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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 | import Mathlib
import OAI.Probability.GaussianRegression.Main
import OAI.Probability.MemoryPrecision.KernelCaps
import OAI.Probability.MemoryPrecision.KernelIntegral
import OAI.Probability.MemoryPrecision.KernelVersion
import OAI.Probability.MemoryPrecision.SeedAverage
import OAI.Probability.MemoryPrecision.StochasticMass
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 NoiselessRegression
namespace Policy
variable {d M T : ℕ} (K : Policy d M T)
lemma borel_sphere_bound (hK : K.Borel) (hd : 16 ≤ d) {ε α : ℝ} (hε : 0 < ε)
(hαlo : ((d:ℝ)-1)/2 ≤ α) (hαhi : α ≤ (d:ℝ)/2) :
(∫⁻ z : MemoryPrecision.Kernel.Sphere d × Rows d T,
K.pathValue (K.outputTest (chordTest ε)) z.1 (records z.1 z.2)
∂(UniformSphere d).prod (RowLaw d T)) ≤
(T+1:ℝ≥0∞)*ENNReal.ofReal
((3*4^d:ℝ)*ε^((d:ℝ)-1-α)*RealStep d (2^M)^BlockCount T (d/8)) := by
have : NeZero d := ⟨by omega⟩
rw [lintegral_prod _ (K.experiment_fixed_measurable hK (measurable_chordTest ε)).aemeasurable]
apply K.integral_pathValue_bound (K.outputTest (chordTest ε)) Subtype.val hK
(K.outputTest_measurable (measurable_chordTest ε)) measurable_subtype_coe (UniformSphere d)
intro r hr j
have hd' : (16:ℝ) ≤ d := by exact_mod_cast hd
have hb := stoch_sphere_mass hd (show 1 ≤ 2^M from one_le_pow₀ (by norm_num))
(K.marginal_measurable hK) (fun n j obs => (K.marginal_sum n j obs).le)
(K.outputTest_measurable (measurable_chordTest ε))
(K.outputTest_le_one (chordTest_le_one ε))
(H := (3*4^d)*ε^((d:ℝ)-1-α)) (by positivity) hαlo hαhi
(K.output_chord_sphere_local (by omega) hε (by linarith) (by linarith))
(BlockCount T (d/8)) r (hr.trans (blockCount_mul_ge T (by omega))) 0 j
exact sphere_mass_bound hb
lemma completed_sphere_bound (hK : K.Completed) (hd : 16 ≤ d) {ε α : ℝ} (hε : 0 < ε)
(hαlo : ((d:ℝ)-1)/2 ≤ α) (hαhi : α ≤ (d:ℝ)/2) :
(∫⁻ z : MemoryPrecision.Kernel.Sphere d × Rows d T,
K.pathValue (K.outputTest (chordTest ε)) z.1 (records z.1 z.2)
∂(UniformSphere d).prod (RowLaw d T)) ≤
(T+1:ℝ≥0∞)*ENNReal.ofReal
((3*4^d:ℝ)*ε^((d:ℝ)-1-α)*RealStep d (2^M)^BlockCount T (d/8)) := by
have : NeZero d := ⟨by omega⟩
have he := K.version_pathValue_ae ((UniformSphere d).prod (RowLaw d T)) hK Prod.fst
(fun z : MemoryPrecision.Kernel.Sphere d × Rows d T => records z.1 z.2)
(fun i => show Measure.QuasiMeasurePreserving _ _ _ from
sphere_sample_observation_quasi (by simpa using (show 3 ≤ d by omega)) i) (chordTest ε)
rw [← lintegral_congr_ae he]
exact (K.version hK).borel_sphere_bound (K.version_borel hK) hd hε hαlo hαhi
lemma chord_real_bound {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 α : ℝ} (hε : 0 < ε)
(hαlo : ((d:ℝ)-1)/2 ≤ α) (hαhi : α ≤ (d:ℝ)/2) :
ChordSuccess L ρ (UniformSphere d) ε ≤
(T+1:ℝ≥0∞)*ENNReal.ofReal
((3*4^d:ℝ)*ε^((d:ℝ)-1-α)*RealStep d (2^M)^BlockCount T (d/8)) := by
have : NeZero d := ⟨by omega⟩
apply seed_average_bound (UniformSphere d) (RowLaw d T) ρ
(hL.experiment (chordTest ε) (measurable_chordTest ε) (chordTest_le_one ε))
filter_upwards [hL.completed] with ω hω
exact (L ω).completed_sphere_bound hω hd hε hαlo hαhi
lemma zero_fixed_sphere_bound {d M : ℕ} (K : Policy d M 0) (hd : 1 ≤ d)
{ε : ℝ} (hε : 0 < ε) :
(∫⁻ z : MemoryPrecision.Kernel.Sphere d × Rows d 0,
K.pathValue (K.outputTest (chordTest ε)) z.1 (records z.1 z.2)
∂(UniformSphere d).prod (RowLaw d 0)) ≤ ENNReal.ofReal ((3*4^d:ℝ)*ε^(d-1)) := by
have : NeZero d := ⟨by omega⟩
have he (s : MemoryPrecision.Kernel.Sphere d) (obs : Observations d 0) :
K.pathValue (K.outputTest (chordTest ε)) s obs =
∑ a : Action M, K.initial a * K.outputTest (chordTest ε) 0 a.2 s := by
unfold pathValue
simp only [eval, ite_self]
simp_rw [he]
have hm : Measurable (fun s : MemoryPrecision.Kernel.Sphere d =>
∑ a : Action M, K.initial a * K.outputTest (chordTest ε) 0 a.2 s) :=
Finset.measurable_sum _ (fun a _ => (K.outputTest_measurable (measurable_chordTest ε) 0 a.2).const_mul _)
have hm' : Measurable (fun z : MemoryPrecision.Kernel.Sphere d × Rows d 0 =>
∑ a : Action M, K.initial a * K.outputTest (chordTest ε) 0 a.2 z.1) :=
hm.comp measurable_fst
rw [lintegral_prod _ hm'.aemeasurable]
simp only [lintegral_const, measure_univ, mul_one]
rw [lintegral_finsetSum _ (fun a _ =>
(K.outputTest_measurable (measurable_chordTest ε) 0 a.2).const_mul _)]
calc
_ = ∑ a : Action M, K.initial a *
(∫⁻ s, K.outputTest (chordTest ε) 0 a.2 s ∂UniformSphere d) := by
apply Finset.sum_congr rfl
intro a _
exact lintegral_const_mul _ (K.outputTest_measurable (measurable_chordTest ε) 0 a.2)
_ ≤ ∑ a : Action M, K.initial a * ENNReal.ofReal ((3*4^d:ℝ)*ε^(d-1)) :=
Finset.sum_le_sum (fun a _ => mul_le_mul' le_rfl (K.output_chord_sphere_total hd hε 0 a.2))
_ = _ := by rw [← Finset.sum_mul, K.initial_sum, one_mul]
lemma chord_zero_bound {d M : ℕ} {Ω : Type u} [MeasurableSpace Ω] (L : Ω → Policy d M 0)
(ρ : Measure Ω) [IsProbabilityMeasure ρ] (hL : Admissible L ρ (UniformSphere d))
(hd : 1 ≤ d) {ε : ℝ} (hε : 0 < ε) :
ChordSuccess L ρ (UniformSphere d) ε ≤ ENNReal.ofReal ((3*4^d:ℝ)*ε^(d-1)) := by
have : NeZero d := ⟨by omega⟩
apply seed_average_bound (UniformSphere d) (RowLaw d 0) ρ
(hL.experiment (chordTest ε) (measurable_chordTest ε) (chordTest_le_one ε))
exact Filter.Eventually.of_forall (fun ω => (L ω).zero_fixed_sphere_bound hd hε)
end Policy
end MemoryPrecision.Kernel
end
end OAI
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