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b317718 | 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 | import OAI.Probability.GaussianReplacement.StreamSphere
import OAI.Probability.GaussianRegression.Blocks
namespace OAI
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
noncomputable section
universe u
namespace NoiselessRegression
open InnerProductSpace
open scoped Pointwise
namespace Learner
lemma run_congr_labels {d M T : ℕ} {Ω : Type u} (L : Learner d M T Ω)
(s t : UnitSphere d) (ω : Ω) (x : Samples d T)
(h : ∀ i, ⟪x i,s.val⟫ = ⟪x i,t.val⟫) (n : ℕ) :
L.run s ω x n = L.run t ω x n := by
induction n with
| zero => rfl
| succ n ih =>
simp only [run, ih]
split
· rfl
· split
· rw [h]
· rfl
lemma estimate_congr_labels {d M T : ℕ} {Ω : Type u} (L : Learner d M T Ω)
(s t : UnitSphere d) (ω : Ω) (x : Samples d T)
(h : ∀ i, ⟪x i,s.val⟫ = ⟪x i,t.val⟫) : L.estimate s ω x = L.estimate t ω x := by
simp only [estimate, terminal, run_congr_labels L s t ω x h]
lemma angularSuccess_nullMeasurable {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω]
(L : Learner d M T Ω) (ρ : Measure Ω) (ε : ℝ) (hL : L.Admissible ρ) :
NullMeasurableSet {z : UnitSphere d × (Ω × Samples d T) |
angularError z.1 (L.estimate z.1 z.2.1 z.2.2) ≤ ε}
((uniformSphere d).prod (ρ.prod (sampleLaw d T))) := by
apply nullMeasurableSet_le _ aemeasurable_const
exact Real.continuous_arccos.measurable.comp_aemeasurable
((measurable_subtype_coe.comp_aemeasurable hL.uniform_experiment).inner
(measurable_subtype_coe.comp measurable_fst).aemeasurable)
lemma linear_success_bound {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω]
(L : Learner d M T Ω) (ρ : Measure Ω) [IsProbabilityMeasure ρ]
(hd : 0 < d) (hT : 4*T ≤ d) {ε : ℝ} (hε : ε ≤ 1/10) (hL : L.Admissible ρ) :
L.uniformSuccess ρ ε ≤ ENNReal.ofReal (21/32 : ℝ) := by
classical
let : NeZero d := ⟨hd.ne'⟩
have hmeas := ae_fiber_aemeasurable hL.uniform_experiment
unfold uniformSuccess
refine (prod_null_set_bound (L.angularSuccess_nullMeasurable ρ ε hL)
(c := ENNReal.ofReal (21/32 : ℝ)) ?_).trans_eq (by simp)
filter_upwards [hmeas] with z hz
let U : Submodule ℝ (Vector d) := Submodule.span ℝ (Set.range z.2)
have hdim : Module.finrank ℝ U ≤ T := by
have h := finrank_span_le_card (R := ℝ) (Set.range z.2)
apply h.trans
rw [Set.toFinset_range]
exact (Finset.card_image_le).trans_eq (by simp)
have hU : 4 * Module.finrank ℝ U ≤ Module.finrank ℝ (Vector d) := by
simpa only [Vector, finrank_euclideanSpace_fin] using (Nat.mul_le_mul_left 4 hdim).trans hT
change sphereLaw {s : UnitSphere d | angularError s (L.estimate s z.1 z.2) ≤ ε} ≤ _
apply sphere_reflection_success_bound U hU
· exact nullMeasurableSet_le
(Real.continuous_arccos.measurable.comp_aemeasurable
((measurable_subtype_coe.comp_aemeasurable hz).inner measurable_subtype_coe.aemeasurable))
aemeasurable_const
· intro s hs ht
let t := sphereIsometry U.reflection s
have he : L.estimate s z.1 z.2 = L.estimate t z.1 z.2 := by
apply L.estimate_congr_labels
intro i
have hi : z.2 i ∈ U := Submodule.subset_span (Set.mem_range_self i)
have he := U.reflection.inner_map_map (z.2 i) s.val
rw [(Submodule.reflection_eq_self_iff (z.2 i)).2 hi] at he
exact he.symm
have hs' : dist s.val (L.estimate s z.1 z.2).val ≤ ε := by
apply (chord_le_arccos s (L.estimate s z.1 z.2)).trans
simpa only [Set.mem_ofPred_eq, angularError, real_inner_comm] using hs
have ht' : dist t.val (L.estimate t z.1 z.2).val ≤ ε := by
apply (chord_le_arccos t (L.estimate t z.1 z.2)).trans
simpa only [Set.mem_ofPred_eq, angularError, real_inner_comm] using ht
calc
_ = dist s.val t.val := rfl
_ ≤ dist s.val (L.estimate s z.1 z.2).val + dist (L.estimate s z.1 z.2).val t.val := dist_triangle _ _ _
_ ≤ ε + ε := by rw [he, dist_comm (L.estimate t z.1 z.2).val t.val] at *; exact add_le_add hs' ht'
_ ≤ 1/5 := by linarith
end Learner
section Block
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
[FiniteDimensional ℝ E] [MeasurableSpace E] [borelSpaceE : BorelSpace E]
omit borelSpaceE in
lemma blockValue_le_one_no_borel {J : Type*} [Fintype J] {m : ℕ} {h : J → E → ℝ≥0∞}
(hh : ∀ j s, h j s ≤ 1) (φ : ((Fin m → E) × (Fin m → ℝ)) → J) (s : E) :
blockValue h φ s ≤ 1 := by
exact (lintegral_mono (fun A => hh _ s)).trans_eq (by simp)
end Block
end NoiselessRegression
end
end OAI
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