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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 | import OAI.Probability.GaussianRegression.Precision
namespace OAI
/-!
Quadratic and subquadratic memory lower bounds for Gaussian regression.
-/
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
open InnerProductSpace
open scoped Pointwise
noncomputable section
universe u
namespace NoiselessRegression
theorem fixed_quadratic_memory : FixedQuadraticMemory.{u} := by
intro A hA
refine ⟨1/(4*(400000+100*A)), by positivity, 32, ?_⟩
intro d hd M T hM ε hε hε10 Ω _ ρ hρ L hL hs
let : IsProbabilityMeasure ρ := hρ
have hT : d ≤ 4*T := by
by_contra h
have hb := hs.trans (L.linear_success_bound ρ (by omega) (by omega) hε10 hL)
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)
(L.streaming_success_bound_real ρ hL (by omega) hε ha hq hqa hs)
section LowerFubini
variable {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
{μ : Measure X} {ν : Measure Y} [SFinite μ] [SFinite ν]
lemma prod_null_set_lower_bound {S : Set (X × Y)} (hS : NullMeasurableSet S (μ.prod ν))
{c : ℝ≥0∞} (h : ∀ᵐ x ∂μ, c ≤ ν {y | (x,y) ∈ S}) :
c * μ univ ≤ (μ.prod ν) S := by
have hm : AEMeasurable (S.indicator (fun _ => (1 : ℝ≥0∞))) (μ.prod ν) :=
aemeasurable_const.indicator₀ hS
rw [← one_mul ((μ.prod ν) S), ← lintegral_indicator_const₀ hS 1, lintegral_prod _ hm]
have hc : c * μ univ = ∫⁻ _ : X, c ∂μ := (lintegral_const c).symm
rw [hc]
apply lintegral_mono_ae
filter_upwards [h, ae_fiber_aemeasurable hm.prod_swap] with x hx hmx
have hs : NullMeasurableSet {y | (x,y) ∈ S} ν := by
have heq : {y | (x,y) ∈ S} = {y | S.indicator (fun _ => (1 : ℝ≥0∞)) (x,y) = 1} := by
ext y
by_cases hy : (x,y) ∈ S <;> simp [hy]
rw [heq]
exact nullMeasurableSet_eq_fun hmx aemeasurable_const
change c ≤ ∫⁻ y, ({y | (x,y) ∈ S}).indicator (fun _ => (1 : ℝ≥0∞)) y ∂ν
simpa [lintegral_indicator_const₀ hs] using hx
end LowerFubini
namespace Learner
lemma uniformSuccess_of_pointwise {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 : 0 < d) (ε : ℝ) {c : ℝ≥0∞}
(hs : ∀ s, c ≤ L.success ρ ε s) : c ≤ L.uniformSuccess ρ ε := by
let : NeZero d := ⟨hd.ne'⟩
have hb := prod_null_set_lower_bound (L.angularSuccess_nullMeasurable ρ ε hL)
(c := c) (Filter.Eventually.of_forall hs)
simpa only [measure_univ, mul_one, uniformSuccess] using hb
end Learner
theorem subquadratic_memory : SubquadraticMemory.{u} := by
obtain ⟨c, hc, d₁, hfixed⟩ := fixed_quadratic_memory (1 : ℝ) (by norm_num)
refine ⟨c, hc, ?_⟩
intro M hM
obtain ⟨d₂, hd₂⟩ := eventually_atTop.1 hM.eventuallyLE
refine ⟨max d₁ (max d₂ 1), ?_⟩
intro d hd T ε hε hε10 Ω _ ρ hρ L hL hs
let : 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
apply hfixed d (by omega) (M d) T hmem ε hε hε10 Ω ρ hρ L hL
rcases hs with hs | hs
· exact hs
· exact L.uniformSuccess_of_pointwise ρ hL (by omega) ε hs
theorem main : SubquadraticMemory.{u} ∧ FixedQuadraticMemory.{u} :=
⟨subquadratic_memory, fixed_quadratic_memory⟩
end NoiselessRegression
end
end OAI
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