| 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 |
| |
