File size: 5,308 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 | import Mathlib
import OAI.Probability.GaussianRegression.Main
import OAI.Probability.MemoryPrecision.SuccessArithmetic
import OAI.Probability.MemoryPrecision.ZeroHorizon
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
open NoiselessRegression
lemma chord_real_coefficient {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω]
(L : Learner d M T Ω) (ρ : Measure Ω) [IsProbabilityMeasure ρ]
(hL : L.Admissible ρ) (hd : 16 ≤ d) {ε : ℝ} (hε : 0 < ε)
{α : ℝ} (hαlo : ((d:ℝ)-1)/2 ≤ α) (hαhi : α ≤ (d:ℝ)/2) :
uniformChordSuccess L ρ 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 (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 : Learner d M T Ω) (ρ : Measure Ω) [IsProbabilityMeasure ρ]
(hL : L.Admissible ρ) (hd : 16 ≤ d) {ε : ℝ} (hε : 0 < ε) (hε1 : ε ≤ 1/10) :
uniformChordSuccess L ρ 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 (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 : Learner d M T Ω) (ρ : Measure Ω) [IsProbabilityMeasure ρ]
(hL : L.Admissible ρ) (hd : 16 ≤ d) {ε : ℝ} (hε : 0 < ε) :
uniformChordSuccess L ρ 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)
def EllipsoidSuccessStatement : Prop :=
∃ C C₀ : ℝ, 1 ≤ C ∧ 1 ≤ C₀ ∧
∀ d M T : ℕ, 16 ≤ d → ∀ ε : ℝ, 0 < ε → ε ≤ 1/10 →
∀ (Ω : Type u) [MeasurableSpace Ω] (ρ : Measure Ω), IsProbabilityMeasure ρ →
∀ L : Learner d M T Ω, L.Admissible ρ →
L.uniformSuccess ρ ε ≤ min 1 (ENNReal.ofReal
(4^d * ε^(((d:ℝ)-1)/2) *
(C^d * (((d/8+2)*2^M : ℕ):ℝ)^(1/((d/8:ℕ):ℝ)))^(BlockCount T (d/8)))) ∧
uniformChordSuccess L ρ ε ≤ min 1 (ENNReal.ofReal
(C₀^d * ε^((d:ℝ)/2-1) *
(C^d * (((2*(d/8)+4)*2^M : ℕ):ℝ)^(1/((d/8:ℕ):ℝ)))^(BlockCount T (d/8)))) ∧
{z : UnitSphere d × (Ω × Samples d T) |
Learner.angularError z.1 (L.estimate z.1 z.2.1 z.2.2) ≤ ε} ⊆
{z | dist (L.estimate z.1 z.2.1 z.2.2).val z.1.val ≤ ε}
theorem ellipsoid_success : EllipsoidSuccessStatement := by
refine ⟨SuccessConstant, 12, 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 ?_ ?_, le_min ?_ ?_, angular_event_subset_chord L ε⟩
· exact prob_le_one
· exact (angular_le_chord L ρ ε).trans (chord_half_sphere_bound L ρ hL hd hε hε1)
· exact prob_le_one
· exact chord_half_ambient_bound L ρ hL hd hε
end MemoryPrecision
end
end OAI
|