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e320e82 | 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 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 | import OAI.Combinatorics.ExpectationThreshold.Basic
import OAI.Combinatorics.ExpectationThreshold.ProbabilitySubsetProduct
import Mathlib.Algebra.Order.GroupWithZero.Basic
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring
namespace OAI
/-!
# The upper expectation bound for rounding
The score sums over nonempty subsets of the original ground set. Its moments
come from the actual coordinate product law. Nonemptiness saves a factor of
one fifth in the fractional-cover cost.
-/
open scoped BigOperators
namespace TalagrandThreshold
noncomputable section
variable {α : Type*} [Fintype α] [DecidableEq α]
/-- The nonnegative subset-product score used by the rounding argument. -/
def productScore (g : Finset α → ℝ) (Y : α → ℝ) : ℝ :=
∑ S ∈ (Finset.univ.erase ∅ : Finset (Finset α)), g S * ∏ x ∈ S, Y x
theorem productScore_nonneg {g : Finset α → ℝ} {Y : α → ℝ}
(hg : ∀ S, 0 ≤ g S) (hY : ∀ x, 0 ≤ Y x) : 0 ≤ productScore g Y := by
exact Finset.sum_nonneg fun S _ =>
mul_nonneg (hg S) (Finset.prod_nonneg fun x _ => hY x)
variable {Ω : Type*} [Fintype Ω]
/-- Independence of the coordinates gives the full subset-product expectation. -/
theorem expectation_productScore (ν : FiniteLaw Ω) (g : Finset α → ℝ) (Y : Ω → ℝ) :
(FiniteLaw.pi (fun _ : α => ν)).expectation
(fun ω => productScore g (fun x => Y (ω x))) =
∑ S ∈ (Finset.univ.erase ∅ : Finset (Finset α)),
g S * (ν.expectation Y) ^ S.card := by
classical
calc
_ = ∑ S ∈ (Finset.univ.erase ∅ : Finset (Finset α)),
(FiniteLaw.pi (fun _ : α => ν)).expectation
(fun ω => g S * ∏ x ∈ S, Y (ω x)) := by
unfold FiniteLaw.expectation productScore
simp only [Finset.mul_sum]
exact Finset.sum_comm
_ = _ := by
apply Finset.sum_congr rfl
intro S _
rw [FiniteLaw.expectation_smul, FiniteLaw.expectation_pi_subset_product_const]
/-- A nonempty product gains at least one factor of one fifth. -/
theorem one_fifth_pow_le {r : ℝ} (hr : 0 ≤ r) {k : ℕ} (hk : 0 < k) :
(r / 5) ^ k ≤ (1 / 5) * r ^ k := by
have hsmall : (1 / 5 : ℝ) ^ k ≤ 1 / 5 :=
pow_le_of_le_one (by norm_num) (by norm_num) (Nat.ne_of_gt hk)
calc
(r / 5) ^ k = (1 / 5 : ℝ) ^ k * r ^ k := by
rw [← mul_pow]
congr 1
ring
_ ≤ (1 / 5) * r ^ k := mul_le_mul_of_nonneg_right hsmall (pow_nonneg hr k)
/-- The product score costs at most one fifth of the fractional-cover cost. -/
theorem expectation_productScore_le (ν : FiniteLaw Ω) (g : Finset α → ℝ)
(Y : Ω → ℝ) {r : ℝ} (hg : ∀ S, 0 ≤ g S) (hY : ∀ ω, 0 ≤ Y ω)
(hr : 0 ≤ r) (hmean : ν.expectation Y ≤ r / 5) :
(FiniteLaw.pi (fun _ : α => ν)).expectation
(fun ω => productScore g (fun x => Y (ω x))) ≤
(1 / 5) * fracCost r g := by
classical
have hmean0 := ν.expectation_nonneg hY
rw [expectation_productScore]
calc
(∑ S ∈ (Finset.univ.erase ∅ : Finset (Finset α)),
g S * (ν.expectation Y) ^ S.card) ≤
∑ S ∈ (Finset.univ.erase ∅ : Finset (Finset α)),
(1 / 5) * (g S * r ^ S.card) := by
apply Finset.sum_le_sum
intro S hS
have hne : S.Nonempty :=
Finset.nonempty_iff_ne_empty.mpr (Finset.mem_erase.mp hS).1
have hpower : (ν.expectation Y) ^ S.card ≤ (1 / 5) * r ^ S.card :=
(pow_le_pow_left₀ hmean0 hmean S.card).trans
(one_fifth_pow_le hr (Finset.card_pos.mpr hne))
calc
g S * (ν.expectation Y) ^ S.card ≤ g S * ((1 / 5) * r ^ S.card) :=
mul_le_mul_of_nonneg_left hpower (hg S)
_ = (1 / 5) * (g S * r ^ S.card) := by ring
_ = (1 / 5) * ∑ S ∈ (Finset.univ.erase ∅ : Finset (Finset α)),
g S * r ^ S.card := (Finset.mul_sum _ _ _).symm
_ ≤ (1 / 5) * fracCost r g := by
apply mul_le_mul_of_nonneg_left _ (by norm_num : (0 : ℝ) ≤ 1 / 5)
unfold fracCost
exact Finset.sum_le_sum_of_subset_of_nonneg (Finset.erase_subset _ _)
(fun S _ _ => mul_nonneg (hg S) (pow_nonneg hr S.card))
/-- The budget one half gives the upper bound one tenth for the same score. -/
theorem expectation_productScore_le_tenth (ν : FiniteLaw Ω) (g : Finset α → ℝ)
(Y : Ω → ℝ) {r : ℝ} (hg : ∀ S, 0 ≤ g S) (hY : ∀ ω, 0 ≤ Y ω)
(hr : 0 ≤ r) (hmean : ν.expectation Y ≤ r / 5) (hcost : fracCost r g ≤ 1 / 2) :
(FiniteLaw.pi (fun _ : α => ν)).expectation
(fun ω => productScore g (fun x => Y (ω x))) ≤ 1 / 10 := by
calc
_ ≤ (1 / 5) * fracCost r g := expectation_productScore_le ν g Y hg hY hr hmean
_ ≤ (1 / 5 : ℝ) * (1 / 2) :=
mul_le_mul_of_nonneg_left hcost (by norm_num : (0 : ℝ) ≤ 1 / 5)
_ = 1 / 10 := by norm_num
end
end TalagrandThreshold
end OAI
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