| import Mathlib |
| import OAI.Computability.MaxCut.Estimates.Degree |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace OptimalMaxCut.BooleanFourthMoment |
| open scoped BigOperators |
| open Finset MaxCutGames.Foundations.Hastad |
|
|
| theorem sum_cube_succ {n : ℕ} (F : Cube (Fin (n + 1)) → ℝ) : |
| ∑ x, F x = ∑ b : Bool, ∑ y : Cube (Fin n), F (Fin.cons b y) := by |
| rw [← (Fin.consEquiv (fun _ : Fin (n + 1) => Bool)).sum_comp] |
| exact Fintype.sum_prod_type _ |
|
|
| theorem expect_cube_succ {n : ℕ} (F : Cube (Fin (n + 1)) → ℝ) : |
| (𝔼 x, F x) = 𝔼 b : Bool, 𝔼 y : Cube (Fin n), F (Fin.cons b y) := by |
| calc |
| _ = 𝔼 p : Bool × Cube (Fin n), F (Fin.cons p.1 p.2) := |
| (Fintype.expect_equiv (Fin.consEquiv (fun _ : Fin (n + 1) => Bool)) _ F (fun _ => rfl)).symm |
| _ = _ := by rw [← Finset.univ_product_univ, Finset.expect_product] |
| |
| theorem expect_bool (F : Bool → ℝ) : (𝔼 b, F b) = (F false + F true) / 2 := by |
| rw [Fintype.expect_eq_sum_div_card] |
| simp [add_comm] |
| |
| noncomputable def polynomial {I : Type*} [Fintype I] [DecidableEq I] |
| (c : Cube I → ℝ) (x : Cube I) : ℝ := ∑ s, c s * walsh s x |
| |
| noncomputable def energy {I : Type*} [Fintype I] [DecidableEq I] |
| (c : Cube I → ℝ) : ℝ := |
| ∑ s, (∏ i, if s i then (3 : ℝ) else 1) * c s ^ 2 |
| |
| theorem energy_nonneg {I : Type*} [Fintype I] [DecidableEq I] (c : Cube I → ℝ) : |
| 0 ≤ energy c := by |
| apply Finset.sum_nonneg |
| intro s _ |
| apply mul_nonneg _ (sq_nonneg _) |
| apply Finset.prod_nonneg |
| intro i _ |
| split_ifs <;> norm_num |
| |
| theorem walsh_cons {n : ℕ} (s x : Cube (Fin n)) (a b : Bool) : |
| walsh (Fin.cons a s) (Fin.cons b x) = (if a then bitSign b else 1) * walsh s x := by |
| cases a <;> simp [walsh, Fin.prod_univ_succ, bitSign] |
| |
| theorem polynomial_cons {n : ℕ} (c : Cube (Fin (n + 1)) → ℝ) (x : Cube (Fin n)) (b : Bool) : |
| polynomial c (Fin.cons b x) = polynomial (fun s => c (Fin.cons false s)) x + |
| bitSign b * polynomial (fun s => c (Fin.cons true s)) x := by |
| unfold polynomial |
| rw [sum_cube_succ, Fintype.sum_bool] |
| simp only [walsh_cons, Bool.false_eq_true, ↓reduceIte, one_mul] |
| rw [Finset.mul_sum, add_comm] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro s _ |
| ring |
| |
| theorem energy_cons {n : ℕ} (c : Cube (Fin (n + 1)) → ℝ) : |
| energy c = energy (fun s => c (Fin.cons false s)) + |
| 3 * energy (fun s => c (Fin.cons true s)) := by |
| unfold energy |
| rw [sum_cube_succ, Fintype.sum_bool] |
| simp only [Fin.prod_univ_succ, Fin.cons_zero, Fin.cons_succ, Bool.false_eq_true, ↓reduceIte, |
| one_mul, mul_assoc] |
| rw [← Finset.mul_sum] |
| ring |
| |
| theorem fourth_moment_cons {n : ℕ} (c : Cube (Fin (n + 1)) → ℝ) : |
| (𝔼 x, polynomial c x ^ 4) = |
| (𝔼 x, polynomial (fun s => c (Fin.cons false s)) x ^ 4) + |
| 6 * (𝔼 x, polynomial (fun s => c (Fin.cons false s)) x ^ 2 * |
| polynomial (fun s => c (Fin.cons true s)) x ^ 2) + |
| (𝔼 x, polynomial (fun s => c (Fin.cons true s)) x ^ 4) := by |
| rw [expect_cube_succ, Finset.expect_comm] |
| simp_rw [expect_bool] |
| simp only [polynomial_cons, bitSign, Bool.false_eq_true, ↓reduceIte, one_mul, neg_one_mul] |
| simp_rw [show ∀ a b : ℝ, ((a + b) ^ 4 + (a + -b) ^ 4) / 2 = a ^ 4 + 6 * (a ^ 2 * b ^ 2) + b ^ 4 by |
| intro a b; ring] |
| rw [Finset.expect_add_distrib, Finset.expect_add_distrib, ← Finset.mul_expect] |
| |
| theorem fourth_moment_bound (n : ℕ) (c : Cube (Fin n) → ℝ) : |
| (𝔼 x, polynomial c x ^ 4) ≤ energy c ^ 2 := by |
| induction n with |
| | zero => simp [polynomial, energy, walsh]; exact le_of_eq (by ring) |
| | succ n ih => |
| let c₀ : Cube (Fin n) → ℝ := fun s => c (Fin.cons false s) |
| let c₁ : Cube (Fin n) → ℝ := fun s => c (Fin.cons true s) |
| have h₀ := ih c₀ |
| have h₁ := ih c₁ |
| have he₀ := energy_nonneg c₀ |
| have he₁ := energy_nonneg c₁ |
| have hm₀ : 0 ≤ 𝔼 x, polynomial c₀ x ^ 4 := by positivity |
| have hm₁ : 0 ≤ 𝔼 x, polynomial c₁ x ^ 4 := by positivity |
| have hc := Finset.expect_mul_sq_le_sq_mul_sq Finset.univ |
| (fun x => polynomial c₀ x ^ 2) (fun x => polynomial c₁ x ^ 2) |
| simp only [← pow_mul, Nat.reduceMul] at hc |
| have hc' : (𝔼 x, polynomial c₀ x ^ 2 * polynomial c₁ x ^ 2) ^ 2 ≤ |
| (energy c₀ * energy c₁) ^ 2 := by |
| calc _ ≤ _ := hc |
| _ ≤ energy c₀ ^ 2 * energy c₁ ^ 2 := mul_le_mul h₀ h₁ hm₁ (sq_nonneg _) |
| _ = _ := by ring |
| have hc'' : (𝔼 x, polynomial c₀ x ^ 2 * polynomial c₁ x ^ 2) ≤ energy c₀ * energy c₁ := |
| (sq_le_sq₀ (by positivity) (mul_nonneg he₀ he₁)).mp hc' |
| rw [fourth_moment_cons, energy_cons] |
| change (𝔼 x, polynomial c₀ x ^ 4) + |
| 6 * (𝔼 x, polynomial c₀ x ^ 2 * polynomial c₁ x ^ 2) + |
| (𝔼 x, polynomial c₁ x ^ 4) ≤ (energy c₀ + 3 * energy c₁) ^ 2 |
| nlinarith [sq_nonneg (energy c₁)] |
|
|
| theorem weight_eq_degree {I : Type*} [Fintype I] [DecidableEq I] (s : Cube I) : |
| (∏ i, if s i then (3 : ℝ) else 1) = 3 ^ Analytic.degree s := by |
| simp [Finset.prod_ite, Analytic.degree] |
|
|
| theorem energy_truncated {I : Type*} [Fintype I] [DecidableEq I] |
| (K : ℕ) (c : Cube I → ℝ) (hc : ∀ s, K < Analytic.degree s → c s = 0) : |
| energy c ≤ 3 ^ K * ∑ s, c s ^ 2 := by |
| unfold energy |
| rw [Finset.mul_sum] |
| apply Finset.sum_le_sum |
| intro s _ |
| rw [weight_eq_degree] |
| by_cases hs : Analytic.degree s ≤ K |
| · exact mul_le_mul_of_nonneg_right (pow_le_pow_right₀ (by norm_num) hs) (sq_nonneg _) |
| · rw [hc s (lt_of_not_ge hs)]; simp |
|
|
| theorem coefficient_polynomial {I : Type*} [Fintype I] [DecidableEq I] |
| (c : Cube I → ℝ) (a : Cube I) : coefficient (polynomial c) a = c a := by |
| unfold coefficient polynomial |
| simp only [Finset.sum_mul] |
| rw [Finset.expect_sum_comm] |
| simp_rw [mul_assoc, ← Finset.mul_expect, walsh_orthogonality] |
| simp |
|
|
| theorem polynomial_second_moment {I : Type*} [Fintype I] [DecidableEq I] (c : Cube I → ℝ) : |
| (𝔼 x, polynomial c x ^ 2) = ∑ s, c s ^ 2 := by |
| rw [← walsh_parseval] |
| simp only [coefficient_polynomial] |
|
|
| theorem degree_fourth_moment_bound {n K : ℕ} (c : Cube (Fin n) → ℝ) |
| (hc : ∀ s, K < Analytic.degree s → c s = 0) : |
| (𝔼 x, polynomial c x ^ 4) ≤ (3 ^ K * (𝔼 x, polynomial c x ^ 2)) ^ 2 := by |
| rw [polynomial_second_moment] |
| exact (fourth_moment_bound n c).trans |
| ((sq_le_sq₀ (energy_nonneg c) (by positivity)).mpr (energy_truncated K c hc)) |
|
|
| theorem degree_third_moment_bound {n K : ℕ} (c : Cube (Fin n) → ℝ) |
| (hc : ∀ s, K < Analytic.degree s → c s = 0) {τ : ℝ} (hτ : 0 ≤ τ) |
| (hsmall : (𝔼 x, polynomial c x ^ 2) ≤ τ ^ 2) : |
| (𝔼 x, |polynomial c x| ^ 3) ≤ 3 ^ K * τ * (𝔼 x, polynomial c x ^ 2) := by |
| have hsecond : 0 ≤ 𝔼 x, polynomial c x ^ 2 := by positivity |
| have hfourth := degree_fourth_moment_bound c hc |
| have hCS := Finset.expect_mul_sq_le_sq_mul_sq Finset.univ |
| (fun x => |polynomial c x|) (fun x => polynomial c x ^ 2) |
| have hp (a : ℝ) : |a| * a ^ 2 = |a| ^ 3 := by rw [← sq_abs]; ring |
| simp_rw [hp, sq_abs, ← pow_mul, Nat.reduceMul] at hCS |
| have hbound : (𝔼 x, |polynomial c x| ^ 3) ^ 2 ≤ |
| (3 ^ K * τ * (𝔼 x, polynomial c x ^ 2)) ^ 2 := by |
| calc |
| _ ≤ (𝔼 x, polynomial c x ^ 2) * (𝔼 x, polynomial c x ^ 4) := hCS |
| _ ≤ (𝔼 x, polynomial c x ^ 2) * (3 ^ K * (𝔼 x, polynomial c x ^ 2)) ^ 2 := |
| mul_le_mul_of_nonneg_left hfourth hsecond |
| _ = (3 ^ K * (𝔼 x, polynomial c x ^ 2)) ^ 2 * (𝔼 x, polynomial c x ^ 2) := by ring |
| _ ≤ (3 ^ K * (𝔼 x, polynomial c x ^ 2)) ^ 2 * τ ^ 2 := |
| mul_le_mul_of_nonneg_left hsmall (sq_nonneg _) |
| _ = _ := by ring |
| exact (sq_le_sq₀ (by positivity) (by positivity)).mp hbound |
|
|
| theorem walsh_flip {I : Type*} [Fintype I] [DecidableEq I] |
| (s x : Cube I) (i : I) : |
| walsh s (Function.update x i (!x i)) = (if s i then -1 else 1) * walsh s x := by |
| classical |
| have hp (j : I) : bitSign (s j && Function.update x i (!x i) j) = |
| (if j = i then (if s i then (-1 : ℝ) else 1) else 1) * bitSign (s j && x j) := by |
| by_cases h : j = i |
| · subst j; cases s i <;> cases x i <;> simp [bitSign] |
| · simp [h] |
| simp only [walsh, hp, Finset.prod_mul_distrib] |
| simp |
|
|
| noncomputable def discreteDiff {I : Type*} [Fintype I] [DecidableEq I] |
| (f : Cube I → ℝ) (i : I) (x : Cube I) : ℝ := (f x - f (Function.update x i (!x i))) / 2 |
|
|
| theorem polynomial_diff {I : Type*} [Fintype I] [DecidableEq I] |
| (c : Cube I → ℝ) (i : I) : |
| discreteDiff (polynomial c) i = polynomial (fun s => if s i then c s else 0) := by |
| funext x |
| simp only [discreteDiff, polynomial, walsh_flip, ← Finset.sum_sub_distrib, Finset.sum_div] |
| apply Finset.sum_congr rfl |
| intro s _ |
| cases s i <;> simp |
|
|
| theorem total_diff_second {I : Type*} [Fintype I] [DecidableEq I] |
| (c : Cube I → ℝ) : |
| (∑ i, 𝔼 x, discreteDiff (polynomial c) i x ^ 2) = |
| ∑ s, (Analytic.degree s : ℝ) * c s ^ 2 := by |
| simp only [polynomial_diff, polynomial_second_moment] |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro s _ |
| simp only [ite_pow, zero_pow (by omega : 2 ≠ 0)] |
| rw [← Finset.sum_filter] |
| simp [Analytic.degree] |
|
|
| theorem total_diff_second_truncated {I : Type*} [Fintype I] [DecidableEq I] |
| (K : ℕ) (c : Cube I → ℝ) (hc : ∀ s, K < Analytic.degree s → c s = 0) : |
| (∑ i, 𝔼 x, discreteDiff (polynomial c) i x ^ 2) ≤ |
| K * (𝔼 x, polynomial c x ^ 2) := by |
| rw [total_diff_second, polynomial_second_moment, Finset.mul_sum] |
| apply Finset.sum_le_sum |
| intro s _ |
| by_cases hs : Analytic.degree s ≤ K |
| · exact mul_le_mul_of_nonneg_right (by exact_mod_cast hs) (sq_nonneg _) |
| · rw [hc s (lt_of_not_ge hs)]; simp |
|
|
| theorem total_diff_third_truncated {n K : ℕ} (c : Cube (Fin n) → ℝ) |
| (hc : ∀ s, K < Analytic.degree s → c s = 0) {τ : ℝ} (hτ : 0 ≤ τ) |
| (hsmall : ∀ i, (𝔼 x, discreteDiff (polynomial c) i x ^ 2) ≤ τ ^ 2) : |
| (∑ i, 𝔼 x, |discreteDiff (polynomial c) i x| ^ 3) ≤ |
| 3 ^ K * τ * K * (𝔼 x, polynomial c x ^ 2) := by |
| calc |
| _ ≤ ∑ i, 3 ^ K * τ * (𝔼 x, discreteDiff (polynomial c) i x ^ 2) := by |
| apply Finset.sum_le_sum |
| intro i _ |
| have hi := hsmall i |
| rw [polynomial_diff] at hi ⊢ |
| exact degree_third_moment_bound _ (fun s hs => by rw [hc s hs]; split_ifs <;> rfl) |
| hτ hi |
| _ = 3 ^ K * τ * (∑ i, 𝔼 x, discreteDiff (polynomial c) i x ^ 2) := by rw [Finset.mul_sum] |
| _ ≤ 3 ^ K * τ * (K * (𝔼 x, polynomial c x ^ 2)) := |
| mul_le_mul_of_nonneg_left (total_diff_second_truncated K c hc) (by positivity) |
| _ = _ := by ring |
|
|
| end OptimalMaxCut.BooleanFourthMoment |
|
|
| end |
|
|
| end OAI |
|
|
