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dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 | import Mathlib
namespace OAI
section
section
noncomputable section
namespace LogConcaveSampling.FinitePicard
open scoped NNReal BigOperators
variable {I E X : Type*} [Fintype I] [NormedAddCommGroup E] [NormedSpace ℝ E]
[PseudoMetricSpace X]
def correction (w : I → I → ℝ) (φ : I → E → E) (z : I → E) : I → E :=
fun i => ∑j,w i j • φ j (z j)
def step (w : I → I → ℝ) (φ : I → E → E) (a z : I → E) : I → E :=
a+correction w φ z
lemma correction_lipschitz (w : I → I → ℝ) (φ : I → E → E) {A K : ℝ≥0}
(hw : ∀i,∑j,|w i j|≤A) (hφ : ∀i,LipschitzWith K (φ i)) :
LipschitzWith (A*K) (correction w φ) := by
classical
apply LipschitzWith.of_dist_le_mul
intro z v
apply (dist_pi_le_iff (by positivity)).mpr
intro i
simp only [correction,dist_eq_norm,←Finset.sum_sub_distrib,←smul_sub]
calc
_ ≤ ∑j,‖w i j • (φ j (z j)-φ j (v j))‖ := norm_sum_le _ _
_ ≤ ∑j,|w i j| *((K:ℝ)*dist z v) := by
apply Finset.sum_le_sum
intro j _
rw [norm_smul,Real.norm_eq_abs]
apply mul_le_mul_of_nonneg_left _ (abs_nonneg _)
simpa only [dist_eq_norm] using ((hφ j).dist_le_mul (z j) (v j)).trans
(mul_le_mul_of_nonneg_left (dist_le_pi_dist z v j) (by positivity))
_ = (∑j,|w i j|)*((K:ℝ)*dist z v) := by rw [Finset.sum_mul]
_ ≤ (A:ℝ)*((K:ℝ)*dist z v) := mul_le_mul_of_nonneg_right (hw i) (by positivity)
_ = _ := by simp only [NNReal.coe_mul,dist_eq_norm]; ring
lemma step_lipschitz (w : I → I → ℝ) (φ : I → E → E) (a : I → E) {A K : ℝ≥0}
(hw : ∀i,∑j,|w i j|≤A) (hφ : ∀i,LipschitzWith K (φ i)) :
LipschitzWith (A*K) (step w φ a) := by
apply LipschitzWith.of_dist_le_mul
intro z v
simpa only [step,dist_add_left] using (correction_lipschitz w φ hw hφ).dist_le_mul z v
theorem iterate_defect {Y : Type*} [PseudoMetricSpace Y] {q : ℝ≥0}
{G : Y → Y} (hG : LipschitzWith q G) (hq : (q:ℝ)≤1/2)
(a e : Y) {D : ℝ} (hD : 0≤D) (he : dist (G e) e≤D) (n : ℕ) :
dist (G^[n] a) e≤2*D+(q:ℝ)^n*dist a e := by
induction n with
| zero => simp only [Function.iterate_zero_apply,pow_zero,one_mul]; linarith
| succ n ih =>
rw [Function.iterate_succ_apply']
calc
_ ≤ dist (G (G^[n] a)) (G e)+dist (G e) e := dist_triangle _ _ _
_ ≤ (q:ℝ)*dist (G^[n] a) e+D := add_le_add (hG.dist_le_mul _ _) he
_ ≤ (q:ℝ)*(2*D+(q:ℝ)^n*dist a e)+D := add_le_add (mul_le_mul_of_nonneg_left ih q.coe_nonneg) le_rfl
_ ≤ 2*D+(q:ℝ)^(n+1)*dist a e := by
rw [pow_succ]
nlinarith [mul_le_mul_of_nonneg_right hq hD]
def nodes (w : I → I → ℝ) (φ : I → E → E) (a : X → I → E) : ℕ → X → I → E
| 0 => a
| n+1 => fun x => step w φ (a x) (nodes w φ a n x)
omit [PseudoMetricSpace X] in
lemma nodes_eq_iterate (w : I → I → ℝ) (φ : I → E → E) (a : X → I → E)
(n : ℕ) (x : X) : nodes w φ a n x=(step w φ (a x))^[n] (a x) := by
induction n with
| zero => rfl
| succ n ih => simp only [nodes,ih,Function.iterate_succ_apply']
theorem nodes_lipschitz (w : I → I → ℝ) (φ : I → E → E) (a : X → I → E)
{A K B : ℝ≥0} (hw : ∀i,∑j,|w i j|≤A) (hφ : ∀i,LipschitzWith K (φ i))
(ha : LipschitzWith B a) (hq : (A*K:ℝ≥0)≤1/2) (n : ℕ) :
LipschitzWith (2*B) (nodes w φ a n) := by
induction n with
| zero => exact ha.weaken (by nlinarith)
| succ n ih =>
change LipschitzWith (2*B) (fun x => a x+correction w φ (nodes w φ a n x))
apply (ha.add ((correction_lipschitz w φ hw hφ).comp ih)).weaken
nlinarith
theorem nodes_correction_lipschitz (w : I → I → ℝ) (φ : I → E → E) (a : X → I → E)
{A K B : ℝ≥0} (hw : ∀i,∑j,|w i j|≤A) (hφ : ∀i,LipschitzWith K (φ i))
(ha : LipschitzWith B a) (hq : (A*K:ℝ≥0)≤1/2) (n : ℕ) :
LipschitzWith (A*K*(2*B)) (fun x => nodes w φ a (n+1) x-a x) := by
convert (correction_lipschitz w φ hw hφ).comp (nodes_lipschitz w φ a hw hφ ha hq n) using 1
funext x
simp only [nodes,step,add_sub_cancel_left,Function.comp_apply]
end LogConcaveSampling.FinitePicard
end
end
end
end OAI
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