| import OAI.Computability.UniqueGames.Games.FactorizationLemmas |
| import OAI.Computability.UniqueGames.Games.Variation |
|
|
| namespace OAI |
|
|
| section |
|
|
| /-! Exact variation decomposition for the two choices of revealed endpoint. |
| The support of the actual reveal profiles makes both branch identities valid, |
| including zero-probability revealed endpoints. -/ |
| namespace UniqueGamesTheorem.Foundations.Repetition |
| open scoped BigOperators |
| noncomputable section |
| variable {S X Y : Type*} [Fintype S] [Fintype X] [Fintype Y] |
| [DecidableEq X] [DecidableEq Y] |
|
|
| def maskedJoint (p : S × (X × Y) → ℝ) : (S × (X ⊕ Y)) × (X × Y) → ℝ := fun z => |
| (if z.1.2 = Sum.inl z.2.1 then p (z.1.1, z.2) / 2 else 0) + |
| (if z.1.2 = Sum.inr z.2.2 then p (z.1.1, z.2) / 2 else 0) |
|
|
| def maskedModel (μ : Games.FiniteDistribution (X × Y)) (p : S × (X × Y) → ℝ) : |
| (S × (X ⊕ Y)) × (X × Y) → ℝ := fun z => |
| Information.firstMarginal (maskedJoint p) z.1 * (revealProfile μ z.1.2).weight z.2 |
|
|
| def leftRevealModel (μ : Games.FiniteDistribution (X × Y)) (p : S × (X × Y) → ℝ) : |
| S × (X × Y) → ℝ := fun z => |
| (∑ y, p (z.1, (z.2.1, y))) * (revealProfile μ (Sum.inl z.2.1)).weight z.2 |
|
|
| def rightRevealModel (μ : Games.FiniteDistribution (X × Y)) (p : S × (X × Y) → ℝ) : |
| S × (X × Y) → ℝ := fun z => |
| (∑ x, p (z.1, (x, z.2.2))) * (revealProfile μ (Sum.inr z.2.2)).weight z.2 |
|
|
| theorem sum_maskedJoint (p : S × (X × Y) → ℝ) : |
| (∑ z, maskedJoint p z) = ∑ z, p z := by |
| classical |
| have hrow (s : S) (q : X × Y) : |
| (∑ r : X ⊕ Y, maskedJoint p ((s,r),q)) = p (s,q) := by |
| simp [maskedJoint] |
| calc |
| _ = ∑ s : S, ∑ r : X ⊕ Y, ∑ q : X × Y, maskedJoint p ((s,r),q) := by |
| simp only [Fintype.sum_prod_type] |
| _ = ∑ s : S, ∑ q : X × Y, p (s,q) := by |
| apply Finset.sum_congr rfl |
| intro s _ |
| rw [Finset.sum_comm] |
| simp_rw [hrow] |
| _ = _ := (Fintype.sum_prod_type _).symm |
|
|
| theorem maskedJoint_isProbability_iff (p : S × (X × Y) → ℝ) : |
| Information.IsProbability (maskedJoint p) ↔ Information.IsProbability p := by |
| classical |
| constructor |
| · intro h |
| constructor |
| · intro z |
| have hp := h.1 ((z.1,Sum.inl z.2.1),z.2) |
| simp [maskedJoint] at hp |
| linarith |
| · rw [← sum_maskedJoint] |
| exact h.2 |
| · intro h |
| constructor |
| · intro z |
| have hp : 0 ≤ p (z.1.1,z.2) / 2 := div_nonneg (h.1 _) (by norm_num) |
| dsimp [maskedJoint] |
| split_ifs <;> linarith |
| · rw [sum_maskedJoint] |
| exact h.2 |
|
|
| omit [Fintype S] in |
| theorem maskedJoint_firstMarginal_inl (p : S × (X × Y) → ℝ) (s : S) (x : X) : |
| Information.firstMarginal (maskedJoint p) (s, Sum.inl x) = |
| (∑ y, p (s, (x, y))) / 2 := by |
| simp [Information.firstMarginal, maskedJoint, Fintype.sum_prod_type, |
| div_eq_mul_inv, Finset.sum_mul] |
| rw [Finset.sum_comm] |
| simp |
|
|
| omit [Fintype S] in |
| theorem maskedJoint_firstMarginal_inr (p : S × (X × Y) → ℝ) (s : S) (y : Y) : |
| Information.firstMarginal (maskedJoint p) (s, Sum.inr y) = |
| (∑ x, p (s, (x, y))) / 2 := by |
| simp [Information.firstMarginal, maskedJoint, Fintype.sum_prod_type, |
| div_eq_mul_inv, Finset.sum_mul] |
|
|
| private theorem abs_half_sub_half_mul (a b c : ℝ) : |
| |a / 2 - b / 2 * c| = |a - b * c| / 2 := by |
| rw [show a / 2 - b / 2 * c = (a - b * c) / 2 by ring, abs_div] |
| norm_num |
|
|
| omit [Fintype S] in |
| theorem abs_masked_inl (μ : Games.FiniteDistribution (X × Y)) |
| (p : S × (X × Y) → ℝ) (s : S) (u x : X) (y : Y) : |
| |maskedJoint p ((s, Sum.inl u), (x, y)) - maskedModel μ p ((s, Sum.inl u), (x, y))| = |
| if x = u then |p (s, (x, y)) - leftRevealModel μ p (s, (x, y))| / 2 else 0 := by |
| by_cases hx : x = u |
| · subst x |
| simp only [maskedModel, maskedJoint_firstMarginal_inl] |
| simpa [maskedJoint, leftRevealModel] using |
| abs_half_sub_half_mul (p (s, (u, y))) (∑ v, p (s, (u, v))) |
| ((revealProfile μ (Sum.inl u)).weight (u, y)) |
| · have hs := revealProfile_inl_support μ u (x, y) hx |
| simp [maskedJoint, maskedModel, hs, hx, Ne.symm hx] |
|
|
| omit [Fintype S] in |
| theorem abs_masked_inr (μ : Games.FiniteDistribution (X × Y)) |
| (p : S × (X × Y) → ℝ) (s : S) (v : Y) (x : X) (y : Y) : |
| |maskedJoint p ((s, Sum.inr v), (x, y)) - maskedModel μ p ((s, Sum.inr v), (x, y))| = |
| if y = v then |p (s, (x, y)) - rightRevealModel μ p (s, (x, y))| / 2 else 0 := by |
| by_cases hy : y = v |
| · subst y |
| simp only [maskedModel, maskedJoint_firstMarginal_inr] |
| simpa [maskedJoint, rightRevealModel] using |
| abs_half_sub_half_mul (p (s, (x, v))) (∑ u, p (s, (u, v))) |
| ((revealProfile μ (Sum.inr v)).weight (x, v)) |
| · have hs := revealProfile_inr_support μ v (x, y) hy |
| simp [maskedJoint, maskedModel, hs, hy, Ne.symm hy] |
|
|
| theorem masked_totalVariation_identity (μ : Games.FiniteDistribution (X × Y)) |
| (p : S × (X × Y) → ℝ) : |
| 2 * Information.totalVariation (maskedJoint p) (maskedModel μ p) = |
| Information.totalVariation p (leftRevealModel μ p) + |
| Information.totalVariation p (rightRevealModel μ p) := by |
| have hleft (s : S) : |
| (∑ u : X, ∑ q : X × Y, |
| |maskedJoint p ((s, Sum.inl u), q) - maskedModel μ p ((s, Sum.inl u), q)|) = |
| (∑ q : X × Y, |p (s, q) - leftRevealModel μ p (s, q)|) / 2 := by |
| simp_rw [Fintype.sum_prod_type, abs_masked_inl] |
| simp [Finset.sum_ite_irrel, div_eq_mul_inv, Finset.sum_mul] |
| have hright (s : S) : |
| (∑ v : Y, ∑ q : X × Y, |
| |maskedJoint p ((s, Sum.inr v), q) - maskedModel μ p ((s, Sum.inr v), q)|) = |
| (∑ q : X × Y, |p (s, q) - rightRevealModel μ p (s, q)|) / 2 := by |
| have hcollapse : |
| (∑ v : Y, ∑ q : X × Y, |
| |maskedJoint p ((s, Sum.inr v), q) - maskedModel μ p ((s, Sum.inr v), q)|) = |
| ∑ v : Y, ∑ x : X, |p (s, (x, v)) - rightRevealModel μ p (s, (x, v))| / 2 := by |
| simp [Fintype.sum_prod_type, abs_masked_inr] |
| rw [hcollapse, Finset.sum_comm] |
| simp [Fintype.sum_prod_type, div_eq_mul_inv, Finset.sum_mul] |
| have hmass : |
| (∑ z : (S × (X ⊕ Y)) × (X × Y), |maskedJoint p z - maskedModel μ p z|) = |
| ((∑ z : S × (X × Y), |p z - leftRevealModel μ p z|) + |
| (∑ z : S × (X × Y), |p z - rightRevealModel μ p z|)) / 2 := by |
| calc |
| _ = ∑ s : S, |
| ((∑ u : X, ∑ q : X × Y, |maskedJoint p ((s, Sum.inl u), q) - maskedModel μ p ((s, Sum.inl u), q)|) + |
| (∑ v : Y, ∑ q : X × Y, |maskedJoint p ((s, Sum.inr v), q) - maskedModel μ p ((s, Sum.inr v), q)|)) := by |
| simp only [Fintype.sum_prod_type, Fintype.sum_sum_type] |
| _ = ∑ s : S, |
| ((∑ q : X × Y, |p (s, q) - leftRevealModel μ p (s, q)|) / 2 + |
| (∑ q : X × Y, |p (s, q) - rightRevealModel μ p (s, q)|) / 2) := by |
| simp_rw [hleft, hright] |
| _ = _ := by |
| simp [Fintype.sum_prod_type, div_eq_mul_inv, Finset.sum_add_distrib, Finset.sum_mul, add_mul] |
| unfold Information.totalVariation |
| rw [hmass] |
| ring |
|
|
| theorem leftRevealModel_totalVariation_le (μ : Games.FiniteDistribution (X × Y)) |
| (p : S × (X × Y) → ℝ) : |
| Information.totalVariation p (leftRevealModel μ p) ≤ |
| 2 * Information.totalVariation (maskedJoint p) (maskedModel μ p) := by |
| rw [masked_totalVariation_identity] |
| linarith [Information.totalVariation_nonneg p (rightRevealModel μ p)] |
|
|
| theorem rightRevealModel_totalVariation_le (μ : Games.FiniteDistribution (X × Y)) |
| (p : S × (X × Y) → ℝ) : |
| Information.totalVariation p (rightRevealModel μ p) ≤ |
| 2 * Information.totalVariation (maskedJoint p) (maskedModel μ p) := by |
| rw [masked_totalVariation_identity] |
| linarith [Information.totalVariation_nonneg p (leftRevealModel μ p)] |
|
|
| end |
| end UniqueGamesTheorem.Foundations.Repetition |
|
|
| end |
|
|
| end OAI |
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|
