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a2ee19a | 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 | import Mathlib
import OAI.Probability.Perceptron.Model
namespace OAI
noncomputable section
open Set Filter
open scoped Topology BigOperators
namespace SphericalPerceptronFreeEnergy
lemma box_coordinate_comparison {I : Type*} [Fintype I] [DecidableEq I]
{F : (I → ℝ) → ℝ} {L : I → ℝ} {a b : ℝ}
(hderiv : ∀ u : I → ℝ, (∀ i, u i ∈ Icc a b) → ∀ j,
∃ D : ℝ, HasDerivAt (fun r => F (u+Pi.single j r)) D 0 ∧ |D| ≤ L j)
{u : I → ℝ} (hu : ∀ i, u i ∈ Icc a b) (j : I)
{x y : ℝ} (hx : x ∈ Icc a b) (hy : y ∈ Icc a b) :
|F (Function.update u j y)-F (Function.update u j x)| ≤ L j*|y-x| := by
have hd (s : ℝ) (hs : s ∈ Icc a b) :
∃ D : ℝ, HasDerivAt (fun r => F (Function.update u j r)) D s ∧ |D| ≤ L j := by
have huw : ∀ i, Function.update u j s i ∈ Icc a b := by
intro i
by_cases hi : i=j
· subst i; simpa using hs
· simpa [Function.update_of_ne hi] using hu i
obtain ⟨D,hD,hDb⟩ := hderiv (Function.update u j s) huw j
refine ⟨D,?_,hDb⟩
have hsub : HasDerivAt (fun r : ℝ => r-s) 1 s := (hasDerivAt_id s).sub_const s
have hc := hD.comp_of_eq s hsub (sub_self s).symm
simp only [mul_one] at hc
apply hc.congr_of_eventuallyEq
apply Filter.Eventually.of_forall
intro r
change F (Function.update u j r) = F (Function.update u j s + Pi.single j (r-s))
congr 1
funext i
by_cases hi : i=j
· subst i
simp
· simp [Function.update_of_ne hi,Pi.single_eq_of_ne hi]
let D := fun s => if hs : s ∈ Icc a b then (hd s hs).choose else 0
have hD (s) (hs : s ∈ Icc a b) :
HasDerivWithinAt (fun r => F (Function.update u j r)) (D s) (Icc a b) s := by
dsimp [D]
rw [dite_eq_left hs]
exact (hd s hs).choose_spec.1.hasDerivWithinAt
have hDb (s) (hs : s ∈ Icc a b) : ‖D s‖ ≤ L j := by
dsimp [D]
rw [dite_eq_left hs]
exact (hd s hs).choose_spec.2
simpa only [Real.norm_eq_abs] using
Convex.norm_image_sub_le_of_norm_hasDerivWithin_le hD hDb (convex_Icc a b) hx hy
lemma box_sum_coordinate_comparison {I : Type*} [Fintype I] [DecidableEq I]
{F : (I → ℝ) → ℝ} {L : I → ℝ} {a b : ℝ}
(hderiv : ∀ u : I → ℝ, (∀ i, u i ∈ Icc a b) → ∀ j,
∃ D : ℝ, HasDerivAt (fun r => F (u+Pi.single j r)) D 0 ∧ |D| ≤ L j)
{u v : I → ℝ} (hu : ∀ i, u i ∈ Icc a b) (hv : ∀ i, v i ∈ Icc a b) :
|F v-F u| ≤ ∑ j, L j*|v j-u j| := by
let w := fun s : Finset I => fun i => if i ∈ s then v i else u i
have hw (s : Finset I) (i : I) : w s i ∈ Icc a b := by
dsimp [w]
split_ifs
· exact hv i
· exact hu i
have hh (s : Finset I) : |F (w s)-F u| ≤ ∑ j ∈ s, L j*|v j-u j| := by
induction s using Finset.induction_on with
| empty => simp [w]
| @insert j s hj ih =>
have h0 : Function.update (w s) j (u j)=w s := by
funext i
by_cases hi : i=j
· subst i; simp [w,hj]
· simp [Function.update_of_ne hi]
have h1 : Function.update (w s) j (v j)=w (insert j s) := by
funext i
by_cases hi : i=j
· subst i; simp [w]
· simp [w,hi]
have hc := box_coordinate_comparison hderiv (hw s) j (hu j) (hv j)
rw [h0,h1] at hc
rw [Finset.sum_insert hj]
exact (abs_sub_le (F (w (insert j s))) (F (w s)) (F u)).trans (add_le_add hc ih)
simpa [w] using hh Finset.univ
end SphericalPerceptronFreeEnergy
end
end OAI
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