File size: 3,043 Bytes
e63eb7f | 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 | import OAI.Combinatorics.EditApproximation.Refinement.ComputedCoarseWidthCost
import OAI.Combinatorics.EditApproximation.Probability.FreshCoordinateCost
import OAI.Combinatorics.EditApproximation.Coarse.CoarseGlobalSeed
namespace OAI
/-! First-request work for a complete independent coarse source. The
query is adaptive; each entry is charged only on its first appearance.
No conditioning on seed correctness occurs in this estimate. -/
namespace EditApproximation
open MeasureTheory
section FirstRequest
variable {ι α : Type*} [Fintype ι] [DecidableEq ι] [DecidableEq α]
(N B : ℕ) [NeZero B] [MeasurableSpace (Fin B)] [MeasurableSingletonClass (Fin B)]
(depth : ι → ℕ) (x y : ι → List α) (query : FiniteQuery ι)
noncomputable def coarseFirstRequestWork (draw : CoarseFullSeedDraws B depth x y) : ℝ :=
∑ i, selectionIndicator (i ∈ query.keys
(fun j => (coarseFullSeedTable N B depth x y draw j : ℚ))) *
(coarseInitialArrayWork B (depth i) (initialThresholdQ N) (initialLambda N) (x i) (y i) (draw i) : ℝ)
theorem coarseFirstRequestWork_expected_le
(hB : B = (integerParameters N).B) (hN : 0 < N) (hell : 200 ≤ inputSmallLog N)
(hdepth : ∀ i, depth i = computedCoarseDepth N (coarseEntryWidth (x i) (y i)))
(hsize : ∀ i, coarseEntryWidth (x i) (y i) ≤ N)
(hinverse : coarseInverseAllowanceFactor B (initialThresholdQ N) (initialLambda N) ≤
(inputHeight N : ℝ) ^ 8)
(budget : ℝ)
(hbudget : ∀ draw : CoarseFullSeedDraws B depth x y,
((query.cost (fun i => (coarseFullSeedTable N B depth x y draw i : ℚ))
(fun i => (x i).length) : ℕ) : ℝ) ≤ budget) :
(∫ draw, coarseFirstRequestWork N B depth x y query draw
∂coarseFullSeedMeasure B depth x y) ≤ budget * computedCoarseWidthOverhead N := by
let μ := fun i => coarseInitialMeasure B (depth i) (coarseEntryWidth (x i) (y i))
let answer := fun i draw =>
(coarseInitialValue B (depth i) (initialThresholdQ N) (initialLambda N) N (x i) (y i) draw : ℚ)
let cost := fun i draw =>
(coarseInitialArrayWork B (depth i) (initialThresholdQ N) (initialLambda N) (x i) (y i) draw : ℝ)
have hcost (i) : (∫ value, cost i value ∂μ i) ≤
((x i).length : ℝ) * computedCoarseWidthOverhead N :=
computed_coarse_expected_source_width N B hB hN hell hinverse (x i) (y i)
(depth i) (hdepth i) (hsize i)
have hselected (draw : CoarseFullSeedDraws B depth x y) :
(∑ i, selectionIndicator (i ∈ query.keys (fun j => answer j (draw j))) *
((x i).length : ℝ)) ≤ budget := by
have hs := sum_selected_nat_weights_le (query.keys (fun j => answer j (draw j)))
(fun i => (x i).length)
have hb := hbudget draw
rw [FiniteQuery.cost_eq_sum] at hb
exact hs.trans hb
have h := finiteQuery_fresh_expected_cost_le μ query answer cost
(fun i => ((x i).length : ℝ)) (computedCoarseWidthOverhead N) budget
(Nat.cast_nonneg _) hcost hselected
exact h
end FirstRequest
end EditApproximation
end OAI
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