| import Mathlib |
| import OAI.Probability.SKGap.Localization.TapPotential |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open scoped BigOperators |
| open Real Matrix Set Filter Function |
| open scoped Topology |
| namespace SKGap |
| section Basins |
| variable {X : Type*} [TopologicalSpace X] (T : X → X) |
|
|
| lemma attractionBasin_eq_iUnion {p : X} {U : Set X} (hU : IsOpen U) (hp : p ∈ U) |
| (hatt : U ⊆ attractionBasin T p) : |
| attractionBasin T p = ⋃ k : ℕ, (T^[k]) ⁻¹' U := by |
| ext x |
| constructor |
| · intro hx |
| have h := hx.eventually (hU.mem_nhds hp) |
| obtain ⟨k, hk⟩ := h.exists |
| exact mem_iUnion.mpr ⟨k, hk⟩ |
| · intro hx |
| obtain ⟨k, hk⟩ := mem_iUnion.mp hx |
| exact (mem_attractionBasin_iterate_iff T p x k).mp (hatt hk) |
|
|
| lemma isOpen_attractionBasin (hT : Continuous T) {p : X} {U : Set X} |
| (hU : IsOpen U) (hp : p ∈ U) (hatt : U ⊆ attractionBasin T p) : |
| IsOpen (attractionBasin T p) := by |
| rw [attractionBasin_eq_iUnion T hU hp hatt] |
| exact isOpen_iUnion (fun k => hU.preimage (hT.iterate k)) |
|
|
| lemma disjoint_attractionBasins [T2Space X] {p q : X} (hpq : p ≠ q) : |
| Disjoint (attractionBasin T p) (attractionBasin T q) := by |
| rw [disjoint_left] |
| intro x hx hy |
| exact hpq (tendsto_nhds_unique hx hy) |
|
|
| end Basins |
|
|
| theorem lyapunov_unique_fixed_point {X : Type*} [TopologicalSpace X] [CompactSpace X] |
| [ConnectedSpace X] [T2Space X] |
| (T : X → X) (hT : Continuous T) (φ : X → ℝ) (hφ : Continuous φ) |
| (hstrict : ∀ x, T x ≠ x → φ (T x) < φ x) |
| (hattract : ∀ p, T p = p → ∃ U : Set X, IsOpen U ∧ p ∈ U ∧ |
| U ⊆ attractionBasin T p) : ∃! p, T p = p := by |
| have hopen (p) (hp : T p = p) : IsOpen (attractionBasin T p) := by |
| obtain ⟨U, hU, hpU, hatt⟩ := hattract p hp |
| exact isOpen_attractionBasin T hT hU hpU hatt |
| have hcover (x : X) : ∃ p, T p = p ∧ x ∈ attractionBasin T p := by |
| let U := ⋃ p : {p // T p = p}, attractionBasin T p.val |
| have hU : IsOpen U := isOpen_iUnion (fun p => hopen p.val p.property) |
| have hfixed (p) (hp : T p = p) : p ∈ U := |
| mem_iUnion.mpr ⟨⟨p, hp⟩, fixed_mem_attractionBasin T hp⟩ |
| obtain ⟨k, hk⟩ := lyapunov_eventually_hits T hT φ hφ hstrict hU hfixed x |
| obtain ⟨p, hp⟩ := mem_iUnion.mp hk |
| exact ⟨p.val, p.property, (mem_attractionBasin_iterate_iff T p.val x k).mp hp⟩ |
| obtain ⟨p, hp⟩ := lyapunov_fixed_point T φ hφ hstrict |
| refine ⟨p, hp, fun q hq => ?_⟩ |
| by_contra hqp |
| let V := ⋃ q : {q // T q = q ∧ q ≠ p}, attractionBasin T q.val |
| have hV : IsOpen V := isOpen_iUnion (fun q => hopen q.val q.property.1) |
| have hdisj : Disjoint (attractionBasin T p) V := by |
| rw [disjoint_iUnion_right] |
| intro q |
| exact disjoint_attractionBasins T q.property.2.symm |
| have hcov : (univ : Set X) ⊆ attractionBasin T p ∪ V := by |
| intro x _ |
| obtain ⟨r, hr, hx⟩ := hcover x |
| by_cases he : r = p |
| · subst r |
| exact Or.inl hx |
| · exact Or.inr (mem_iUnion.mpr ⟨⟨r, hr, he⟩, hx⟩) |
| have hconn := IsPreconnected.subset_or_subset (hopen p hp) hV hdisj hcov |
| isPreconnected_univ |
| rcases hconn with hconn | hconn |
| · have hqp' := tendsto_nhds_unique (hconn (mem_univ q)) |
| (fixed_mem_attractionBasin T hq) |
| exact hqp hqp'.symm |
| · exact disjoint_left.mp hdisj (fixed_mem_attractionBasin T hp) (hconn (mem_univ p)) |
|
|
| theorem local_contraction_attracts {X : Type*} [MetricSpace X] |
| {T : X → X} {p : X} (hp : T p = p) {K : NNReal} (hK : K < 1) |
| {U : Set X} (hU : U ∈ 𝓝 p) (hlip : LipschitzOnWith K T U) : |
| ∃ V : Set X, IsOpen V ∧ p ∈ V ∧ V ⊆ attractionBasin T p := by |
| obtain ⟨ε, hε, hball⟩ := Metric.mem_nhds_iff.mp hU |
| have hpU : p ∈ U := hball (Metric.mem_ball_self hε) |
| have hmaps : MapsTo T (Metric.ball p ε) (Metric.ball p ε) := by |
| intro x hx |
| have h := hlip.dist_le_mul x (hball hx) p hpU |
| rw [hp] at h |
| exact h.trans_lt ((mul_le_of_le_one_left (dist_nonneg : 0 ≤ dist x p) |
| (show (K:ℝ) ≤ 1 by exact_mod_cast hK.le)).trans_lt hx) |
| refine ⟨Metric.ball p ε, Metric.isOpen_ball, Metric.mem_ball_self hε, ?_⟩ |
| intro x hx |
| have hbd (n : ℕ) : dist (T^[n] x) p ≤ (K:ℝ)^n * dist x p := by |
| induction n with |
| | zero => simp |
| | succ n ih => |
| rw [iterate_succ_apply'] |
| have h := hlip.dist_le_mul (T^[n] x) (hball (hmaps.iterate n hx)) p hpU |
| rw [hp] at h |
| exact h.trans (by simpa only [pow_succ', mul_assoc] using |
| mul_le_mul_of_nonneg_left ih K.coe_nonneg) |
| have ht := (tendsto_pow_atTop_nhds_zero_of_lt_one K.coe_nonneg |
| (show (K:ℝ) < 1 by exact_mod_cast hK)).mul_const (dist x p) |
| rw [zero_mul] at ht |
| exact Metric.tendsto_atTop.mpr (fun δ hδ => by |
| obtain ⟨N, hN⟩ := eventually_atTop.mp (ht.eventually (gt_mem_nhds hδ)) |
| exact ⟨N, fun n hn => (hbd n).trans_lt (hN n hn)⟩) |
|
|
| theorem strict_derivative_attracts {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {T : E → E} {p : E} {D : E →L[ℝ] E} |
| (hp : T p = p) (hD : HasStrictFDerivAt T D p) (hnorm : ‖D‖ < 1) : |
| ∃ U : Set E, IsOpen U ∧ p ∈ U ∧ U ⊆ attractionBasin T p := by |
| have hn : ‖D‖₊ < 1 := by exact_mod_cast hnorm |
| obtain ⟨K, hDK, hK⟩ := exists_between hn |
| obtain ⟨U, hU, hlip⟩ := hD.exists_lipschitzOnWith_of_nnnorm_lt K hDK |
| exact local_contraction_attracts hp hK hU hlip |
|
|
| end SKGap |
|
|
| noncomputable section |
| open Set Function Real |
| open scoped Topology NNReal |
| namespace SKGap |
| section GradientDescent |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
|
|
| theorem lipschitz_gradient_remainder {φ : E → ℝ} {G : E → E} {s : Set E} |
| (hs : Convex ℝ s) (hgrad : ∀ x ∈ s, HasFDerivAt φ (innerSL ℝ (G x)) x) |
| {L : ℝ≥0} (hG : LipschitzOnWith L G s) {x y : E} (hx : x ∈ s) (hy : y ∈ s) : |
| |φ y - φ x - inner ℝ (G x) (y-x)| ≤ (L:ℝ)*‖y-x‖^2 := by |
| let d := y-x |
| let ℓ : ℝ →ᵃ[ℝ] E := AffineMap.lineMap x y |
| let f := fun t : ℝ => φ (ℓ t) - t * inner ℝ (G x) d |
| let f' := fun t : ℝ => inner ℝ (G (ℓ t) - G x) d |
| have hℓ (t : ℝ) : ℓ t = x + t • d := by |
| simp only [ℓ, d, AffineMap.lineMap_apply_module] |
| module |
| have hmem (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) : ℓ t ∈ s := |
| hs.mapsTo_lineMap hx hy ht |
| have hf (t : ℝ) (ht : t ∈ Icc (0:ℝ) 1) : |
| HasDerivWithinAt f (f' t) (Icc (0:ℝ) 1) t := by |
| have hd := ((hgrad (ℓ t) (hmem t ht)).comp_hasDerivAt t |
| AffineMap.hasDerivAt_lineMap).sub |
| ((hasDerivAt_id t).mul_const (inner ℝ (G x) d)) |
| convert! (hd.hasDerivWithinAt (s := Icc (0:ℝ) 1)) using 1 |
| simp only [f', innerSL_apply_apply, inner_sub_left, d, one_mul] |
| have hb (t : ℝ) (ht : t ∈ Ico (0:ℝ) 1) : |
| ‖f' t‖ ≤ (L:ℝ)*‖d‖^2 := by |
| have hdiff : ‖ℓ t-x‖ ≤ ‖d‖ := by |
| rw [hℓ, add_sub_cancel_left, norm_smul, Real.norm_eq_abs, abs_of_nonneg ht.1] |
| exact mul_le_of_le_one_left (norm_nonneg _) ht.2.le |
| have h := (hG.norm_sub_le (hmem t ⟨ht.1, ht.2.le⟩) hx).trans |
| (mul_le_mul_of_nonneg_left hdiff L.coe_nonneg) |
| calc |
| ‖f' t‖ ≤ ‖G (ℓ t)-G x‖*‖d‖ := norm_inner_le_norm _ _ |
| _ ≤ ((L:ℝ)*‖d‖)*‖d‖ := mul_le_mul_of_nonneg_right h (norm_nonneg d) |
| _ = _ := by ring |
| have h := norm_image_sub_le_of_norm_deriv_le_segment_01' hf hb |
| simpa only [f, ℓ, AffineMap.lineMap_apply_zero, AffineMap.lineMap_apply_one, |
| zero_mul, sub_zero, one_mul, Real.norm_eq_abs, d, sub_right_comm] using h |
|
|
| theorem gradient_step_strict_descent {φ : E → ℝ} {G : E → E} {s : Set E} |
| (hs : Convex ℝ s) (hgrad : ∀ x ∈ s, HasFDerivAt φ (innerSL ℝ (G x)) x) |
| {L : ℝ≥0} (hG : LipschitzOnWith L G s) {ε : ℝ} (hε : 0 < ε) |
| (hsmall : (L:ℝ)*ε < 1) {x : E} (hx : x ∈ s) (hstep : x-ε • G x ∈ s) |
| (hGx : G x ≠ 0) : φ (x-ε • G x) < φ x := by |
| have h := (le_abs_self _).trans (lipschitz_gradient_remainder hs hgrad hG hx hstep) |
| have he : x-ε • G x-x = -ε • G x := by module |
| rw [he, inner_smul_right, real_inner_self_eq_norm_sq, |
| norm_smul, Real.norm_eq_abs, abs_of_neg (neg_neg_of_pos hε)] at h |
| have hn : 0 < ‖G x‖^2 := sq_pos_of_pos (norm_pos_iff.mpr hGx) |
| have hc : 0 < ε*(1-(L:ℝ)*ε)*‖G x‖^2 := by positivity |
| nlinarith only [h, hc] |
|
|
| theorem gradient_step_derivative_sq_bound {D : E →L[ℝ] E} {c M ε : ℝ} |
| (_hc : 0 ≤ c) (_hM : 0 ≤ M) (hε : 0 ≤ ε) |
| (hcoercive : ∀ v, c*‖v‖^2 ≤ inner ℝ v (D v)) (hbound : ‖D‖ ≤ M) |
| (v : E) : |
| ‖v-ε • D v‖^2 ≤ (1-2*ε*c+ε^2*M^2)*‖v‖^2 := by |
| have hD : ‖D v‖ ≤ M*‖v‖ := D.le_of_opNorm_le hbound v |
| have hD2 : ‖D v‖^2 ≤ M^2*‖v‖^2 := by |
| nlinarith [norm_nonneg (D v), mul_nonneg _hM (norm_nonneg v)] |
| have h := mul_le_mul_of_nonneg_left (hcoercive v) (show 0 ≤ 2*ε by positivity) |
| have h' := mul_le_mul_of_nonneg_left hD2 (sq_nonneg ε) |
| rw [norm_sub_sq_real, inner_smul_right, norm_smul, Real.norm_eq_abs, |
| abs_of_nonneg hε] |
| nlinarith only [h, h'] |
|
|
| end GradientDescent |
| end SKGap |
|
|
| end |
| end |
| end |
|
|
| end OAI |
|
|
