nat_ge |x| |
| filter_upwards [eventually_ge_atTop N] with n hn |
| have hb : |x| ≤ (n:ℝ) := hN.trans (by exact_mod_cast hn) |
| have hx := abs_le.mp hb |
| simp only [realClip, min_eq_right hx.2, max_eq_right hx.1] |
| exact (tendsto_congr' he).2 tendsto_const_nhds |
|
|
| def clippedBC {f : ℝ → ℝ} (hf : Continuous f) (n : ℕ) : ℝ →ᵇ ℝ := |
| BoundedContinuousFunction.ofNormedAddCommGroup (fun x => realClip n (f x)) |
| (by unfold realClip; fun_prop) n (fun x => by |
| rw [Real.norm_eq_abs, abs_le]; exact realClip_bounds n (f x)) |
|
|
| lemma entropyTestBound_continuous {P Q : Measure ℝ} [IsProbabilityMeasure P] |
| [IsProbabilityMeasure Q] {D : ℝ} (hD : EntropyTestBound P Q D) |
| {f : ℝ → ℝ} (hf : Continuous f) (hi : Integrable f P) |
| (he : Integrable (fun x => exp (f x)) Q) : |
| (∫ x, f x ∂P) - log (∫ x, exp (f x) ∂Q) ≤ D := by |
| have hP : Tendsto (fun n => ∫ x, clippedBC hf n x ∂P) atTop (𝓝 (∫ x, f x ∂P)) := by |
| apply tendsto_integral_of_dominated_convergence (fun x => |f x|) |
| · intro n; exact (clippedBC hf n).continuous.aestronglyMeasurable |
| · exact hi.abs |
| · intro n; exact ae_of_all _ (fun x => by |
| change ‖realClip n (f x)‖ ≤ |f x| |
| rw [Real.norm_eq_abs]; exact realClip_abs n (f x)) |
| · exact ae_of_all _ (fun x => tendsto_realClip (f x)) |
| have hQ : Tendsto (fun n => ∫ x, exp (clippedBC hf n x) ∂Q) atTop (𝓝 (∫ x, exp (f x) ∂Q)) := by |
| apply tendsto_integral_of_dominated_convergence (fun x => 1+exp (f x)) |
| · intro n; exact (by fun_prop : Continuous (fun x => exp (clippedBC hf n x))).aestronglyMeasurable |
| · exact (integrable_const 1).add he |
| · intro n; exact ae_of_all _ (fun x => by |
| rw [Real.norm_eq_abs, abs_of_pos (exp_pos _)] |
| calc |
| _ ≤ exp (max 0 (f x)) := exp_le_exp.mpr (realClip_le n (f x)) |
| _ ≤ 1+exp (f x) := by |
| by_cases hx : 0 ≤ f x |
| · rw [max_eq_right hx]; linarith |
| · rw [max_eq_left (le_of_not_ge hx), exp_zero]; linarith [exp_pos (f x)]) |
| · exact ae_of_all _ (fun x => continuous_exp.continuousAt.tendsto.comp (tendsto_realClip (f x))) |
| have hpos : 0 < ∫ x, exp (f x) ∂Q := integral_pos_iff_support_of_nonneg |
| (fun x => (exp_pos (f x)).le) he |>.mpr (by simp [Function.support, (exp_pos _).ne']) |
| exact le_of_tendsto (hP.sub ((continuousAt_log hpos.ne').tendsto.comp hQ)) |
| (Eventually.of_forall (fun n => hD (clippedBC hf n))) |
|
|
| lemma entropyTestBound_normalized {P Q : Measure ℝ} [IsProbabilityMeasure P] |
| [IsProbabilityMeasure Q] {D : ℝ} (hD : EntropyTestBound P Q D) |
| {f : ℝ → ℝ} (hf : Continuous f) (hi : Integrable f P) |
| (he : Integrable (fun x => exp (f x)) Q) (hZ : ∫ x, exp (f x) ∂Q = 1) : |
| (∫ x, f x ∂P) ≤ D := by |
| simpa only [hZ, log_one, sub_zero] using entropyTestBound_continuous hD hf hi he |
|
|
| lemma integrable_exp_boundedContinuous {Q : Measure ℝ} [IsFiniteMeasure Q] (f : ℝ →ᵇ ℝ) (t : ℝ) : |
| Integrable (fun x => exp (t*f x)) Q := by |
| apply Integrable.of_bound (by fun_prop) (exp (|t| * ‖f‖)) |
| exact ae_of_all _ (fun x => by |
| rw [Real.norm_eq_abs, abs_of_pos (exp_pos _)] |
| apply exp_le_exp.mpr |
| calc |
| t*f x ≤ |t*f x| := le_abs_self _ |
| _ = |t| * |f x| := abs_mul _ _ |
| _ ≤ |t| * ‖f‖ := mul_le_mul_of_nonneg_left (by simpa only [Real.norm_eq_abs] using f.norm_coe_le_norm x) (abs_nonneg _)) |
|
|
| lemma boundedContinuous_mgf_deriv {Q : Measure ℝ} [IsProbabilityMeasure Q] (f : ℝ →ᵇ ℝ) : |
| HasDerivAt (fun t : ℝ => ∫ x, exp (t*f x) ∂Q) (∫ x, f x ∂Q) 0 := by |
| have hset : integrableExpSet (fun x => f x) Q = univ := by |
| ext t; exact iff_true_intro (integrable_exp_boundedContinuous f t) |
| have h := hasDerivAt_mgf (X := fun x => f x) (μ := Q) (t := 0) (by simp [hset]) |
| convert! h using 1; simp |
|
|
| lemma entropyTestBound_zero_eq {P Q : Measure ℝ} [IsProbabilityMeasure P] |
| [IsProbabilityMeasure Q] (hD : EntropyTestBound P Q 0) : P = Q := by |
| apply ext_of_forall_integral_eq_of_IsFiniteMeasure |
| intro f |
| let g : ℝ → ℝ := fun t => log (∫ x, exp (t*f x) ∂Q) - t*(∫ x, f x ∂P) |
| have hg0 : g 0 = 0 := by simp [g] |
| have hmin : IsLocalMin g 0 := by |
| apply Filter.Eventually.of_forall |
| intro t |
| have h := hD (t • f) |
| simp only [BoundedContinuousFunction.smul_apply, smul_eq_mul, integral_const_mul] at h |
| rw [hg0] |
| dsimp [g] |
| exact (show 0 ≤ log (∫ x, exp (t*f x) ∂Q)-t*(∫ x, f x ∂P) by linarith) |
| have hder : HasDerivAt g ((∫ x, f x ∂Q)-(∫ x, f x ∂P)) 0 := by |
| have h := (boundedContinuous_mgf_deriv (Q := Q) f).log (by simp) |
| convert! h.sub ((hasDerivAt_id (0:ℝ)).mul_const (∫ x, f x ∂P)) using 1; simp |
| have hz := hmin.hasDerivAt_eq_zero hder |
| linarith |
|
|
| lemma entropyTestBound_nonneg {P Q : Measure ℝ} [IsProbabilityMeasure P] |
| [IsProbabilityMeasure Q] {D : ℝ} (hD : EntropyTestBound P Q D) : 0 ≤ D := by |
| simpa using hD 0 |
|
|
| end SKGap |
|
|
| end |
| end |
| end |
| end |
|
|
| end OAI |
|
|
