File size: 6,804 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 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 | import Mathlib
import OAI.Probability.GaussianRegression.Main
import OAI.Probability.MemoryPrecision.EllipsoidSuccess
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
lemma chord_real_coefficient {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω]
(L : Ω → Policy d M T) (ρ : Measure Ω) [IsProbabilityMeasure ρ]
(hL : Admissible L ρ (UniformSphere d)) (hd : 16 ≤ d) {ε : ℝ} (hε : 0 < ε)
{α : ℝ} (hαlo : ((d:ℝ)-1)/2 ≤ α) (hαhi : α ≤ (d:ℝ)/2) :
ChordSuccess L ρ (UniformSphere d) ε ≤ ENNReal.ofReal
((T+1:ℝ) * ((3*4^d:ℝ)*ε^((d:ℝ)-1-α) * RealStep d (2^M)^BlockCount T (d/8))) := by
apply (Policy.chord_real_bound L ρ hL hd hε hαlo hαhi).trans_eq
rw [ENNReal.ofReal_mul (by positivity : (0:ℝ) ≤ T+1)]
congr 1
simp [ENNReal.ofReal_add]
lemma chord_half_sphere_bound {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω]
(L : Ω → Policy d M T) (ρ : Measure Ω) [IsProbabilityMeasure ρ]
(hL : Admissible L ρ (UniformSphere d)) (hd : 16 ≤ d) {ε : ℝ} (hε : 0 < ε) (hε1 : ε ≤ 1/10) :
ChordSuccess L ρ (UniformSphere d) ε ≤ ENNReal.ofReal
(4^d * ε^(((d:ℝ)-1)/2) *
(SuccessConstant^d * (((d/8+2)*2^M:ℕ):ℝ)^(1/((d/8:ℕ):ℝ)))^BlockCount T (d/8)) := by
by_cases hT : T = 0
· subst T
simp only [blockCount_zero, pow_zero, mul_one]
exact (Policy.chord_zero_bound L ρ hL (by omega) hε).trans
(ENNReal.ofReal_le_ofReal (terminal_angular_zero_bound hd hε hε1))
· have hb := chord_real_coefficient L ρ hL hd hε (le_refl (((d:ℝ)-1)/2)) (by linarith)
have he : (d:ℝ)-1-((d:ℝ)-1)/2 = ((d:ℝ)-1)/2 := by ring
rw [he] at hb
apply hb.trans (ENNReal.ofReal_le_ofReal ?_)
have hN : 1 ≤ 2^M := one_le_pow₀ (by norm_num)
have hNK : 2^M ≤ (d/8+2)*2^M := by
simp
have hs := stopping_step_bound_three hd hN hNK (by omega : 0 < T)
calc
_ = (4^d * ε^(((d:ℝ)-1)/2)) * (3*((T+1:ℝ)*RealStep d (2^M)^BlockCount T (d/8))) := by ring
_ ≤ _ := mul_le_mul_of_nonneg_left hs (by positivity)
lemma chord_half_ambient_bound {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω]
(L : Ω → Policy d M T) (ρ : Measure Ω) [IsProbabilityMeasure ρ]
(hL : Admissible L ρ (UniformSphere d)) (hd : 16 ≤ d) {ε : ℝ} (hε : 0 < ε) :
ChordSuccess L ρ (UniformSphere d) ε ≤ ENNReal.ofReal
(12^d * ε^((d:ℝ)/2-1) *
(SuccessConstant^d * (((2*(d/8)+4)*2^M:ℕ):ℝ)^(1/((d/8:ℕ):ℝ)))^BlockCount T (d/8)) := by
have hb := chord_real_coefficient L ρ hL hd hε (by linarith : ((d:ℝ)-1)/2 ≤ (d:ℝ)/2) le_rfl
have he : (d:ℝ)-1-(d:ℝ)/2 = (d:ℝ)/2-1 := by ring
rw [he] at hb
apply hb.trans (ENNReal.ofReal_le_ofReal ?_)
have hN : 1 ≤ 2^M := one_le_pow₀ (by norm_num)
have hNK : 2^M ≤ (2*(d/8)+4)*2^M := by
simp
have hR := realStep_pos d hN
have hs := stopping_step_bound hd hN hNK T
have hSp := stepConstant_pos
have hK : 2*StepConstant ≤ SuccessConstant := by dsimp [SuccessConstant]; linarith
have hroot : 0 ≤ (((2*(d/8)+4)*2^M:ℕ):ℝ)^(1/((d/8:ℕ):ℝ)) := by positivity
have hbase := mul_le_mul_of_nonneg_right (pow_le_pow_left₀ (by positivity : 0 ≤ 2*StepConstant) hK d) hroot
have hpow := pow_le_pow_left₀ (by positivity : 0 ≤ (2*StepConstant)^d *
(((2*(d/8)+4)*2^M:ℕ):ℝ)^(1/((d/8:ℕ):ℝ))) hbase (BlockCount T (d/8))
calc
_ = ((3*4^d:ℝ)*ε^((d:ℝ)/2-1))*((T+1:ℝ)*RealStep d (2^M)^BlockCount T (d/8)) := by ring
_ ≤ _ := mul_le_mul
(mul_le_mul_of_nonneg_right (cap_constant_le (by omega : 1 ≤ d)) (by positivity))
(hs.trans hpow) (by positivity) (by positivity)
end MemoryPrecision.Kernel
namespace MemoryPrecision.Kernel
open Policy
def BlockCount (T p : ℕ) : ℕ := ⌈(T:ℝ)/(p:ℝ)⌉₊
def SourceEllipsoidStatement : Prop :=
let two : ℝ := @OfNat.ofNat ℝ 2 (@instOfNatAtLeastTwo ℝ 2 Real.commRing.toNatCast
(Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 0)))
let four : ℝ := @OfNat.ofNat ℝ 4 (@instOfNatAtLeastTwo ℝ 4 Real.commRing.toNatCast
(Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 2)))
let ten : ℝ := @OfNat.ofNat ℝ 10 (@instOfNatAtLeastTwo ℝ 10 Real.commRing.toNatCast
(Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 8)))
∃ C C₀ : ℝ, 1 ≤ C ∧ 1 ≤ C₀ ∧
∀ d M T : ℕ, 16 ≤ d → ∀ ε : ℝ, 0 < ε → ε ≤ 1/ten →
∀ (Ω : Type u) [MeasurableSpace Ω] (ρ : Measure Ω), IsProbabilityMeasure ρ →
∀ K : Ω → Policy d M T, Policy.Admissible K ρ (UniformSphere d) →
Policy.Success K ρ (UniformSphere d) ε ≤ min 1 (ENNReal.ofReal
(four^d * ε^(((d:ℝ)-1)/two) *
(C^d * (((d/8+2)*2^M : ℕ):ℝ)^(1/((d/8:ℕ):ℝ)))^(BlockCount T (d/8)))) ∧
Policy.ChordSuccess K ρ (UniformSphere d) ε ≤ min 1 (ENNReal.ofReal
(C₀^d * ε^((d:ℝ)/two-1) *
(C^d * (((2*(d/8)+4)*2^M : ℕ):ℝ)^(1/((d/8:ℕ):ℝ)))^(BlockCount T (d/8)))) ∧
{x : Sphere d × Sphere d | Real.arccos (@inner ℝ (Vector d)
(@InnerProductSpace.toInner ℝ (Vector d) _
(PiLp.seminormedAddCommGroup
(@OfNat.ofNat ℝ≥0∞ 2
(@instOfNatAtLeastTwo ℝ≥0∞ 2 ENNReal.instAddCommMonoidWithOne.toNatCast
(Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 0))))
(fun _ : Fin d => ℝ))
(PiLp.innerProductSpace (𝕜 := ℝ) (fun _ : Fin d => ℝ)))
x.2.val x.1.val) ≤ ε} ⊆
{x | @Dist.dist (Vector d)
(PiLp.instDist (p := @OfNat.ofNat ℝ≥0∞ 2
(@instOfNatAtLeastTwo ℝ≥0∞ 2 ENNReal.instAddCommMonoidWithOne.toNatCast
(Mathlib.Meta.NormNum.instAtLeastTwo (nat_lit 0))))
(α := fun _ : Fin d => ℝ)) x.2.val x.1.val ≤ ε}
theorem source_ellipsoid_success : SourceEllipsoidStatement := by
refine ⟨MemoryPrecision.SuccessConstant, 12, MemoryPrecision.successConstant_one, by norm_num, ?_⟩
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) ?_,
le_min (Policy.chordSuccess_le_one L ρ (UniformSphere d) ε) ?_, ?_⟩
· exact (Policy.success_le_chord L ρ (UniformSphere d) ε).trans
(chord_half_sphere_bound L ρ hL hd hε hε1)
· exact chord_half_ambient_bound L ρ hL hd hε
· intro x hx
exact (NoiselessRegression.chord_le_arccos x.2 x.1).trans hx
end MemoryPrecision.Kernel
end
end OAI
|