| import OAI.Computability.PerfectCompleteness.Construction.OwnBucketBias |
|
|
| namespace OAI |
|
|
| section |
|
|
| namespace PerfectCompleteness.OwnLawHeavyList |
|
|
| open QuarterBalance |
| open HiddenBucketBias (recursiveLaw) |
| open UniqueGamesTheorem.Foundations.Games |
| open scoped TensorProduct Classical |
|
|
| noncomputable section |
|
|
| variable {V : Type*} [AddCommGroup V] [Module F2 V] [Fintype V] |
|
|
| local instance dualFintype : Fintype (Module.Dual F2 V) := |
| Fintype.ofInjective (fun f : Module.Dual F2 V => (f : V → F2)) DFunLike.coe_injective |
|
|
| def character (f : Module.Dual F2 V) (x : V) : ℝ := sign (f x) |
|
|
| omit [Fintype V] in |
| theorem character_sq (f : Module.Dual F2 V) (x : V) : character f x ^ 2 = 1 := by |
| unfold character sign |
| split <;> norm_num |
|
|
| omit [Fintype V] in |
| theorem character_pair (f g : Module.Dual F2 V) (x : V) : |
| character f x * character g x = character (f - g) x := by |
| simp only [character, sub_eq_add_neg, |
| LinearMap.add_apply, LinearMap.neg_apply, ZMod.neg_eq_self_mod_two, sign_add] |
|
|
| def heavyList (μ : FiniteDistribution V) (target : V → ℝ) (threshold : ℝ) : |
| Finset (Module.Dual F2 V) := |
| SmallBias.heavySet μ character target threshold |
|
|
| theorem mem_heavyList (μ : FiniteDistribution V) (target : V → ℝ) (threshold : ℝ) |
| (f : Module.Dual F2 V) : |
| f ∈ heavyList μ target threshold ↔ |
| threshold ≤ |μ.expectation (fun x => target x * character f x)| := by |
| simp only [heavyList, SmallBias.heavySet, Finset.mem_filter, Finset.mem_univ, true_and] |
|
|
| theorem heavyList_card_le (μ : FiniteDistribution V) (target : V → ℝ) |
| (threshold bias : ℝ) (hthreshold : 0 < threshold) (hbias : 0 ≤ bias) |
| (hsmall : bias ≤ threshold ^ 2 / 2) (htarget : ∀ x, target x ^ 2 ≤ 1) |
| (hdual : ∀ f : Module.Dual F2 V, f ≠ 0 → |
| |μ.expectation (fun x => sign (f x))| ≤ bias) : |
| ((heavyList μ target threshold).card : ℝ) ≤ 2 / threshold ^ 2 := by |
| apply SmallBias.heavySet_card_le μ character target threshold bias |
| hthreshold hbias hsmall htarget (fun f x => (character_sq f x).le) |
| intro f g hfg |
| simp only [character_pair] |
| exact hdual (f - g) (sub_ne_zero.mpr hfg) |
|
|
| universe u |
| variable {branch : Nat → Nat} {n m width : Nat} |
|
|
| theorem own_heavyList_card_le (W : Submodule F2 (Fin width → F2)) |
| (repeats : Nat → Nat) (hrep : RecursiveSamplerBias.RepetitionsBalanced repeats) |
| (p : DescendantSpaces.Path branch n m) |
| (A : RecursiveSpaces.Slots branch n → Type u) [∀ s, Finite (A s)] |
| (target : (W ⊗[F2] RecursiveSpaces.space F2 branch n A) → ℝ) |
| (threshold : ℝ) (hthreshold : 0 < threshold) |
| (hsmall : (7 / 8 : ℝ) ^ repeats n ≤ threshold ^ 2 / 2) |
| (htarget : ∀ x, target x ^ 2 ≤ 1) : |
| ((heavyList (recursiveLaw W repeats p A) target threshold).card : ℝ) ≤ |
| 2 / threshold ^ 2 := by |
| let : Fintype (RecursiveSpaces.space F2 branch n A) := Fintype.ofFinite _ |
| apply heavyList_card_le (recursiveLaw W repeats p A) target threshold |
| ((7 / 8 : ℝ) ^ repeats n) hthreshold (pow_nonneg (by norm_num) _) hsmall htarget |
| exact fun f hf => OwnBucketBias.bias_le W repeats hrep p A f hf |
|
|
| end |
| end PerfectCompleteness.OwnLawHeavyList |
|
|
| end |
|
|
| end OAI |
|
|
