| 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 [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 [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,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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|
