File size: 3,797 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 | import Mathlib
import OAI.Probability.GaussianRegression.Main
import OAI.Probability.MemoryPrecision.RealStream
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 NoiselessRegression.Learner
lemma estimate_zero {d M : ℕ} {Ω : Type u} (L : Learner d M 0 Ω)
(s : UnitSphere d) (ω : Ω) (x : Samples d 0) :
L.estimate s ω x = L.output 0 ω (L.initialChoice ω).2 := by
by_cases h : (L.initialChoice ω).1 = true <;> simp [estimate, terminal, run, start, h]
lemma frozen_chord_zero {d M : ℕ} {Ω : Type u} (L : Learner d M 0 Ω)
(ω : Ω) (hd : 1 ≤ d) {ε : ℝ} (hε : 0 < ε) :
((uniformSphere d).prod (sampleLaw d 0))
{z | dist z.1.val (L.estimate z.1 ω z.2).val ≤ ε} ≤
ENNReal.ofReal ((3*4^d:ℝ)*ε^(d-1)) := by
let : NeZero d := ⟨by omega⟩
let y := (L.output 0 ω (L.initialChoice ω).2).val
simp_rw [estimate_zero]
change ((uniformSphere d).prod (sampleLaw d 0))
(Prod.fst ⁻¹' {s : UnitSphere d | dist s.val y ≤ ε}) ≤ _
rw [measurePreserving_fst.measure_preimage
(measurableSet_le (measurable_subtype_coe.dist measurable_const) measurable_const).nullMeasurableSet]
have h := ambientSphereLaw_closedBall (E := Space d) y hε (d-1) (by simp)
rw [ambientSphereLaw, Measure.map_apply measurable_subtype_coe Metric.isClosed_closedBall.measurableSet] at h
simpa [uniformSphere, sphereLaw, ENNReal.ofReal_mul, finrank_euclideanSpace_fin,
Metric.closedBall, Set.preimage_ofPred_eq] using h
lemma chord_zero_bound {d M : ℕ} {Ω : Type u} [MeasurableSpace Ω]
(L : Learner d M 0 Ω) (ρ : Measure Ω) [IsProbabilityMeasure ρ]
(hL : L.Admissible ρ) (hd : 1 ≤ d) {ε : ℝ} (hε : 0 < ε) :
uniformChordSuccess L ρ ε ≤ ENNReal.ofReal ((3*4^d:ℝ)*ε^(d-1)) := by
let : NeZero d := ⟨by omega⟩
have hp := ((MeasurePreserving.id (uniformSphere d)).prod
(Measure.measurePreserving_swap (μ := sampleLaw d 0) (ν := ρ))).comp
(measurePreserving_prodAssoc (uniformSphere d) (sampleLaw d 0) ρ)
let S := {z : UnitSphere d × (Ω × Samples d 0) |
dist (L.estimate z.1 z.2.1 z.2.2).val z.1.val ≤ ε}
have hS : NullMeasurableSet S ((uniformSphere d).prod (ρ.prod (sampleLaw d 0))) :=
nullMeasurableSet_le
((measurable_subtype_coe.comp_aemeasurable hL.uniform_experiment).dist
(measurable_subtype_coe.comp measurable_fst).aemeasurable) aemeasurable_const
have he := hp.measure_preimage hS
have hb := prod_null_set_bound (hS.preimage hp.quasiMeasurePreserving)
(c := ENNReal.ofReal ((3*4^d:ℝ)*ε^(d-1)))
(by
filter_upwards [] with ω
change ((uniformSphere d).prod (sampleLaw d 0))
{z | dist (L.estimate z.1 ω z.2).val z.1.val ≤ ε} ≤ _
simpa only [dist_comm] using frozen_chord_zero L ω hd hε)
simpa only [he, measure_univ, mul_one, S, uniformChordSuccess] using hb
lemma angular_event_subset_chord {d M T : ℕ} {Ω : Type u} (L : Learner d M T Ω) (ε : ℝ) :
{z : UnitSphere d × (Ω × Samples d T) |
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} := by
intro z hz
exact le_trans (chord_le_arccos (L.estimate z.1 z.2.1 z.2.2) z.1) hz
lemma angular_le_chord {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω]
(L : Learner d M T Ω) (ρ : Measure Ω) (ε : ℝ) :
L.uniformSuccess ρ ε ≤ uniformChordSuccess L ρ ε :=
measure_mono (angular_event_subset_chord L ε)
end MemoryPrecision
end
end OAI
|