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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 | import OAI.Probability.PosteriorReplicas.Regression.Conclusions
namespace OAI
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace
noncomputable section
universe u
namespace PosteriorRegression
namespace Learner
structure UniformAdmissible {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω]
(L : Learner d M T Ω) (ρ : Measure Ω) : Prop where
transition : ∀ᵐ ω ∂ρ, ∀ t a,
AEMeasurable (fun obs => (L.transition t ω a obs).2) (observationReference d)
uniform_experiment : AEMeasurable
(fun z : UnitSphere d × (Ω × Samples d T) => L.estimate z.1 z.2.1 z.2.2)
((uniformSphere d).prod (ρ.prod (sampleLaw d T)))
lemma Admissible.uniformAdmissible {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω]
{L : Learner d M T Ω} {ρ : Measure Ω} [IsProbabilityMeasure ρ]
(hL : L.Admissible ρ) : L.UniformAdmissible ρ := by
have : SigmaFinite (observationReference d) := by unfold observationReference; infer_instance
refine ⟨?_,hL.uniform_experiment⟩
simp only [ae_all_iff]
intro t a
filter_upwards [ae_fiber_aemeasurable (hL.transition t a).prod_swap] with ω hω
exact measurable_snd.comp_aemeasurable hω
lemma UniformAdmissible.angularSuccess_nullMeasurable {d M T : ℕ} {Ω : Type u}
[MeasurableSpace Ω] {L : Learner d M T Ω} {ρ : Measure Ω}
(hL : L.UniformAdmissible ρ) (ε : ℝ) :
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 UniformAdmissible.linear_success_bound {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω]
{L : Learner d M T Ω} {ρ : Measure Ω} [IsProbabilityMeasure ρ]
(hL : L.UniformAdmissible ρ) (hd : 0 < d) (hT : 4*T ≤ d) {ε : ℝ} (hε : ε ≤ 1/10) :
L.uniformSuccess ρ ε ≤ ENNReal.ofReal (21/32 : ℝ) := by
classical
have : NeZero d := ⟨hd.ne'⟩
have hmeas := ae_fiber_aemeasurable hL.uniform_experiment
unfold uniformSuccess
refine (prod_null_set_bound (hL.angularSuccess_nullMeasurable ε)
(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
lemma UniformAdmissible.streaming_success_bound {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω]
{L : Learner d M T Ω} {ρ : Measure Ω} [IsProbabilityMeasure ρ]
(hL : L.UniformAdmissible ρ) (hd : 3 ≤ d) {ε : ℝ} (hε : 0 < ε)
{a q : ℕ} (ha : 2*a ≤ d-1) (hq : 2 ≤ q) (hqa : 2*q ≤ a) :
L.uniformSuccess ρ ε ≤
(T+1 : ℝ≥0∞) * (ENNReal.ofReal ((3 : ℝ)*4^d) * ENNReal.ofReal (ε^a)) *
streamFactor (E := Vector d) (J := State M) q ^ (T/q+1) := by
have : NeZero d := ⟨by omega⟩
have hp := ((MeasurePreserving.id (uniformSphere d)).prod
(Measure.measurePreserving_swap (μ := sampleLaw d T) (ν := ρ))).comp
(measurePreserving_prodAssoc (uniformSphere d) (sampleLaw d T) ρ)
let S := {z : UnitSphere d × (Ω × Samples d T) |
angularError z.1 (L.estimate z.1 z.2.1 z.2.2) ≤ ε}
have hS : NullMeasurableSet S ((uniformSphere d).prod (ρ.prod (sampleLaw d T))) :=
hL.angularSuccess_nullMeasurable ε
have he := hp.measure_preimage hS
have hb := prod_null_set_bound (hS.preimage hp.quasiMeasurePreserving)
(c := (T+1 : ℝ≥0∞) * (ENNReal.ofReal ((3 : ℝ)*4^d) * ENNReal.ofReal (ε^a)) *
streamFactor (E := Vector d) (J := State M) q ^ (T/q+1))
(by
filter_upwards [hL.transition] with ω hω
exact L.frozen_success_bound ω hd hω hε ha hq hqa)
simpa only [he, measure_univ, mul_one, S, uniformSuccess] using hb
lemma UniformAdmissible.streaming_success_bound_real {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω]
{L : Learner d M T Ω} {ρ : Measure Ω} [IsProbabilityMeasure ρ]
(hL : L.UniformAdmissible ρ) (hd : 3 ≤ d) {ε : ℝ} (hε : 0 < ε)
{a q : ℕ} (ha : 2*a ≤ d-1) (hq : 2 ≤ q) (hqa : 2*q ≤ a)
(hs : (2/3 : ℝ≥0∞) ≤ L.uniformSuccess ρ ε) :
(2/3 : ℝ) ≤ ((T : ℝ)+1) * ((3 : ℝ)*4^d * ε^a) *
((32768 : ℝ)^d * ((2 : ℝ)^M)^(1/(q : ℝ)))^(T/q+1) := by
have hb := hs.trans (hL.streaming_success_bound hd hε ha hq hqa)
have ht : (T+1 : ℝ≥0∞) *
(ENNReal.ofReal ((3 : ℝ)*4^d) * ENNReal.ofReal (ε^a)) *
streamFactor (E := Vector d) (J := State M) q ^ (T/q+1) ≠ ∞ := by
apply ENNReal.mul_ne_top
· apply ENNReal.mul_ne_top (by simp)
exact ENNReal.mul_ne_top ENNReal.ofReal_ne_top ENNReal.ofReal_ne_top
· exact ENNReal.pow_ne_top (streamFactor_ne_top q)
have hr := ENNReal.toReal_mono ht hb
simpa [ENNReal.toReal_div, ENNReal.toReal_ofNat, ENNReal.toReal_mul,
ENNReal.toReal_add, ENNReal.toReal_natCast, ENNReal.toReal_one,
ENNReal.toReal_ofReal (by positivity : (0 : ℝ) ≤ 3*4^d),
ENNReal.toReal_ofReal (by positivity : 0 ≤ ε^a), streamFactor,
ENNReal.toReal_pow, ← ENNReal.toReal_rpow, finrank_euclideanSpace_fin,
Fintype.card_fin, State, Nat.cast_pow, Nat.cast_ofNat, Vector] using hr
end Learner
def UniformSubquadraticMemory : Prop :=
∃ c : ℝ, 0 < c ∧ ∀ M : ℕ → ℕ,
Asymptotics.IsLittleO atTop (fun d => (M d : ℝ)) (fun d => (d : ℝ) ^ 2) →
∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d → ∀ T : ℕ, ∀ ε : ℝ,
0 < ε → ε ≤ 1 / 10 →
∀ (Ω : Type u) [MeasurableSpace Ω] (ρ : Measure Ω), IsProbabilityMeasure ρ →
∀ L : Learner d (M d) T Ω, L.UniformAdmissible ρ →
(2 / 3 : ℝ≥0∞) ≤ L.uniformSuccess ρ ε →
c * (d : ℝ) * Real.log (1 / ε) ≤ (T : ℝ)
lemma source_fixed_quadratic_memory {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω]
{A : ℝ} (hA : 0 < A) (hd : 32 ≤ d) (hM : (M:ℝ) ≤ A*(d:ℝ)^2)
{ε : ℝ} (hε : 0<ε) (hε10 : ε≤1/10)
(ρ : Measure Ω) [IsProbabilityMeasure ρ] (L : Learner d M T Ω)
(hL : L.UniformAdmissible ρ) (hs : (2/3 : ℝ≥0∞) ≤ L.uniformSuccess ρ ε) :
(1 / (4*(400000+100*A))) * (d : ℝ) * Real.log (1/ε) ≤ (T : ℝ) := by
have hT : d ≤ 4*T := by
by_contra h
have hb := hs.trans (hL.linear_success_bound (by omega) (by omega) hε10)
have hr := ENNReal.toReal_mono ENNReal.ofReal_ne_top hb
norm_num [ENNReal.toReal_div] at hr
let q := (d-1)/4
let a := 2*q
have hq : 2 ≤ q := by dsimp [q]; omega
have hqa : 2*q ≤ a := le_rfl
have ha : 2*a ≤ d-1 := by dsimp [a, q]; omega
have hdq : d ≤ 8*q := by dsimp [q]; omega
have hda : d ≤ 4*a := by dsimp [a]; omega
exact precision_from_stream_bound hd hT (by omega) hdq hda hM hA hε (by linarith)
(hL.streaming_success_bound_real (by omega) hε ha hq hqa hs)
theorem source_subquadratic_memory : UniformSubquadraticMemory.{u} := by
refine ⟨1/(4*(400000+100*(1:ℝ))),by positivity,?_⟩
intro M hM
obtain ⟨d₂, hd₂⟩ := eventually_atTop.1 hM.eventuallyLE
refine ⟨max 32 d₂, ?_⟩
intro d hd T ε hε hε10 Ω _ ρ hρ L hL hs
have : IsProbabilityMeasure ρ := hρ
have hmem : (M d : ℝ) ≤ 1 * (d : ℝ)^2 := by
have hm := hd₂ d (by omega)
simpa only [Real.norm_eq_abs, abs_of_nonneg (Nat.cast_nonneg (M d) : (0 : ℝ) ≤ M d),
abs_of_nonneg (sq_nonneg (d : ℝ)), one_mul] using hm
exact source_fixed_quadratic_memory (by norm_num) (by omega) hmem hε hε10 ρ L hL hs
end PosteriorRegression
end
end OAI
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