File size: 3,502 Bytes
a2ee19a | 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 | import Mathlib
import OAI.Probability.GaussianRegression.Main
import OAI.Probability.MemoryPrecision.KernelEllipsoid
import OAI.Probability.MemoryPrecision.StreamArithmetic
import OAI.Probability.MemoryPrecision.StreamZero
namespace OAI
open MeasureTheory ProbabilityTheory Set Filter EuclideanGeometry
open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise
noncomputable section
universe u v
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise
namespace MemoryPrecision.Kernel
open MemoryPrecision Policy
def SourceStreamStatement : Prop :=
∃ C : ℝ, 0 < C ∧ ∃ d₀ : ℕ, 0 < d₀ ∧
∀ d M T : ℕ, d₀ ≤ d → ∀ ε : ℝ, 0 < ε → ε ≤ 1/10 →
∀ (Ω : Type u) [MeasurableSpace Ω] (ρ : Measure Ω), IsProbabilityMeasure ρ →
∀ L : Ω → MemoryPrecision.Kernel.Policy d M T,
MemoryPrecision.Kernel.Policy.Admissible L ρ (MemoryPrecision.Kernel.UniformSphere d) →
MemoryPrecision.Kernel.Policy.Success L ρ (MemoryPrecision.Kernel.UniformSphere d) ε ≤ min 1 (ENNReal.ofReal
((2*ε*Real.exp (C*(1+(M:ℝ)/(d:ℝ)^2)*(MemoryPrecision.Kernel.BlockCount T ((d-1)/8):ℝ)))^(((d:ℝ)-1)/2)))
lemma source_stream_explicit :
∀ d M T : ℕ, 32 ≤ d → ∀ ε : ℝ, 0 < ε → ε ≤ 1/10 →
∀ (Ω : Type u) [MeasurableSpace Ω] (ρ : Measure Ω), IsProbabilityMeasure ρ →
∀ L : Ω → Policy d M T, Admissible L ρ (UniformSphere d) →
Success L ρ (UniformSphere d) ε ≤ min 1 (ENNReal.ofReal
((2*ε*Real.exp (StreamConstant*(1+(M:ℝ)/(d:ℝ)^2)*(BlockCount T ((d-1)/8):ℝ)))^(((d:ℝ)-1)/2))) := by
intro d M T hd ε hε hε1 Ω _ ρ hρ L hL
have : IsProbabilityMeasure ρ := hρ
let : NeZero d := ⟨by omega⟩
refine le_min (Policy.success_le_one L ρ (UniformSphere d) ε) ?_
apply (Policy.success_le_chord L ρ (UniformSphere d) ε).trans
by_cases hT : T = 0
· subst T
simp only [BlockCount, Nat.ceil_zero, zero_div, Nat.cast_zero, mul_zero, Real.exp_zero, mul_one]
exact (Policy.chord_zero_bound L ρ hL (by omega) hε).trans
(ENNReal.ofReal_le_ofReal (zero_cap_precise hd hε hε1))
· apply (chord_half_sphere_bound L ρ hL (by omega) hε hε1).trans
apply ENNReal.ofReal_le_ofReal
let a := ((d:ℝ)-1)/2
let X := StreamConstant*(1+(M:ℝ)/(d:ℝ)^2)*(BlockCount T ((d-1)/8):ℝ)
have ha : 0 ≤ a := by
dsimp [a]
have : (32:ℝ) ≤ d := by exact_mod_cast hd
linarith
have he : Real.exp (a*X) = (Real.exp X)^a := by rw [← Real.exp_mul, mul_comm]
have h2 : ε^a ≤ (2*ε)^a := Real.rpow_le_rpow hε.le (by linarith) ha
calc
_ = ε^a * ((4:ℝ)^d * (SuccessConstant^d *
(((d/8+2)*2^M:ℕ):ℝ)^(1/((d/8:ℕ):ℝ)))^BlockCount T (d/8)) := by dsimp [a, BlockCount, MemoryPrecision.BlockCount]; ring
_ ≤ ε^a * Real.exp (a*X) := by
apply mul_le_mul_of_nonneg_left _ (by positivity)
simpa only [a, X, BlockCount, MemoryPrecision.BlockCount, mul_assoc] using stream_full_factor hd (by omega : 0 < T)
_ ≤ (2*ε)^a * Real.exp (a*X) := mul_le_mul_of_nonneg_right h2 (by positivity)
_ = (2*ε*Real.exp X)^a := by
rw [he]
exact (Real.mul_rpow (by positivity : (0:ℝ) ≤ 2*ε) (Real.exp_pos X).le).symm
theorem source_stream_success : SourceStreamStatement :=
⟨StreamConstant, streamConstant_pos, 32, by norm_num, source_stream_explicit⟩
end MemoryPrecision.Kernel
end
end OAI
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